Daily Papers

Large Language Model (LLM) techniques play an increasingly important role in Natural Language to SQL (NL2SQL) translation. LLMs trained by extensive corpora have strong natural language understanding and basic SQL generation abilities without additional tuning specific to NL2SQL tasks. Existing LLMs-based NL2SQL approaches try to improve the translation by enhancing the LLMs with an emphasis on user intention understanding. However, LLMs sometimes fail to generate appropriate SQL due to their lack of knowledge in organizing complex logical operator composition. A promising method is to input the LLMs with demonstrations, which include known NL2SQL translations from various databases. LLMs can learn to organize operator compositions from the input demonstrations for the given task. In this paper, we propose PURPLE (Pre-trained models Utilized to Retrieve Prompts for Logical Enhancement), which improves accuracy by retrieving demonstrations containing the requisite logical operator composition for the NL2SQL task on hand, thereby guiding LLMs to produce better SQL translation. PURPLE achieves a new state-of-the-art performance of 80.5% exact-set match accuracy and 87.8% execution match accuracy on the validation set of the popular NL2SQL benchmark Spider. PURPLE maintains high accuracy across diverse benchmarks, budgetary constraints, and various LLMs, showing robustness and cost-effectiveness.

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

Large language models excel at pattern matching, yet often fall short in systematic compositional generalization. We propose the coverage principle: a data-centric framework showing that models relying primarily on pattern matching for compositional tasks cannot reliably generalize beyond substituting fragments that yield identical results when used in the same contexts. We demonstrate that this framework has a strong predictive power for the generalization capabilities of Transformers. First, we derive and empirically confirm that the training data required for two-hop generalization grows at least quadratically with the token set size, and the training data efficiency does not improve with 20x parameter scaling. Second, for compositional tasks with path ambiguity where one variable affects the output through multiple computational paths, we show that Transformers learn context-dependent state representations that undermine both performance and interoperability. Third, Chain-of-Thought supervision improves training data efficiency for multi-hop tasks but still struggles with path ambiguity. Finally, we outline a mechanism-based taxonomy that distinguishes three ways neural networks can generalize: structure-based (bounded by coverage), property-based (leveraging algebraic invariances), and shared-operator (through function reuse). This conceptual lens contextualizes our results and highlights where new architectural ideas are needed to achieve systematic compositionally. Overall, the coverage principle provides a unified lens for understanding compositional reasoning, and underscores the need for fundamental architectural or training innovations to achieve truly systematic compositionality.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference queries become tractable, such as model counting and most probable configuration. Recent works have explored analyzing probabilistic and causal inference queries as compositions of basic operators to derive tractability conditions. In this paper, we take an algebraic perspective for compositional inference, and show that a large class of queries - including marginal MAP, probabilistic answer set programming inference, and causal backdoor adjustment - correspond to a combination of basic operators over semirings: aggregation, product, and elementwise mapping. Using this framework, we uncover simple and general sufficient conditions for tractable composition of these operators, in terms of circuit properties (e.g., marginal determinism, compatibility) and conditions on the elementwise mappings. Applying our analysis, we derive novel tractability conditions for many such compositional queries. Our results unify tractability conditions for existing problems on circuits, while providing a blueprint for analysing novel compositional inference queries.

Since their introduction, diffusion models have quickly become the prevailing approach to generative modeling in many domains. They can be interpreted as learning the gradients of a time-varying sequence of log-probability density functions. This interpretation has motivated classifier-based and classifier-free guidance as methods for post-hoc control of diffusion models. In this work, we build upon these ideas using the score-based interpretation of diffusion models, and explore alternative ways to condition, modify, and reuse diffusion models for tasks involving compositional generation and guidance. In particular, we investigate why certain types of composition fail using current techniques and present a number of solutions. We conclude that the sampler (not the model) is responsible for this failure and propose new samplers, inspired by MCMC, which enable successful compositional generation. Further, we propose an energy-based parameterization of diffusion models which enables the use of new compositional operators and more sophisticated, Metropolis-corrected samplers. Intriguingly we find these samplers lead to notable improvements in compositional generation across a wide set of problems such as classifier-guided ImageNet modeling and compositional text-to-image generation.

We extend JAX with the capability to automatically differentiate higher-order functions (functionals and operators). By representing functions as a generalization of arrays, we seamlessly use JAX's existing primitive system to implement higher-order functions. We present a set of primitive operators that serve as foundational building blocks for constructing several key types of functionals. For every introduced primitive operator, we derive and implement both linearization and transposition rules, aligning with JAX's internal protocols for forward and reverse mode automatic differentiation. This enhancement allows for functional differentiation in the same syntax traditionally use for functions. The resulting functional gradients are themselves functions ready to be invoked in python. We showcase this tool's efficacy and simplicity through applications where functional derivatives are indispensable. The source code of this work is released at https://github.com/sail-sg/autofd .

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a distributed universal approximation theorem guaranteeing that any Lipschitz non-linear operator between L^2([0,1]^d) spaces can be approximated uniformly over the Sobolev unit ball therein, to any given varepsilon>0 accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of O(varepsilon^{-1}). Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies varepsilon. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of L^2([0,1]^d).

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

We present a feasibility-seeking approach to neural network training. This mathematical optimization framework is distinct from conventional gradient-based loss minimization and uses projection operators and iterative projection algorithms. We reformulate training as a large-scale feasibility problem: finding network parameters and states that satisfy local constraints derived from its elementary operations. Training then involves projecting onto these constraints, a local operation that can be parallelized across the network. We introduce PJAX, a JAX-based software framework that enables this paradigm. PJAX composes projection operators for elementary operations, automatically deriving the solution operators for the feasibility problems (akin to autodiff for derivatives). It inherently supports GPU/TPU acceleration, provides a familiar NumPy-like API, and is extensible. We train diverse architectures (MLPs, CNNs, RNNs) on standard benchmarks using PJAX, demonstrating its functionality and generality. Our results show that this approach is as a compelling alternative to gradient-based training, with clear advantages in parallelism and the ability to handle non-differentiable operations.

A sequence of operators T_n from a Hilbert space {mathfrak H} to Hilbert spaces {mathfrak K}_n which is nondecreasing in the sense of contractive domination is shown to have a limit which is still a linear operator T from {mathfrak H} to a Hilbert space {mathfrak K}. Moreover, the closability or closedness of T_n is preserved in the limit. The closures converge likewise and the connection between the limits is investigated. There is no similar way of dealing directly with linear relations. However, the sequence of closures is still nondecreasing and then the convergence is governed by the monotonicity principle. There are some related results for nonincreasing sequences.

Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing and reasoning tasks. However, their performance in the foundational domain of arithmetic remains unsatisfactory. When dealing with arithmetic tasks, LLMs often memorize specific examples rather than learning the underlying computational logic, limiting their ability to generalize to new problems. In this paper, we propose a Composable Arithmetic Execution Framework (CAEF) that enables LLMs to learn to execute step-by-step computations by emulating Turing Machines, thereby gaining a genuine understanding of computational logic. Moreover, the proposed framework is highly scalable, allowing composing learned operators to significantly reduce the difficulty of learning complex operators. In our evaluation, CAEF achieves nearly 100% accuracy across seven common mathematical operations on the LLaMA 3.1-8B model, effectively supporting computations involving operands with up to 100 digits, a level where GPT-4o falls short noticeably in some settings.

Large Language Models (LLMs) have exhibited significant potential in performing diverse tasks, including the ability to call functions or use external tools to enhance their performance. While current research on function calling by LLMs primarily focuses on single-turn interactions, this paper addresses the overlooked necessity for LLMs to engage in multi-turn function calling--critical for handling compositional, real-world queries that require planning with functions but not only use functions. To facilitate this, we introduce an approach, BUTTON, which generates synthetic compositional instruction tuning data via bottom-up instruction construction and top-down trajectory generation. In the bottom-up phase, we generate simple atomic tasks based on real-world scenarios and build compositional tasks using heuristic strategies based on atomic tasks. Corresponding functions are then developed for these compositional tasks. The top-down phase features a multi-agent environment where interactions among simulated humans, assistants, and tools are utilized to gather multi-turn function calling trajectories. This approach ensures task compositionality and allows for effective function and trajectory generation by examining atomic tasks within compositional tasks. We produce a dataset BUTTONInstruct comprising 8k data points and demonstrate its effectiveness through extensive experiments across various LLMs.

As a common image editing operation, image composition (object insertion) aims to combine the foreground from one image and another background image, resulting in a composite image. However, there are many issues that could make the composite images unrealistic. These issues can be summarized as the inconsistency between foreground and background, which includes appearance inconsistency (e.g., incompatible illumination), geometry inconsistency (e.g., unreasonable size), and semantic inconsistency (e.g., mismatched semantic context). Image composition task could be decomposed into multiple sub-tasks, in which each sub-task targets at one or more issues. Specifically, object placement aims to find reasonable scale, location, and shape for the foreground. Image blending aims to address the unnatural boundary between foreground and background. Image harmonization aims to adjust the illumination statistics of foreground. Shadow (resp., reflection) generation aims to generate plausible shadow (resp., reflection) for the foreground. These sub-tasks can be executed sequentially or parallelly to acquire realistic composite images. To the best of our knowledge, there is no previous survey on image composition (object insertion). In this paper, we conduct comprehensive survey over the sub-tasks and combinatorial task of image composition (object insertion). For each one, we summarize the existing methods, available datasets, and common evaluation metrics. We have also contributed the first image composition toolbox libcom, which assembles 10+ image composition related functions (e.g., image blending, image harmonization, object placement, shadow generation, generative composition). The ultimate goal of this toolbox is solving all the problems related to image composition with simple `import libcom'.

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

The Cambrian explosion of easily accessible pre-trained diffusion models suggests a demand for methods that combine multiple different pre-trained diffusion models without incurring the significant computational burden of re-training a larger combined model. In this paper, we cast the problem of combining multiple pre-trained diffusion models at the generation stage under a novel proposed framework termed superposition. Theoretically, we derive superposition from rigorous first principles stemming from the celebrated continuity equation and design two novel algorithms tailor-made for combining diffusion models in SuperDiff. SuperDiff leverages a new scalable It\^o density estimator for the log likelihood of the diffusion SDE which incurs no additional overhead compared to the well-known Hutchinson's estimator needed for divergence calculations. We demonstrate that SuperDiff is scalable to large pre-trained diffusion models as superposition is performed solely through composition during inference, and also enjoys painless implementation as it combines different pre-trained vector fields through an automated re-weighting scheme. Notably, we show that SuperDiff is efficient during inference time, and mimics traditional composition operators such as the logical OR and the logical AND. We empirically demonstrate the utility of using SuperDiff for generating more diverse images on CIFAR-10, more faithful prompt conditioned image editing using Stable Diffusion, and improved unconditional de novo structure design of proteins. https://github.com/necludov/super-diffusion

We investigate the construction of latent spaces through self-supervised learning to support semantically meaningful operations. Analogous to operational amplifiers, these "operational latent spaces" (OpLaS) not only demonstrate semantic structure such as clustering but also support common transformational operations with inherent semantic meaning. Some operational latent spaces are found to have arisen "unintentionally" in the progress toward some (other) self-supervised learning objective, in which unintended but still useful properties are discovered among the relationships of points in the space. Other spaces may be constructed "intentionally" by developers stipulating certain kinds of clustering or transformations intended to produce the desired structure. We focus on the intentional creation of operational latent spaces via self-supervised learning, including the introduction of rotation operators via a novel "FiLMR" layer, which can be used to enable ring-like symmetries found in some musical constructions.

