text: This paper presents OptNet, a network architecture that integrates optimization problems (here, specifically in the form of quadratic programs) as individual layers in larger end-to-end trainable deep networks. These layers encode constraints and complex dependencies between the hidden states that traditional convolutional and fully-connected layers often cannot capture. In this paper, we explore the foundations for such an architecture: we show how techniques from sensitivity analysis, bilevel optimization, and implicit differentiation can be used to exactly differentiate through these layers and with respect to layer parameters; we develop a highly efficient solver for these layers that exploits fast GPU-based batch solves within a primal-dual interior point method, and which provides backpropagation gradients with virtually no additional cost on top of the solve; and we highlight the application of these approaches in several problems. In one notable example, we show that the method is capable of learning to play mini-Sudoku (4x4) given just input and output games, with no a priori information about the rules of the game; this highlights the ability of our architecture to learn hard constraints better than other neural architectures.
  @cite_18: 
  @cite_14: This paper presents the input convex neural network architecture. These are scalar-valued (potentially deep) neural networks with constraints on the network parameters such that the output of the network is a convex function of (some of) the inputs. The networks allow for efficient inference via optimization over some inputs to the network given others, and can be applied to settings including structured prediction, data imputation, reinforcement learning, and others. In this paper we lay the basic groundwork for these models, proposing methods for inference, optimization and learning, and analyze their representational power. We show that many existing neural network architectures can be made input-convex with only minor modification, and develop specialized optimization algorithms tailored to this setting. Finally, we highlight the performance of the methods on multi-label prediction, image completion, and reinforcement learning problems, where we show improvement over the existing state of the art in many cases.
  @cite_8: We introduce structured prediction energy networks (SPENs), a flexible framework for structured prediction. A deep architecture is used to define an energy function of candidate labels, and then predictions are produced by using back-propagation to iteratively optimize the energy with respect to the labels. This deep architecture captures dependencies between labels that would lead to intractable graphical models, and performs structure learning by automatically learning discriminative features of the structured output. One natural application of our technique is multi-label classification, which traditionally has required strict prior assumptions about the interactions between labels to ensure tractable learning and prediction. We are able to apply SPENs to multi-label problems with substantially larger label sets than previous applications of structured prediction, while modeling high-order interactions using minimal structural assumptions. Overall, deep learning provides remarkable tools for learning features of the inputs to a prediction problem, and this work extends these techniques to learning features of structured outputs. Our experiments provide impressive performance on a variety of benchmark multi-label classification tasks, demonstrate that our technique can be used to provide interpretable structure learning, and illuminate fundamental trade-offs between feed-forward and iterative structured prediction.
  @cite_6: Imputing missing values in high dimensional time-series is a difficult problem. This paper presents a strategy for training energy-based graphical models for imputation directly, bypassing difficulties probabilistic approaches would face. The training strategy is inspired by recent work on optimization-based learning (Domke, 2012) and allows complex neural models with convolutional and recurrent structures to be trained for imputation tasks. In this work, we use this training strategy to derive learning rules for three substantially different neural architectures. Inference in these models is done by either truncated gradient descent or variational mean-field iterations. In our experiments, we found that the training methods outperform the Contrastive Divergence learning algorithm. Moreover, the training methods can easily handle missing values in the training data itself during learning. We demonstrate the performance of this learning scheme and the three models we introduce on one artificial and two real-world data sets.
  @cite_19: 
  @cite_23: We introduce the multi-prediction deep Boltzmann machine (MP-DBM). The MP-DBM can be seen as a single probabilistic model trained to maximize a variational approximation to the generalized pseudolikelihood, or as a family of recurrent nets that share parameters and approximately solve different inference problems. Prior methods of training DBMs either do not perform well on classification tasks or require an initial learning pass that trains the DBM greedily, one layer at a time. The MP-DBM does not require greedy layerwise pretraining, and outperforms the standard DBM at classification, classification with missing inputs, and mean field prediction tasks.1
  @cite_13: Structured Prediction Energy Networks (SPENs) are a simple, yet expressive family of structured prediction models (Belanger and McCallum, 2016). An energy function over candidate structured outputs is given by a deep network, and predictions are formed by