Lara is a key-value algebra that aims at unifying linear and relational algebra with three types of operation abstraction. The study of Lara's expressive ability reports that it can represent relational algebra and most linear algebra operations. However, several essential computations, such as matrix inversion and determinant, cannot be expressed in Lara. Lara cannot represent global and iterative computation, either. This article proposes IterLara, extending Lara with iterative operators, to provide an algebraic model that unifies operations in general-purpose computing, like big data, AI, scientific computing, and database. We study the expressive ability of Lara and IterLara and prove that IterLara with aggregation functions can represent matrix inversion, determinant. Besides, we demonstrate that IterLara with no limitation of function utility is Turing complete. We also propose the Operation Count (OP) as a metric of computation amount for IterLara and ensure that the OP metric is in accordance with the existing computation metrics.

Optics and lenses are abstract categorical gadgets that model systems with bidirectional data flow. In this paper we observe that the denotational definition of optics - identifying two optics as equivalent by observing their behaviour from the outside - is not suitable for operational, software oriented approaches where optics are not merely observed, but built with their internal setups in mind. We identify operational differences between denotationally isomorphic categories of cartesian optics and lenses: their different composition rule and corresponding space-time tradeoffs, positioning them at two opposite ends of a spectrum. With these motivations we lift the existing categorical constructions and their relationships to the 2-categorical level, showing that the relevant operational concerns become visible. We define the 2-category 2-Optic(C) whose 2-cells explicitly track optics' internal configuration. We show that the 1-category Optic(C) arises by locally quotienting out the connected components of this 2-category. We show that the embedding of lenses into cartesian optics gets weakened from a functor to an oplax functor whose oplaxator now detects the different composition rule. We determine the difficulties in showing this functor forms a part of an adjunction in any of the standard 2-categories. We establish a conjecture that the well-known isomorphism between cartesian lenses and optics arises out of the lax 2-adjunction between their double-categorical counterparts. In addition to presenting new research, this paper is also meant to be an accessible introduction to the topic.

We introduce Poseidon, a foundation model for learning the solution operators of PDEs. It is based on a multiscale operator transformer, with time-conditioned layer norms that enable continuous-in-time evaluations. A novel training strategy leveraging the semi-group property of time-dependent PDEs to allow for significant scaling-up of the training data is also proposed. Poseidon is pretrained on a diverse, large scale dataset for the governing equations of fluid dynamics. It is then evaluated on a suite of 15 challenging downstream tasks that include a wide variety of PDE types and operators. We show that Poseidon exhibits excellent performance across the board by outperforming baselines significantly, both in terms of sample efficiency and accuracy. Poseidon also generalizes very well to new physics that is not seen during pretraining. Moreover, Poseidon scales with respect to model and data size, both for pretraining and for downstream tasks. Taken together, our results showcase the surprising ability of Poseidon to learn effective representations from a very small set of PDEs during pretraining in order to generalize well to unseen and unrelated PDEs downstream, demonstrating its potential as an effective, general purpose PDE foundation model. Finally, the Poseidon model as well as underlying pretraining and downstream datasets are open sourced, with code being available at https://github.com/camlab-ethz/poseidon and pretrained models and datasets at https://huggingface.co/camlab-ethz.

In this paper, we apply the self-attention from the state-of-the-art Transformer in Attention Is All You Need for the first time to a data-driven operator learning problem related to partial differential equations. An effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. By employing the operator approximation theory in Hilbert spaces, it is demonstrated for the first time that the softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without softmax, the approximation capacity of a linearized Transformer variant can be proved to be comparable to a Petrov-Galerkin projection layer-wise, and the estimate is independent with respect to the sequence length. A new layer normalization scheme mimicking the Petrov-Galerkin projection is proposed to allow a scaling to propagate through attention layers, which helps the model achieve remarkable accuracy in operator learning tasks with unnormalized data. Finally, we present three operator learning experiments, including the viscid Burgers' equation, an interface Darcy flow, and an inverse interface coefficient identification problem. The newly proposed simple attention-based operator learner, Galerkin Transformer, shows significant improvements in both training cost and evaluation accuracy over its softmax-normalized counterparts.

Deep learning optimizers are often motivated through a mix of convex and approximate second-order theory. We select three such methods -- Adam, Shampoo and Prodigy -- and argue that each method can instead be understood as a squarely first-order method without convexity assumptions. In fact, after switching off exponential moving averages, each method is equivalent to steepest descent under a particular norm. By generalizing this observation, we chart a new design space for training algorithms. Different operator norms should be assigned to different tensors based on the role that the tensor plays within the network. For example, while linear and embedding layers may have the same weight space of R^{mtimes n}, these layers play different roles and should be assigned different norms. We hope that this idea of carefully metrizing the neural architecture might lead to more stable, scalable and indeed faster training.

Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be computed piecewise in terms of the component processes. We study the structure of this compositional rule, noting that it relates to the lens pattern in functional programming. Working in a suitably general axiomatic presentation of a category of Markov kernels, we see how we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a fibred category. We discuss the compositional nature of this, formulated as a functor on the underlying category and explore how this can used for a more type-driven approach to statistical inference.

Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning.

Convolution is a broadly useful operation with applications including signal processing, machine learning, probability, optics, polynomial multiplication, and efficient parsing. Usually, however, this operation is understood and implemented in more specialized forms, hiding commonalities and limiting usefulness. This paper formulates convolution in the common algebraic framework of semirings and semimodules and populates that framework with various representation types. One of those types is the grand abstract template and itself generalizes to the free semimodule monad. Other representations serve varied uses and performance trade-offs, with implementations calculated from simple and regular specifications. Of particular interest is Brzozowski's method for regular expression matching. Uncovering the method's essence frees it from syntactic manipulations, while generalizing from boolean to weighted membership (such as multisets and probability distributions) and from sets to n-ary relations. The classic trie data structure then provides an elegant and efficient alternative to syntax. Pleasantly, polynomial arithmetic requires no additional implementation effort, works correctly with a variety of representations, and handles multivariate polynomials and power series with ease. Image convolution also falls out as a special case.

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-B\'enard convection, we found CoDA-NO to outperform existing methods by over 36%.

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.

Quantum computation is an emerging technology with important potential for solving certain problems pivotal in various scientific and engineering disciplines. This paper introduces an efficient quantum protocol for the explicit construction of the block-encoding for an important class of Hamiltonians. Using the Schrodingerisation technique -- which converts non-conservative PDEs into conservative ones -- this particular class of Hamiltonians is shown to be sufficient for simulating any linear partial differential equations that have coefficients which are polynomial functions. The class of Hamiltonians consist of discretisations of polynomial products and sums of position and momentum operators. This construction is explicit and leverages minimal one- and two-qubit operations. The explicit construction of this block-encoding forms a fundamental building block for constructing the unitary evolution operator for this Hamiltonian. The proposed algorithm exhibits polynomial scaling with respect to the spatial partitioning size, suggesting an exponential speedup over classical finite-difference methods. This work provides an important foundation for building explicit and efficient quantum circuits for solving partial differential equations.

Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring the formation of very high-dimensional matrices and tensors, and therefore being computationally infeasible. In this paper I show how a linear simplification of recursive neural tensor network models can be mapped directly onto the categorical approach, giving a way of computing the required matrices and tensors. This mapping suggests a number of lines of research for both categorical compositional vector space models of meaning and for recursive neural network models of compositionality.

We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.

Automated software engineering has been greatly empowered by the recent advances in Large Language Models (LLMs) for programming. While current benchmarks have shown that LLMs can perform various software engineering tasks like human developers, the majority of their evaluations are limited to short and self-contained algorithmic tasks. Solving challenging and practical programming tasks requires the capability of utilizing diverse function calls as tools to efficiently implement functionalities like data analysis and web development. In addition, using multiple tools to solve a task needs compositional reasoning by accurately understanding complex instructions. Fulfilling both of these characteristics can pose a great challenge for LLMs. To assess how well LLMs can solve challenging and practical programming tasks, we introduce Bench, a benchmark that challenges LLMs to invoke multiple function calls as tools from 139 libraries and 7 domains for 1,140 fine-grained programming tasks. To evaluate LLMs rigorously, each programming task encompasses 5.6 test cases with an average branch coverage of 99%. In addition, we propose a natural-language-oriented variant of Bench, Benchi, that automatically transforms the original docstrings into short instructions only with essential information. Our extensive evaluation of 60 LLMs shows that LLMs are not yet capable of following complex instructions to use function calls precisely, with scores up to 60%, significantly lower than the human performance of 97%. The results underscore the need for further advancements in this area.

Deep Learning (DL) is prevalently used in various industries to improve decision-making and automate processes, driven by the ever-evolving DL libraries and compilers. The correctness of DL systems is crucial for trust in DL applications. As such, the recent wave of research has been studying the automated synthesis of test-cases (i.e., DNN models and their inputs) for fuzzing DL systems. However, existing model generators only subsume a limited number of operators, lacking the ability to pervasively model operator constraints. To address this challenge, we propose NeuRI, a fully automated approach for generating valid and diverse DL models composed of hundreds of types of operators. NeuRI adopts a three-step process: (i) collecting valid and invalid API traces from various sources; (ii) applying inductive program synthesis over the traces to infer the constraints for constructing valid models; and (iii) using hybrid model generation which incorporates both symbolic and concrete operators. Our evaluation shows that NeuRI improves branch coverage of TensorFlow and PyTorch by 24% and 15% over the state-of-the-art model-level fuzzers. NeuRI finds 100 new bugs for PyTorch and TensorFlow in four months, with 81 already fixed or confirmed. Of these, 9 bugs are labelled as high priority or security vulnerability, constituting 10% of all high-priority bugs of the period. Open-source developers regard error-inducing tests reported by us as "high-quality" and "common in practice".

When writing programs, people have the ability to tackle a new complex task by decomposing it into smaller and more familiar subtasks. While it is difficult to measure whether neural program synthesis methods have similar capabilities, we can measure whether they compositionally generalize, that is, whether a model that has been trained on the simpler subtasks is subsequently able to solve more complex tasks. In this paper, we characterize several different forms of compositional generalization that are desirable in program synthesis, forming a meta-benchmark which we use to create generalization tasks for two popular datasets, RobustFill and DeepCoder. We then propose ExeDec, a novel decomposition-based synthesis strategy that predicts execution subgoals to solve problems step-by-step informed by program execution at each step. ExeDec has better synthesis performance and greatly improved compositional generalization ability compared to baselines.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural operators have emerged as a promising technique to solve elliptic PDEs more efficiently by directly mapping the input to solutions. However, existing networks typically cannot handle complex geometries and inhomogeneous boundary values present in the real world. Here we introduce Boundary-Embedded Neural Operators (BENO), a novel neural operator architecture that embeds the complex geometries and inhomogeneous boundary values into the solving of elliptic PDEs. Inspired by classical Green's function, BENO consists of two branches of Graph Neural Networks (GNNs) for interior source term and boundary values, respectively. Furthermore, a Transformer encoder maps the global boundary geometry into a latent vector which influences each message passing layer of the GNNs. We test our model extensively in elliptic PDEs with various boundary conditions. We show that all existing baseline methods fail to learn the solution operator. In contrast, our model, endowed with boundary-embedded architecture, outperforms state-of-the-art neural operators and strong baselines by an average of 60.96\%. Our source code can be found https://github.com/AI4Science-WestlakeU/beno.git.

Learning partial differential equations' (PDEs) solution operators is an essential problem in machine learning. However, there are several challenges for learning operators in practical applications like the irregular mesh, multiple input functions, and complexity of the PDEs' solution. To address these challenges, we propose a general neural operator transformer (GNOT), a scalable and effective transformer-based framework for learning operators. By designing a novel heterogeneous normalized attention layer, our model is highly flexible to handle multiple input functions and irregular meshes. Besides, we introduce a geometric gating mechanism which could be viewed as a soft domain decomposition to solve the multi-scale problems. The large model capacity of the transformer architecture grants our model the possibility to scale to large datasets and practical problems. We conduct extensive experiments on multiple challenging datasets from different domains and achieve a remarkable improvement compared with alternative methods. Our code and data are publicly available at https://github.com/thu-ml/GNOT.

Expert problem-solving is driven by powerful languages for thinking about problems and their solutions. Acquiring expertise means learning these languages -- systems of concepts, alongside the skills to use them. We present DreamCoder, a system that learns to solve problems by writing programs. It builds expertise by creating programming languages for expressing domain concepts, together with neural networks to guide the search for programs within these languages. A ``wake-sleep'' learning algorithm alternately extends the language with new symbolic abstractions and trains the neural network on imagined and replayed problems. DreamCoder solves both classic inductive programming tasks and creative tasks such as drawing pictures and building scenes. It rediscovers the basics of modern functional programming, vector algebra and classical physics, including Newton's and Coulomb's laws. Concepts are built compositionally from those learned earlier, yielding multi-layered symbolic representations that are interpretable and transferrable to new tasks, while still growing scalably and flexibly with experience.

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.

Pre-trained models produce strong generic representations that can be adapted via fine-tuning. The learned weight difference relative to the pre-trained model, known as a task vector, characterises the direction and stride of fine-tuning. The significance of task vectors is such that simple arithmetic operations on them can be used to combine diverse representations from different domains. This paper builds on these properties of task vectors and aims to answer (1) whether components of task vectors, particularly parameter blocks, exhibit similar characteristics, and (2) how such blocks can be used to enhance knowledge composition and transfer. To this end, we introduce aTLAS, an algorithm that linearly combines parameter blocks with different learned coefficients, resulting in anisotropic scaling at the task vector level. We show that such linear combinations explicitly exploit the low intrinsic dimensionality of pre-trained models, with only a few coefficients being the learnable parameters. Furthermore, composition of parameter blocks leverages the already learned representations, thereby reducing the dependency on large amounts of data. We demonstrate the effectiveness of our method in task arithmetic, few-shot recognition and test-time adaptation, with supervised or unsupervised objectives. In particular, we show that (1) learned anisotropic scaling allows task vectors to be more disentangled, causing less interference in composition; (2) task vector composition excels with scarce or no labeled data and is less prone to domain shift, thus leading to better generalisability; (3) mixing the most informative parameter blocks across different task vectors prior to training can reduce the memory footprint and improve the flexibility of knowledge transfer. Moreover, we show the potential of aTLAS as a PEFT method, particularly with less data, and demonstrate that its scalibility.

Automatic Differentiation (AD) has become a dominant technique in ML. AD frameworks have first been implemented for imperative languages using tapes. Meanwhile, functional implementations of AD have been developed, often based on dual numbers, which are close to the formal specification of differentiation and hence easier to prove correct. But these papers have focussed on correctness not efficiency. Recently, it was shown how an approach using dual numbers could be made efficient through the right optimizations. Optimizations are highly dependent on order, as one optimization can enable another. It can therefore be useful to have fine-grained control over the scheduling of optimizations. One method expresses compiler optimizations as rewrite rules, whose application can be combined and controlled using strategy languages. Previous work describes the use of term rewriting and strategies to generate high-performance code in a compiler for a functional language. In this work, we implement dual numbers AD in a functional array programming language using rewrite rules and strategy combinators for optimization. We aim to combine the elegance of differentiation using dual numbers with a succinct expression of the optimization schedule using a strategy language. We give preliminary evidence suggesting the viability of the approach on a micro-benchmark.

Efficient quantum compiling tactics greatly enhance the capability of quantum computers to execute complicated quantum algorithms. Due to its fundamental importance, a plethora of quantum compilers has been designed in past years. However, there are several caveats to current protocols, which are low optimality, high inference time, limited scalability, and lack of universality. To compensate for these defects, here we devise an efficient and practical quantum compiler assisted by advanced deep reinforcement learning (RL) techniques, i.e., data generation, deep Q-learning, and AQ* search. In this way, our protocol is compatible with various quantum machines and can be used to compile multi-qubit operators. We systematically evaluate the performance of our proposal in compiling quantum operators with both inverse-closed and inverse-free universal basis sets. In the task of single-qubit operator compiling, our proposal outperforms other RL-based quantum compilers in the measure of compiling sequence length and inference time. Meanwhile, the output solution is near-optimal, guaranteed by the Solovay-Kitaev theorem. Notably, for the inverse-free universal basis set, the achieved sequence length complexity is comparable with the inverse-based setting and dramatically advances previous methods. These empirical results contribute to improving the inverse-free Solovay-Kitaev theorem. In addition, for the first time, we demonstrate how to leverage RL-based quantum compilers to accomplish two-qubit operator compiling. The achieved results open an avenue for integrating RL with quantum compiling to unify efficiency and practicality and thus facilitate the exploration of quantum advantages.

Many complex tasks can be decomposed into simpler, independent parts. Discovering such underlying compositional structure has the potential to enable compositional generalization. Despite progress, our most powerful systems struggle to compose flexibly. It therefore seems natural to make models more modular to help capture the compositional nature of many tasks. However, it is unclear under which circumstances modular systems can discover hidden compositional structure. To shed light on this question, we study a teacher-student setting with a modular teacher where we have full control over the composition of ground truth modules. This allows us to relate the problem of compositional generalization to that of identification of the underlying modules. In particular we study modularity in hypernetworks representing a general class of multiplicative interactions. We show theoretically that identification up to linear transformation purely from demonstrations is possible without having to learn an exponential number of module combinations. We further demonstrate empirically that under the theoretically identified conditions, meta-learning from finite data can discover modular policies that generalize compositionally in a number of complex environments.

Compositional actions consist of dynamic (verbs) and static (objects) concepts. Humans can easily recognize unseen compositions using the learned concepts. For machines, solving such a problem requires a model to recognize unseen actions composed of previously observed verbs and objects, thus requiring so-called compositional generalization ability. To facilitate this research, we propose a novel Zero-Shot Compositional Action Recognition (ZS-CAR) task. For evaluating the task, we construct a new benchmark, Something-composition (Sth-com), based on the widely used Something-Something V2 dataset. We also propose a novel Component-to-Composition (C2C) learning method to solve the new ZS-CAR task. C2C includes an independent component learning module and a composition inference module. Last, we devise an enhanced training strategy to address the challenges of component variations between seen and unseen compositions and to handle the subtle balance between learning seen and unseen actions. The experimental results demonstrate that the proposed framework significantly surpasses the existing compositional generalization methods and sets a new state-of-the-art. The new Sth-com benchmark and code are available at https://github.com/RongchangLi/ZSCAR_C2C.

We investigate causal computations taking sequences of inputs to sequences of outputs where the nth output depends on the first n inputs only. We model these in category theory via a construction taking a Cartesian category C to another category St(C) with a novel trace-like operation called "delayed trace", which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in St(C) with an implicit guardedness guarantee. When C is equipped with a Cartesian differential operator, we construct a differential operator for St(C) using an abstract version of backpropagation through time, a technique from machine learning based on unrolling of functions. This obtains a swath of properties for backpropagation through time, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

This thesis introduces quantum natural language processing (QNLP) models based on a simple yet powerful analogy between computational linguistics and quantum mechanics: grammar as entanglement. The grammatical structure of text and sentences connects the meaning of words in the same way that entanglement structure connects the states of quantum systems. Category theory allows to make this language-to-qubit analogy formal: it is a monoidal functor from grammar to vector spaces. We turn this abstract analogy into a concrete algorithm that translates the grammatical structure onto the architecture of parameterised quantum circuits. We then use a hybrid classical-quantum algorithm to train the model so that evaluating the circuits computes the meaning of sentences in data-driven tasks. The implementation of QNLP models motivated the development of DisCoPy (Distributional Compositional Python), the toolkit for applied category theory of which the first chapter gives a comprehensive overview. String diagrams are the core data structure of DisCoPy, they allow to reason about computation at a high level of abstraction. We show how they can encode both grammatical structures and quantum circuits, but also logical formulae, neural networks or arbitrary Python code. Monoidal functors allow to translate these abstract diagrams into concrete computation, interfacing with optimised task-specific libraries. The second chapter uses DisCopy to implement QNLP models as parameterised functors from grammar to quantum circuits. It gives a first proof-of-concept for the more general concept of functorial learning: generalising machine learning from functions to functors by learning from diagram-like data. In order to learn optimal functor parameters via gradient descent, we introduce the notion of diagrammatic differentiation: a graphical calculus for computing the gradients of parameterised diagrams.

We give a geometry of interaction model for a typed lambda-calculus endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate with respect to both a distribution-based and a sampling based operational semantics.

This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expressive capabilities of the individual network itself. Specifically, we prove by construction that L_2circ g^{circ r}circ mathcal{L}_1 can approximate 1-Lipschitz continuous functions on [0,1]^d with an error O(r^{-1/d}), where g is realized by a fixed-size ReLU network, mathcal{L}_1 and L_2 are two affine linear maps matching the dimensions, and g^{circ r} denotes the r-times composition of g. Furthermore, we extend such a result to generic continuous functions on [0,1]^d with the approximation error characterized by the modulus of continuity. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent and realized by a fixed-size ReLU network.

Transformer large language models (LLMs) have sparked admiration for their exceptional performance on tasks that demand intricate multi-step reasoning. Yet, these models simultaneously show failures on surprisingly trivial problems. This begs the question: Are these errors incidental, or do they signal more substantial limitations? In an attempt to demystify Transformers, we investigate the limits of these models across three representative compositional tasks -- multi-digit multiplication, logic grid puzzles, and a classic dynamic programming problem. These tasks require breaking problems down into sub-steps and synthesizing these steps into a precise answer. We formulate compositional tasks as computation graphs to systematically quantify the level of complexity, and break down reasoning steps into intermediate sub-procedures. Our empirical findings suggest that Transformers solve compositional tasks by reducing multi-step compositional reasoning into linearized subgraph matching, without necessarily developing systematic problem-solving skills. To round off our empirical study, we provide theoretical arguments on abstract multi-step reasoning problems that highlight how Transformers' performance will rapidly decay with increased task complexity.

Intermediate reasoning or acting steps have successfully improved large language models (LLMs) for handling various downstream natural language processing (NLP) tasks. When applying LLMs for code generation, recent works mainly focus on directing the models to articulate intermediate natural-language reasoning steps, as in chain-of-thought (CoT) prompting, and then output code with the natural language or other structured intermediate steps. However, such output is not suitable for code translation or generation tasks since the standard CoT has different logical structures and forms of expression with the code. In this work, we introduce the universal code (UniCode) as the intermediate representation. It is a description of algorithm steps using a mix of conventions of programming languages, such as assignment operator, conditional operator, and loop. Hence, we collect an instruction dataset UniCoder-Instruct to train our model UniCoder on multi-task learning objectives. UniCoder-Instruct comprises natural-language questions, code solutions, and the corresponding universal code. The alignment between the intermediate universal code representation and the final code solution significantly improves the quality of the generated code. The experimental results demonstrate that UniCoder with the universal code significantly outperforms the previous prompting methods by a large margin, showcasing the effectiveness of the structural clues in pseudo-code.

Attention operators have been applied on both 1-D data like texts and higher-order data such as images and videos. Use of attention operators on high-order data requires flattening of the spatial or spatial-temporal dimensions into a vector, which is assumed to follow a multivariate normal distribution. This not only incurs excessive requirements on computational resources, but also fails to preserve structures in data. In this work, we propose to avoid flattening by assuming the data follow matrix-variate normal distributions. Based on this new view, we develop Kronecker attention operators (KAOs) that operate on high-order tensor data directly. More importantly, the proposed KAOs lead to dramatic reductions in computational resources. Experimental results show that our methods reduce the amount of required computational resources by a factor of hundreds, with larger factors for higher-dimensional and higher-order data. Results also show that networks with KAOs outperform models without attention, while achieving competitive performance as those with original attention operators.

Physics is based on probabilities as fundamental entities of a mathematical description. Expectation values of observables are computed according to the classical statistical rule. The overall probability distribution for one world covers all times. The quantum formalism arises once one focuses on the evolution of the time-local probabilistic information. Wave functions or the density matrix allow the formulation of a general linear evolution law for classical statistics. The quantum formalism for classical statistics is a powerful tool which allows us to implement for generalized Ising models the momentum observable with the associated Fourier representation. The association of operators to observables permits the computation of expectation values in terms of the density matrix by the usual quantum rule. We show that probabilistic cellular automata are quantum systems in a formulation with discrete time steps and real wave functions. With a complex structure the evolution operator for automata can be expressed in terms of a Hamiltonian involving fermionic creation and annihilation operators. The time-local probabilistic information amounts to a subsystem of the overall probabilistic system which is correlated with its environment consisting of the past and future. Such subsystems typically involve probabilistic observables for which only a probability distribution for their possible measurement values is available. Incomplete statistics does not permit to compute classical correlation functions for arbitrary subsystem-observables. Bell's inequalities are not generally applicable.

We introduce a large-scale dataset of math word problems and an interpretable neural math problem solver that learns to map problems to operation programs. Due to annotation challenges, current datasets in this domain have been either relatively small in scale or did not offer precise operational annotations over diverse problem types. We introduce a new representation language to model precise operation programs corresponding to each math problem that aim to improve both the performance and the interpretability of the learned models. Using this representation language, our new dataset, MathQA, significantly enhances the AQuA dataset with fully-specified operational programs. We additionally introduce a neural sequence-to-program model enhanced with automatic problem categorization. Our experiments show improvements over competitive baselines in our MathQA as well as the AQuA dataset. The results are still significantly lower than human performance indicating that the dataset poses new challenges for future research. Our dataset is available at: https://math-qa.github.io/math-QA/

Large monolithic generative models trained on massive amounts of data have become an increasingly dominant approach in AI research. In this paper, we argue that we should instead construct large generative systems by composing smaller generative models together. We show how such a compositional generative approach enables us to learn distributions in a more data-efficient manner, enabling generalization to parts of the data distribution unseen at training time. We further show how this enables us to program and construct new generative models for tasks completely unseen at training. Finally, we show that in many cases, we can discover separate compositional components from data.

Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a monoidal category. More recently, the notion of a learner has been proposed: these provide a compositional way of modelling supervised learning algorithms, and again form the morphisms of a monoidal category. In this paper, we show that the two concepts are tightly linked. We show both that there is a faithful, identity-on-objects symmetric monoidal functor embedding a category of asymmetric lenses into the category of learners, and furthermore there is such a functor embedding the category of learners into a category of symmetric lenses.

We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries.

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and memory expense -- these systems are often forced to rely on autoregressive time-stepping of the neural network to predict future temporal states. While this is effective in managing costs, it can lead to uncontrolled error growth over time and eventual instability. We analyze the sources of this autoregressive error growth using prototypical neural operator models for physical systems and explore ways to mitigate it. We introduce architectural and application-specific improvements that allow for careful control of instability-inducing operations within these models without inflating the compute/memory expense. We present results on several scientific systems that include Navier-Stokes fluid flow, rotating shallow water, and a high-resolution global weather forecasting system. We demonstrate that applying our design principles to neural operators leads to significantly lower errors for long-term forecasts as well as longer time horizons without qualitative signs of divergence compared to the original models for these systems. We open-source our https://github.com/mikemccabe210/stabilizing_neural_operators{code} for reproducibility.

Learning neural operators for solving partial differential equations (PDEs) has attracted great attention due to its high inference efficiency. However, training such operators requires generating a substantial amount of labeled data, i.e., PDE problems together with their solutions. The data generation process is exceptionally time-consuming, as it involves solving numerous systems of linear equations to obtain numerical solutions to the PDEs. Many existing methods solve these systems independently without considering their inherent similarities, resulting in extremely redundant computations. To tackle this problem, we propose a novel method, namely Sorting Krylov Recycling (SKR), to boost the efficiency of solving these systems, thus significantly accelerating data generation for neural operators training. To the best of our knowledge, SKR is the first attempt to address the time-consuming nature of data generation for learning neural operators. The working horse of SKR is Krylov subspace recycling, a powerful technique for solving a series of interrelated systems by leveraging their inherent similarities. Specifically, SKR employs a sorting algorithm to arrange these systems in a sequence, where adjacent systems exhibit high similarities. Then it equips a solver with Krylov subspace recycling to solve the systems sequentially instead of independently, thus effectively enhancing the solving efficiency. Both theoretical analysis and extensive experiments demonstrate that SKR can significantly accelerate neural operator data generation, achieving a remarkable speedup of up to 13.9 times.

Model fusing has always been an important topic, especially in an era where large language models (LLM) and multi-modal language models (MLM) with different architectures, parameter sizes and training pipelines, are being created all the time. In this work, we propose a post-hoc framework, aiming at fusing heterogeneous models off-the-shell, which we call likelihood composition, and the basic idea is to compose multiple models' likelihood distribution when doing a multi-choice visual-question-answering task. Here the core concept, likelihood, is actually the log-probability of the candidate answer. In likelihood composition, we introduce some basic operations: debias, highlight, majority-vote and ensemble. By combining (composing) these basic elements, we get the mixed composition methods: mix-composition. Through conducting comprehensive experiments on 9 VQA datasets and 10 MLMs, we prove the effectiveness of mix-composition compared with simple ensemble or majority-vote methods. In this framework, people can propose new basic composition methods and combine them to get the new mixed composition methods. We hope our proposed likelihood composition can provide a new perspective of fusing heterogeneous models and inspire the exploration under this framework.

Recent advances in large language models (LLMs) and audio language models have significantly improved music generation, particularly in lyrics-to-song generation. However, existing approaches still struggle with the complex composition of songs and the scarcity of high-quality data, leading to limitations in sound quality, musicality, instruction following, and vocal-instrument harmony. To address these challenges, we introduce LeVo, an LM-based framework consisting of LeLM and a music codec. LeLM is capable of parallelly modeling two types of tokens: mixed tokens, which represent the combined audio of vocals and accompaniment to achieve vocal-instrument harmony, and dual-track tokens, which separately encode vocals and accompaniment for high-quality song generation. It employs two decoder-only transformers and a modular extension training strategy to prevent interference between different token types. To further enhance musicality and instruction following, we introduce a multi-preference alignment method based on Direct Preference Optimization (DPO). This method handles diverse human preferences through a semi-automatic data construction process and DPO post-training. Experimental results demonstrate that LeVo consistently outperforms existing methods on both objective and subjective metrics. Ablation studies further justify the effectiveness of our designs. Audio examples are available at https://levo-demo.github.io/.

In this paper, we propose a highly parameter-efficient approach to scaling pre-trained language models (PLMs) to a deeper model depth. Unlike prior work that shares all parameters or uses extra blocks, we design a more capable parameter-sharing architecture based on matrix product operator (MPO). MPO decomposition can reorganize and factorize the information of a parameter matrix into two parts: the major part that contains the major information (central tensor) and the supplementary part that only has a small proportion of parameters (auxiliary tensors). Based on such a decomposition, our architecture shares the central tensor across all layers for reducing the model size and meanwhile keeps layer-specific auxiliary tensors (also using adapters) for enhancing the adaptation flexibility. To improve the model training, we further propose a stable initialization algorithm tailored for the MPO-based architecture. Extensive experiments have demonstrated the effectiveness of our proposed model in reducing the model size and achieving highly competitive performance.

Approximating nonlinear differential equations using a neural network provides a robust and efficient tool for various scientific computing tasks, including real-time predictions, inverse problems, optimal controls, and surrogate modeling. Previous works have focused on embedding dynamical systems into networks through two approaches: learning a single solution operator (i.e., the mapping from input parametrized functions to solutions) or learning the governing system of equations (i.e., the constitutive model relative to the state variables). Both of these approaches yield different representations for the same underlying data or function. Additionally, observing that families of differential equations often share key characteristics, we seek one network representation across a wide range of equations. Our method, called Predicting Operators and Symbolic Expressions (PROSE), learns maps from multimodal inputs to multimodal outputs, capable of generating both numerical predictions and mathematical equations. By using a transformer structure and a feature fusion approach, our network can simultaneously embed sets of solution operators for various parametric differential equations using a single trained network. Detailed experiments demonstrate that the network benefits from its multimodal nature, resulting in improved prediction accuracy and better generalization. The network is shown to be able to handle noise in the data and errors in the symbolic representation, including noisy numerical values, model misspecification, and erroneous addition or deletion of terms. PROSE provides a new neural network framework for differential equations which allows for more flexibility and generality in learning operators and governing equations from data.

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.

Transformers have found extensive applications across various domains due to the powerful fitting capabilities. This success can be partially attributed to their inherent nonlinearity. Thus, in addition to the ReLU function employed in the original transformer architecture, researchers have explored alternative modules such as GeLU and SwishGLU to enhance nonlinearity and thereby augment representational capacity. In this paper, we propose a novel category of polynomial composition activations (PolyCom), designed to optimize the dynamics of transformers. Theoretically, we provide a comprehensive mathematical analysis of PolyCom, highlighting its enhanced expressivity and efficacy relative to other activation functions. Notably, we demonstrate that networks incorporating PolyCom achieve the optimal approximation rate, indicating that PolyCom networks require minimal parameters to approximate general smooth functions in Sobolev spaces. We conduct empirical experiments on the pre-training configurations of large language models (LLMs), including both dense and sparse architectures. By substituting conventional activation functions with PolyCom, we enable LLMs to capture higher-order interactions within the data, thus improving performance metrics in terms of accuracy and convergence rates. Extensive experimental results demonstrate the effectiveness of our method, showing substantial improvements over other activation functions. Code is available at https://github.com/BryceZhuo/PolyCom.

In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood

In large language model (LLM) pretraining, data quality is believed to determine model quality. In this paper, we re-examine the notion of "quality" from the perspective of pre- and post-training co-design. Specifically, we explore the possibility that pre-training on more toxic data can lead to better control in post-training, ultimately decreasing a model's output toxicity. First, we use a toy experiment to study how data composition affects the geometry of features in the representation space. Next, through controlled experiments with Olmo-1B models trained on varying ratios of clean and toxic data, we find that the concept of toxicity enjoys a less entangled linear representation as the proportion of toxic data increases. Furthermore, we show that although toxic data increases the generational toxicity of the base model, it also makes the toxicity easier to remove. Evaluations on Toxigen and Real Toxicity Prompts demonstrate that models trained on toxic data achieve a better trade-off between reducing generational toxicity and preserving general capabilities when detoxifying techniques such as inference-time intervention (ITI) are applied. Our findings suggest that, with post-training taken into account, bad data may lead to good models.

Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership i in S and estimating set intersection sizes |X cap Y| for two sets of symbols X and Y, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

Proximal operators are ubiquitous in inverse problems, commonly appearing as part of algorithmic strategies to regularize problems that are otherwise ill-posed. Modern deep learning models have been brought to bear for these tasks too, as in the framework of plug-and-play or deep unrolling, where they loosely resemble proximal operators. Yet, something essential is lost in employing these purely data-driven approaches: there is no guarantee that a general deep network represents the proximal operator of any function, nor is there any characterization of the function for which the network might provide some approximate proximal. This not only makes guaranteeing convergence of iterative schemes challenging but, more fundamentally, complicates the analysis of what has been learned by these networks about their training data. Herein we provide a framework to develop learned proximal networks (LPN), prove that they provide exact proximal operators for a data-driven nonconvex regularizer, and show how a new training strategy, dubbed proximal matching, provably promotes the recovery of the log-prior of the true data distribution. Such LPN provide general, unsupervised, expressive proximal operators that can be used for general inverse problems with convergence guarantees. We illustrate our results in a series of cases of increasing complexity, demonstrating that these models not only result in state-of-the-art performance, but provide a window into the resulting priors learned from data.

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of {\em labeled} training data. We propose the Deep Generative Neural Operator (DGNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGNO a powerful tool for scientific and engineering applications.

Offline reinforcement learning (RL) is a compelling framework for learning optimal policies from past experiences without additional interaction with the environment. Nevertheless, offline RL inevitably faces the problem of distributional shifts, where the states and actions encountered during policy execution may not be in the training dataset distribution. A common solution involves incorporating conservatism into the policy or the value function to safeguard against uncertainties and unknowns. In this work, we focus on achieving the same objectives of conservatism but from a different perspective. We propose COmpositional COnservatism with Anchor-seeking (COCOA) for offline RL, an approach that pursues conservatism in a compositional manner on top of the transductive reparameterization (Netanyahu et al., 2023), which decomposes the input variable (the state in our case) into an anchor and its difference from the original input. Our COCOA seeks both in-distribution anchors and differences by utilizing the learned reverse dynamics model, encouraging conservatism in the compositional input space for the policy or value function. Such compositional conservatism is independent of and agnostic to the prevalent behavioral conservatism in offline RL. We apply COCOA to four state-of-the-art offline RL algorithms and evaluate them on the D4RL benchmark, where COCOA generally improves the performance of each algorithm. The code is available at https://github.com/runamu/compositional-conservatism.

E(3)-equivariant neural networks have demonstrated success across a wide range of 3D modelling tasks. A fundamental operation in these networks is the tensor product, which interacts two geometric features in an equivariant manner to create new features. Due to the high computational complexity of the tensor product, significant effort has been invested to optimize the runtime of this operation. For example, Luo et al. (2024) recently proposed the Gaunt tensor product (GTP) which promises a significant speedup. In this work, we provide a careful, systematic analysis of a number of tensor product operations. In particular, we emphasize that different tensor products are not performing the same operation. The reported speedups typically come at the cost of expressivity. We introduce measures of expressivity and interactability to characterize these differences. In addition, we realized the original implementation of GTP can be greatly simplified by directly using a spherical grid at no cost in asymptotic runtime. This spherical grid approach is faster on our benchmarks and in actual training of the MACE interatomic potential by 30%. Finally, we provide the first systematic microbenchmarks of the various tensor product operations. We find that the theoretical runtime guarantees can differ wildly from empirical performance, demonstrating the need for careful application-specific benchmarking. Code is available at https://github.com/atomicarchitects/PriceofFreedom.

This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function Delta(lambda) . This can be used to determine the multiplicities of eigenvalues for concrete operators.

We develop tools for explicitly constructing categories enriched over generating data and that compose via ordinary scalar and matrix arithmetic arithmetic operations. We characterize meaningful size maps, weightings, and magnitude that reveal features analogous to outliers that these same notions have previously been shown to reveal in the context of metric spaces. Throughout, we provide examples of such "outlier detection" relevant to the analysis of computer programs, neural networks, cyber-physical systems, and networks of communications channels.

Despite high benchmark scores, Large Language Models (LLMs) often fail simple problem, raising a critical question: Do LLMs learn mathematical principles or merely memorize patterns? Rather than designing increasingly complex benchmarks like recent works, we investigate this using elementary two-integer addition (0 to 2^{64}), probing two core properties: commutativity (A+B=B+A) and compositional generalization (via isomorphic symbolic mappings, e.g., 7 rightarrow y). While state-of-the-art LLMs achieve 73.8-99.8\% accuracy on numerical addition, performance collapses to leq7.5\% under symbolic mapping, indicating failure to generalize learned rules. Non-monotonic performance scaling with digit count and frequent commutativity violations (over 1,700 cases of A+B neq B+A) further support this. Explicitly providing addition rules degrades performance by 81.2\% on average, while self-explanation maintains baseline accuracy, suggesting LLM arithmetic processing is misaligned with human-defined principles. Our findings indicate current LLMs rely on memory pattern over genuine rule learning, highlighting architectural limitations and the need for new approaches to achieve true mathematical reasoning.

Recent studies have drawn attention to the untapped potential of the "star operation" (element-wise multiplication) in network design. While intuitive explanations abound, the foundational rationale behind its application remains largely unexplored. Our study attempts to reveal the star operation's ability to map inputs into high-dimensional, non-linear feature spaces -- akin to kernel tricks -- without widening the network. We further introduce StarNet, a simple yet powerful prototype, demonstrating impressive performance and low latency under compact network structure and efficient budget. Like stars in the sky, the star operation appears unremarkable but holds a vast universe of potential. Our work encourages further exploration across tasks, with codes available at https://github.com/ma-xu/Rewrite-the-Stars.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis. By framing neural networks as operators with fixed points representing desired solutions, we develop a theoretical framework grounded in iterative methods for operator equations. Under defined conditions, we present convergence proofs based on fixed point theory. We demonstrate that popular architectures, such as diffusion models and AlphaFold, inherently employ iterative operator learning. Empirical assessments highlight that performing iterations through network operators improves performance. We also introduce an iterative graph neural network, PIGN, that further demonstrates benefits of iterations. Our work aims to enhance the understanding of deep learning by merging insights from numerical analysis, potentially guiding the design of future networks with clearer theoretical underpinnings and improved performance.

State-of-the-art machine learning methods exhibit limited compositional generalization. At the same time, there is a lack of realistic benchmarks that comprehensively measure this ability, which makes it challenging to find and evaluate improvements. We introduce a novel method to systematically construct such benchmarks by maximizing compound divergence while guaranteeing a small atom divergence between train and test sets, and we quantitatively compare this method to other approaches for creating compositional generalization benchmarks. We present a large and realistic natural language question answering dataset that is constructed according to this method, and we use it to analyze the compositional generalization ability of three machine learning architectures. We find that they fail to generalize compositionally and that there is a surprisingly strong negative correlation between compound divergence and accuracy. We also demonstrate how our method can be used to create new compositionality benchmarks on top of the existing SCAN dataset, which confirms these findings.

Quantizing the activation, weight, and gradient to 4-bit is promising to accelerate neural network training. However, existing 4-bit training methods require custom numerical formats which are not supported by contemporary hardware. In this work, we propose a training method for transformers with all matrix multiplications implemented with the INT4 arithmetic. Training with an ultra-low INT4 precision is challenging. To achieve this, we carefully analyze the specific structures of activation and gradients in transformers to propose dedicated quantizers for them. For forward propagation, we identify the challenge of outliers and propose a Hadamard quantizer to suppress the outliers. For backpropagation, we leverage the structural sparsity of gradients by proposing bit splitting and leverage score sampling techniques to quantize gradients accurately. Our algorithm achieves competitive accuracy on a wide range of tasks including natural language understanding, machine translation, and image classification. Unlike previous 4-bit training methods, our algorithm can be implemented on the current generation of GPUs. Our prototypical linear operator implementation is up to 2.2 times faster than the FP16 counterparts and speeds up the training by up to 35.1%.

Multivariable time series forecasting methods can integrate information from exogenous variables, leading to significant prediction accuracy gains. Transformer architecture has been widely applied in various time series forecasting models due to its ability to capture long-range sequential dependencies. However, a na\"ive application of transformers often struggles to effectively model complex relationships among variables over time. To mitigate against this, we propose a novel architecture, namely the Spectral Operator Neural Network (Sonnet). Sonnet applies learnable wavelet transformations to the input and incorporates spectral analysis using the Koopman operator. Its predictive skill relies on the Multivariable Coherence Attention (MVCA), an operation that leverages spectral coherence to model variable dependencies. Our empirical analysis shows that Sonnet yields the best performance on 34 out of 47 forecasting tasks with an average mean absolute error (MAE) reduction of 1.1% against the most competitive baseline (different per task). We further show that MVCA -- when put in place of the na\"ive attention used in various deep learning models -- can remedy its deficiencies, reducing MAE by 10.7% on average in the most challenging forecasting tasks.

Recently, Mixture-of-Experts (short as MoE) architecture has achieved remarkable success in increasing the model capacity of large-scale language models. However, MoE requires incorporating significantly more parameters than the base model being extended. In this paper, we propose building a parameter-efficient MoE architecture by sharing information among experts. We adopt the matrix product operator (MPO, a tensor decomposition from quantum many-body physics) to reconstruct the parameter matrix in the expert layer and increase model capacity for pre-trained language models by sharing parameters of the central tensor (containing the core information) among different experts while enabling the specificity through the auxiliary tensors (complementing the central tensor) of different experts. To address the unbalanced optimization issue, we further design the gradient mask strategy for the MPO-based MoE architecture. Extensive experiments based on T5 and GPT-2 show improved performance and efficiency of the pre-trained language model (27.2x reduction in total parameters for the superior model performance, compared with the Switch Transformers). Our code is publicly available at https://github.com/RUCAIBox/MPOE.

Query rewriting is a classical technique for transforming complex declarative SQL queries into ``lean'' equivalents that are conducive to (a) faster execution from a performance perspective, and (b) better understanding from a developer perspective. The rewriting is typically achieved via transformation rules, but these rules are limited in scope and difficult to update in a production system. In recent times, LLM-based techniques have also been mooted, but they are prone to both semantic and syntactic errors. We investigate here, how the remarkable cognitive capabilities of LLMs can be leveraged for performant query rewriting while incorporating safeguards and optimizations to ensure correctness and efficiency. Our study shows that these goals can be progressively achieved through incorporation of (a) an ensemble suite of basic prompts, (b) database-sensitive prompts via redundancy removal and selectivity-based rewriting rules, and (c) LLM token probability-guided rewrite paths. Further, a suite of statistical and logic-based tools can be used to guard against errors produced by the model. We have implemented the above LLM-infused techniques in the LITHE system, and evaluated complex analytic queries from multiple benchmarks on contemporary database platforms. The results show significant improvements over SOTA rewriting techniques -- for instance, on TPC-DS, LITHE constructed productive (>1.5x speedup) rewrites for two-thirds of the query suite, delivering four times more coverage than SOTA. Further, the geometric mean of its estimated execution speedups was an order-of-magnitude jump over SOTA performance. In essence, LITHE offers a potent and robust LLM-based intermediary between enterprise applications and database engines.

Triton, a high-level Python-like language designed for building efficient GPU kernels, is widely adopted in deep learning frameworks due to its portability, flexibility, and accessibility. However, programming and parallel optimization still require considerable trial and error from Triton developers. Despite advances in large language models (LLMs) for conventional code generation, these models struggle to generate accurate, performance-optimized Triton code, as they lack awareness of its specifications and the complexities of GPU programming. More critically, there is an urgent need for systematic evaluations tailored to Triton. In this work, we introduce TritonBench, the first comprehensive benchmark for Triton operator generation. TritonBench features two evaluation channels: a curated set of 184 real-world operators from GitHub and a collection of operators aligned with PyTorch interfaces. Unlike conventional code benchmarks prioritizing functional correctness, TritonBench also profiles efficiency performance on widely deployed GPUs aligned with industry applications. Our study reveals that current state-of-the-art code LLMs struggle to generate efficient Triton operators, highlighting a significant gap in high-performance code generation. TritonBench will be available at https://github.com/thunlp/TritonBench.

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which n component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter L. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is mathcal{O}( n + nLvarepsilon^{-1}), which improves upon existing methods by a factor up to n. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Learning complex dynamics driven by partial differential equations directly from data holds great promise for fast and accurate simulations of complex physical systems. In most cases, this problem can be formulated as an operator learning task, where one aims to learn the operator representing the physics of interest, which entails discretization of the continuous system. However, preserving key continuous properties at the discrete level, such as boundary conditions, and addressing physical systems with complex geometries is challenging for most existing approaches. We introduce a family of operator learning architectures, structure-preserving operator networks (SPONs), that allows to preserve key mathematical and physical properties of the continuous system by leveraging finite element (FE) discretizations of the input-output spaces. SPONs are encode-process-decode architectures that are end-to-end differentiable, where the encoder and decoder follows from the discretizations of the input-output spaces. SPONs can operate on complex geometries, enforce certain boundary conditions exactly, and offer theoretical guarantees. Our framework provides a flexible way of devising structure-preserving architectures tailored to specific applications, and offers an explicit trade-off between performance and efficiency, all thanks to the FE discretization of the input-output spaces. Additionally, we introduce a multigrid-inspired SPON architecture that yields improved performance at higher efficiency. Finally, we release a software to automate the design and training of SPON architectures.

Large Language Models (LLMs) have demonstrated remarkable general capabilities, but enhancing skills such as reasoning often demands substantial computational resources and may compromise their generalization. While Parameter-Efficient Fine-Tuning (PEFT) methods offer a more resource-conscious alternative, they typically requires retraining for each LLM backbone due to architectural dependencies. To address these challenges, here we propose Universal Reasoner (UniR) - a single, lightweight, composable, and plug-and-play reasoning module that can be used with any frozen LLM to endow it with specialized reasoning capabilities. Specifically, UniR decomposes the reward into a standalone reasoning module that is trained independently using predefined rewards, effectively translating trajectory-level signals into token-level guidance. Once trained, UniR can be combined with any frozen LLM at inference time by simply adding its output logits to those of the LLM backbone. This additive structure naturally enables modular composition: multiple UniR modules trained for different tasks can be jointly applied by summing their logits, enabling complex reasoning via composition. Experimental results on mathematical reasoning and machine translation tasks show that UniR significantly outperforms existing baseline fine-tuning methods using the Llama3.2 model. Furthermore, UniR demonstrates strong weak-to-strong generalization: reasoning modules trained on smaller models effectively guide much larger LLMs. This makes UniR a cost-efficient, adaptable, and robust solution for enhancing reasoning in LLMs without compromising their core capabilities. Code is open-sourced at https://github.com/hangeol/UniR

Machine learning programs, such as those performing inference, fine-tuning, and training of LLMs, are commonly delegated to untrusted compute providers. To provide correctness guarantees for the client, we propose adapting the cryptographic notion of refereed delegation to the machine learning setting. This approach enables a computationally limited client to delegate a program to multiple untrusted compute providers, with a guarantee of obtaining the correct result if at least one of them is honest. Refereed delegation of ML programs poses two technical hurdles: (1) an arbitration protocol to resolve disputes when compute providers disagree on the output, and (2) the ability to bitwise reproduce ML programs across different hardware setups, For (1), we design Verde, a dispute arbitration protocol that efficiently handles the large scale and graph-based computational model of modern ML programs. For (2), we build RepOps (Reproducible Operators), a library that eliminates hardware "non-determinism" by controlling the order of floating point operations performed on all hardware. Our implementation shows that refereed delegation achieves both strong guarantees for clients and practical overheads for compute providers.

This paper presents Okapi, a new dataset for Natural Language to executable web Application Programming Interfaces (NL2API). This dataset is in English and contains 22,508 questions and 9,019 unique API calls, covering three domains. We define new compositional generalization tasks for NL2API which explore the models' ability to extrapolate from simple API calls in the training set to new and more complex API calls in the inference phase. Also, the models are required to generate API calls that execute correctly as opposed to the existing approaches which evaluate queries with placeholder values. Our dataset is different than most of the existing compositional semantic parsing datasets because it is a non-synthetic dataset studying the compositional generalization in a low-resource setting. Okapi is a step towards creating realistic datasets and benchmarks for studying compositional generalization alongside the existing datasets and tasks. We report the generalization capabilities of sequence-to-sequence baseline models trained on a variety of the SCAN and Okapi datasets tasks. The best model achieves 15\% exact match accuracy when generalizing from simple API calls to more complex API calls. This highlights some challenges for future research. Okapi dataset and tasks are publicly available at https://aka.ms/nl2api/data.

We present semantic correctness proofs of Automatic Differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Finally, we sketch how the analysis extends to other AD methods by considering a continuation-based method.

We study the code generation behavior of instruction-tuned models built on top of code pre-trained language models when they could access an auxiliary function to implement a function. We design several ways to provide auxiliary functions to the models by adding them to the query or providing a response prefix to incorporate the ability to utilize auxiliary functions with the instruction-following capability. Our experimental results show the effectiveness of combining the base models' auxiliary function utilization ability with the instruction following ability. In particular, the performance of adopting our approaches with the open-sourced language models surpasses that of the recent powerful proprietary language models, i.e., gpt-4o.

We present OLMo 2, the next generation of our fully open language models. OLMo 2 includes dense autoregressive models with improved architecture and training recipe, pretraining data mixtures, and instruction tuning recipes. Our modified model architecture and training recipe achieve both better training stability and improved per-token efficiency. Our updated pretraining data mixture introduces a new, specialized data mix called Dolmino Mix 1124, which significantly improves model capabilities across many downstream task benchmarks when introduced via late-stage curriculum training (i.e. specialized data during the annealing phase of pretraining). Finally, we incorporate best practices from T\"ulu 3 to develop OLMo 2-Instruct, focusing on permissive data and extending our final-stage reinforcement learning with verifiable rewards (RLVR). Our OLMo 2 base models sit at the Pareto frontier of performance to compute, often matching or outperforming open-weight only models like Llama 3.1 and Qwen 2.5 while using fewer FLOPs and with fully transparent training data, code, and recipe. Our fully open OLMo 2-Instruct models are competitive with or surpassing open-weight only models of comparable size, including Qwen 2.5, Llama 3.1 and Gemma 2. We release all OLMo 2 artifacts openly -- models at 7B and 13B scales, both pretrained and post-trained, including their full training data, training code and recipes, training logs and thousands of intermediate checkpoints. The final instruction model is available on the Ai2 Playground as a free research demo.

We explore the effects of various plasticity functions on assemblies of neurons. To bridge the gap between experimental and computational theories we make use of a conceptual framework, the Assembly Calculus, which is a formal system for the description of brain function based on assemblies of neurons. The Assembly Calculus includes operations for projecting, associating, and merging assemblies of neurons. Our research is focused on simulating different plasticity functions with Assembly Calculus. Our main contribution is the modification and evaluation of the projection operation. We experiment with Oja's and Spike Time-Dependent Plasticity (STDP) rules and test the effect of various hyper-parameters.

Auxiliary function is a helpful component to improve language model's code generation ability. However, a systematic exploration of how they affect has yet to be done. In this work, we comprehensively evaluate the ability to utilize auxiliary functions encoded in recent code-pretrained language models. First, we construct a human-crafted evaluation set, called HumanExtension, which contains examples of two functions where one function assists the other. With HumanExtension, we design several experiments to examine their ability in a multifaceted way. Our evaluation processes enable a comprehensive understanding of including auxiliary functions in the prompt in terms of effectiveness and robustness. An additional implementation style analysis captures the models' various implementation patterns when they access the auxiliary function. Through this analysis, we discover the models' promising ability to utilize auxiliary functions including their self-improving behavior by implementing the two functions step-by-step. However, our analysis also reveals the model's underutilized behavior to call the auxiliary function, suggesting the future direction to enhance their implementation by eliciting the auxiliary function call ability encoded in the models. We release our code and dataset to facilitate this research direction.

For the evolutionary Stokes problem with dynamic boundary condition we show maximal regularity of weak solutions in time. Due to the characteriation of R-sectorial operators on Hilbert spaces, the proof reduces to finding the correct functional analytic setting and proving that an operator is sectorial, i.e. generates an analytic semigroup.

Despite recent progress made by large language models in code generation, they still struggle with programs that meet complex requirements. Recent work utilizes plan-and-solve decomposition to decrease the complexity and leverage self-tests to refine the generated program. Yet, planning deep-inside requirements in advance can be challenging, and the tests need to be accurate to accomplish self-improvement. To this end, we propose FunCoder, a code generation framework incorporating the divide-and-conquer strategy with functional consensus. Specifically, FunCoder recursively branches off sub-functions as smaller goals during code generation, represented by a tree hierarchy. These sub-functions are then composited to attain more complex objectives. Additionally, we designate functions via a consensus formed by identifying similarities in program behavior, mitigating error propagation. FunCoder outperforms state-of-the-art methods by +9.8% on average in HumanEval, MBPP, xCodeEval and MATH with GPT-3.5 and GPT-4. Moreover, our method demonstrates superiority on smaller models: With FunCoder, StableCode-3b surpasses GPT-3.5 by +18.6% and achieves 97.7% of GPT-4's performance on HumanEval. Further analysis reveals that our proposed dynamic function decomposition is capable of handling complex requirements, and the functional consensus prevails over self-testing in correctness evaluation.

What are the root causes of hallucinations in large language models (LLMs)? We use Communication Complexity to prove that the Transformer layer is incapable of composing functions (e.g., identify a grandparent of a person in a genealogy) if the domains of the functions are large enough; we show through examples that this inability is already empirically present when the domains are quite small. We also point out that several mathematical tasks that are at the core of the so-called compositional tasks thought to be hard for LLMs are unlikely to be solvable by Transformers, for large enough instances and assuming that certain well accepted conjectures in the field of Computational Complexity are true.

Reverse-mode differentiation is used for optimization, but it introduces references, which break the purity of the underlying programs, making them notoriously harder to optimize. We present a reverse-mode differentiation on a purely functional language with array operations. It is the first one to deliver a provably efficient, purely functional, and denotationally correct reverse-mode differentiation. We show that our transformation is semantically correct and verifies the cheap gradient principle. Inspired by PROPs and compilation to categories, we introduce a novel intermediate representation that we call 'unary form'. Our reverse-mode transformation is factored as a compilation scheme through this intermediate representation. We obtain provably efficient gradients by performing general partial evaluation optimizations after our reverse-mode transformation, as opposed to manually derived ones. For simple first-order programs, the obtained output programs resemble static-single-assignment (SSA) code. We emphasize the modularity of our approach and show how our language can easily be enriched with more optimized primitives, as required for some speed-ups in practice.

Language models (LMs) can perform complex reasoning either end-to-end, with hidden latent state, or compositionally, with transparent intermediate state. Composition offers benefits for interpretability and safety, but may need workflow support and infrastructure to remain competitive. We describe iterated decomposition, a human-in-the-loop workflow for developing and refining compositional LM programs. We improve the performance of compositions by zooming in on failing components and refining them through decomposition, additional context, chain of thought, etc. To support this workflow, we develop ICE, an open-source tool for visualizing the execution traces of LM programs. We apply iterated decomposition to three real-world tasks and improve the accuracy of LM programs over less compositional baselines: describing the placebo used in a randomized controlled trial (25% to 65%), evaluating participant adherence to a medical intervention (53% to 70%), and answering NLP questions on the Qasper dataset (38% to 69%). These applications serve as case studies for a workflow that, if automated, could keep ML systems interpretable and safe even as they scale to increasingly complex tasks.

Large language models have demonstrated remarkable capabilities, but their performance is heavily reliant on effective prompt engineering. Automatic prompt optimization (APO) methods are designed to automate this and can be broadly categorized into those targeting instructions (instruction optimization, IO) vs. those targeting exemplars (exemplar selection, ES). Despite their shared objective, these have evolved rather independently, with IO recently receiving more research attention. This paper seeks to bridge this gap by comprehensively comparing the performance of representative IO and ES techniques, both isolation and combination, on a diverse set of challenging tasks. Our findings reveal that intelligently reusing model-generated input-output pairs obtained from evaluating prompts on the validation set as exemplars consistently improves performance over IO methods but is currently under-investigated. We also find that despite the recent focus on IO, how we select exemplars can outweigh how we optimize instructions, with ES strategies as simple as random search outperforming state-of-the-art IO methods with seed instructions without any optimization. Moreover, we observe synergy between ES and IO, with optimal combinations surpassing individual contributions. We conclude that studying exemplar selection as a standalone method and its optimal combination with instruction optimization remains a crucial aspect of APO and deserves greater consideration in future research, even in the era of highly capable instruction-following models.

Program synthesis is challenging largely because of the difficulty of search in a large space of programs. Human programmers routinely tackle the task of writing complex programs by writing sub-programs and then analyzing their intermediate results to compose them in appropriate ways. Motivated by this intuition, we present a new synthesis approach that leverages learning to guide a bottom-up search over programs. In particular, we train a model to prioritize compositions of intermediate values during search conditioned on a given set of input-output examples. This is a powerful combination because of several emergent properties. First, in bottom-up search, intermediate programs can be executed, providing semantic information to the neural network. Second, given the concrete values from those executions, we can exploit rich features based on recent work on property signatures. Finally, bottom-up search allows the system substantial flexibility in what order to generate the solution, allowing the synthesizer to build up a program from multiple smaller sub-programs. Overall, our empirical evaluation finds that the combination of learning and bottom-up search is remarkably effective, even with simple supervised learning approaches. We demonstrate the effectiveness of our technique on two datasets, one from the SyGuS competition and one of our own creation.

Training large models from scratch usually costs a substantial amount of resources. Towards this problem, recent studies such as bert2BERT and LiGO have reused small pretrained models to initialize a large model (termed the ``target model''), leading to a considerable acceleration in training. Despite the successes of these previous studies, they grew pretrained models by mapping partial weights only, ignoring potential correlations across the entire model. As we show in this paper, there are inter- and intra-interactions among the weights of both the pretrained and the target models. As a result, the partial mapping may not capture the complete information and lead to inadequate growth. In this paper, we propose a method that linearly correlates each weight of the target model to all the weights of the pretrained model to further enhance acceleration ability. We utilize multi-linear operators to reduce computational and spacial complexity, enabling acceptable resource requirements. Experiments demonstrate that our method can save 76\% computational costs on DeiT-base transferred from DeiT-small, which outperforms bert2BERT by +12.0\% and LiGO by +20.7\%, respectively.

Memory complexity and data scarcity have so far prohibited learning solution operators of partial differential equations (PDEs) at high resolutions. We address these limitations by introducing a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization, called multi-grid tensorized neural operator (MG-TFNO). MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena, through a decomposition of both the input domain and the operator's parameter space. Our contributions are threefold: i) we enable parallelization over input samples with a novel multi-grid-based domain decomposition, ii) we represent the parameters of the model in a high-order latent subspace of the Fourier domain, through a global tensor factorization, resulting in an extreme reduction in the number of parameters and improved generalization, and iii) we propose architectural improvements to the backbone FNO. Our approach can be used in any operator learning setting. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression. The tensorization combined with the domain decomposition, yields over 150x reduction in the number of parameters and 7x reduction in the domain size without losses in accuracy, while slightly enabling parallelism.

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It performs comparably or favorably to AdamW but with greater memory efficiency. As we can expect from the results of a random search program, Lion incorporates elements from several existing algorithms, including signed momentum, decoupled weight decay, Polak, and Nesterov momentum, but does not fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This lack of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Based on both continuous-time and discrete-time analysis, we demonstrate that Lion is a theoretically novel and principled approach for minimizing a general loss function f(x) while enforcing a bound constraint |x|_infty leq 1/lambda. Lion achieves this through the incorporation of decoupled weight decay, where lambda represents the weight decay coefficient. Our analysis is made possible by the development of a new Lyapunov function for the Lion updates. It applies to a broader family of Lion-kappa algorithms, where the sign(cdot) operator in Lion is replaced by the subgradient of a convex function kappa, leading to the solution of a general composite optimization problem of min_x f(x) + kappa^*(x). Our findings provide valuable insights into the dynamics of Lion and pave the way for further improvements and extensions of Lion-related algorithms.

Advanced diffusion models like RPG, Stable Diffusion 3 and FLUX have made notable strides in compositional text-to-image generation. However, these methods typically exhibit distinct strengths for compositional generation, with some excelling in handling attribute binding and others in spatial relationships. This disparity highlights the need for an approach that can leverage the complementary strengths of various models to comprehensively improve the composition capability. To this end, we introduce IterComp, a novel framework that aggregates composition-aware model preferences from multiple models and employs an iterative feedback learning approach to enhance compositional generation. Specifically, we curate a gallery of six powerful open-source diffusion models and evaluate their three key compositional metrics: attribute binding, spatial relationships, and non-spatial relationships. Based on these metrics, we develop a composition-aware model preference dataset comprising numerous image-rank pairs to train composition-aware reward models. Then, we propose an iterative feedback learning method to enhance compositionality in a closed-loop manner, enabling the progressive self-refinement of both the base diffusion model and reward models over multiple iterations. Theoretical proof demonstrates the effectiveness and extensive experiments show our significant superiority over previous SOTA methods (e.g., Omost and FLUX), particularly in multi-category object composition and complex semantic alignment. IterComp opens new research avenues in reward feedback learning for diffusion models and compositional generation. Code: https://github.com/YangLing0818/IterComp

Diagrams matter. Unfortunately, the deep learning community has no standard method for diagramming architectures. The current combination of linear algebra notation and ad-hoc diagrams fails to offer the necessary precision to understand architectures in all their detail. However, this detail is critical for faithful implementation, mathematical analysis, further innovation, and ethical assurances. I present neural circuit diagrams, a graphical language tailored to the needs of communicating deep learning architectures. Neural circuit diagrams naturally keep track of the changing arrangement of data, precisely show how operations are broadcast over axes, and display the critical parallel behavior of linear operations. A lingering issue with existing diagramming methods is the inability to simultaneously express the detail of axes and the free arrangement of data, which neural circuit diagrams solve. Their compositional structure is analogous to code, creating a close correspondence between diagrams and implementation. In this work, I introduce neural circuit diagrams for an audience of machine learning researchers. After introducing neural circuit diagrams, I cover a host of architectures to show their utility and breed familiarity. This includes the transformer architecture, convolution (and its difficult-to-explain extensions), residual networks, the U-Net, and the vision transformer. I include a Jupyter notebook that provides evidence for the close correspondence between diagrams and code. Finally, I examine backpropagation using neural circuit diagrams. I show their utility in providing mathematical insight and analyzing algorithms' time and space complexities.

Concept-based interpretability methods offer a lens into the internals of foundation models by decomposing their embeddings into high-level concepts. These concept representations are most useful when they are compositional, meaning that the individual concepts compose to explain the full sample. We show that existing unsupervised concept extraction methods find concepts which are not compositional. To automatically discover compositional concept representations, we identify two salient properties of such representations, and propose Compositional Concept Extraction (CCE) for finding concepts which obey these properties. We evaluate CCE on five different datasets over image and text data. Our evaluation shows that CCE finds more compositional concept representations than baselines and yields better accuracy on four downstream classification tasks. Code and data are available at https://github.com/adaminsky/compositional_concepts .

We present Inferflow, an efficient and highly configurable inference engine for large language models (LLMs). With Inferflow, users can serve most of the common transformer models by simply modifying some lines in corresponding configuration files, without writing a single line of source code. Compared with most existing inference engines, Inferflow has some key features. First, by implementing a modular framework of atomic build-blocks and technologies, Inferflow is compositionally generalizable to new models. Second, 3.5-bit quantization is introduced in Inferflow as a tradeoff between 3-bit and 4-bit quantization. Third, hybrid model partitioning for multi-GPU inference is introduced in Inferflow to better balance inference speed and throughput than the existing partition-by-layer and partition-by-tensor strategies.

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

Solely relying on test passing to evaluate Large Language Models (LLMs) for code synthesis may result in unfair assessment or promoting models with data leakage. As an alternative, we introduce CodeMind, a framework designed to gauge the code reasoning abilities of LLMs. CodeMind currently supports three code reasoning tasks: Independent Execution Reasoning (IER), Dependent Execution Reasoning (DER), and Specification Reasoning (SR). The first two evaluate models to predict the execution output of an arbitrary code or code the model could correctly synthesize. The third one evaluates the extent to which LLMs implement the specified expected behavior. Our extensive evaluation of nine LLMs across five benchmarks in two different programming languages using CodeMind shows that LLMs fairly follow control flow constructs and, in general, explain how inputs evolve to output, specifically for simple programs and the ones they can correctly synthesize. However, their performance drops for code with higher complexity, non-trivial logical and arithmetic operators, non-primitive types, and API calls. Furthermore, we observe that, while correlated, specification reasoning (essential for code synthesis) does not imply execution reasoning (essential for broader programming tasks such as testing and debugging): ranking LLMs based on test passing can be different compared to code reasoning.

Fine-tuning large language models (LLMs) with a collection of large and diverse instructions has improved the model's generalization to different tasks, even for unseen tasks. However, most existing instruction datasets include only single instructions, and they struggle to follow complex instructions composed of multiple subtasks (Wang et al., 2023a). In this work, we propose a novel concept of compositional instructions called chain-of-instructions (CoI), where the output of one instruction becomes an input for the next like a chain. Unlike the conventional practice of solving single instruction tasks, our proposed method encourages a model to solve each subtask step by step until the final answer is reached. CoI-tuning (i.e., fine-tuning with CoI instructions) improves the model's ability to handle instructions composed of multiple subtasks. CoI-tuned models also outperformed baseline models on multilingual summarization, demonstrating the generalizability of CoI models on unseen composite downstream tasks.

Large language models (LLMs) with enormous pre-training tokens and parameter amounts emerge abilities, including math reasoning, code generation, and instruction following. These abilities are further enhanced by supervised fine-tuning (SFT). The open-source community has studied on ad-hoc SFT for each ability, while proprietary LLMs are versatile for all abilities. It is important to investigate how to unlock them with multiple abilities via SFT. In this study, we specifically focus on the data composition between mathematical reasoning, code generation, and general human-aligning abilities during SFT. From a scaling perspective, we investigate the relationship between model abilities and various factors including data amounts, data composition ratio, model parameters, and SFT strategies. Our experiments reveal that different abilities exhibit different scaling patterns, and larger models generally show superior performance with the same amount of data. Mathematical reasoning and code generation improve as data amounts increase consistently, while the general ability is enhanced with about a thousand samples and improves slowly. We find data composition results in various abilities improvements with low data amounts, while conflicts of abilities with high data amounts. Our experiments further show that composition data amount impacts performance, while the influence of composition ratio is insignificant. Regarding the SFT strategies, we evaluate sequential learning multiple abilities are prone to catastrophic forgetting. Our proposed Dual-stage Mixed Fine-tuning (DMT) strategy learns specialized abilities first and then learns general abilities with a small amount of specialized data to prevent forgetting, offering a promising solution to learn multiple abilities with different scaling patterns.

We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization of neural networks.

Decompilers are important tools for reverse engineers that help them analyze software at a higher level of abstraction than assembly. Unfortunately, because compilation is lossy, deterministic decompilers produce code that is missing many of the details that make source code readable in the first place, like variable names and types. Neural decompilers, on the other hand, offer the ability to statistically fill in these details. Existing work in neural decompilation, however, suffers from substantial drawbacks that limits its ability to handle real code: it is unable to handle user-defined composite types, which are essential to fully specifying many functions' semantics, or require test cases. In this work, we introduce a new training process to finetune any LLM into a neural decompiler capable of generating the appropriate user-defined types alongside the decompilation. We introduce a new dataset, Realtype, that includes substantially more complicated and realistic types than existing neural decompilation benchmarks. Motivated by the intuition that different parts of data structures can be operated upon by different parts of the program, we show that interprocedural context can help improve neural decompilers' ability to handle user-defined types. We show that our training process yields state-of-the-art results in neural decompilation. We also publicly release the Idioms series of finetuned neural decompilation models in support of open science. In summary, we identify the need for joint code and type prediction, show that it is a hard problem, and take the first steps towards solving it.

Large language models can solve tasks that were not present in the training set. This capability is believed to be due to in-context learning and skill composition. In this work, we study the emergence of in-context learning and skill composition in a collection of modular arithmetic tasks. Specifically, we consider a finite collection of linear modular functions z = a , x + b , y ;mod; p labeled by the vector (a, b) in Z_p^2. We use some of these tasks for pre-training and the rest for out-of-distribution testing. We empirically show that a GPT-style transformer exhibits a transition from in-distribution to out-of-distribution generalization as the number of pre-training tasks increases. We find that the smallest model capable of out-of-distribution generalization requires two transformer blocks, while for deeper models, the out-of-distribution generalization phase is transient, necessitating early stopping. Finally, we perform an interpretability study of the pre-trained models, revealing the highly structured representations in both phases; and discuss the learnt algorithm.

We propose Super-resolution Neural Operator (SRNO), a deep operator learning framework that can resolve high-resolution (HR) images at arbitrary scales from the low-resolution (LR) counterparts. Treating the LR-HR image pairs as continuous functions approximated with different grid sizes, SRNO learns the mapping between the corresponding function spaces. From the perspective of approximation theory, SRNO first embeds the LR input into a higher-dimensional latent representation space, trying to capture sufficient basis functions, and then iteratively approximates the implicit image function with a kernel integral mechanism, followed by a final dimensionality reduction step to generate the RGB representation at the target coordinates. The key characteristics distinguishing SRNO from prior continuous SR works are: 1) the kernel integral in each layer is efficiently implemented via the Galerkin-type attention, which possesses non-local properties in the spatial domain and therefore benefits the grid-free continuum; and 2) the multilayer attention architecture allows for the dynamic latent basis update, which is crucial for SR problems to "hallucinate" high-frequency information from the LR image. Experiments show that SRNO outperforms existing continuous SR methods in terms of both accuracy and running time. Our code is at https://github.com/2y7c3/Super-Resolution-Neural-Operator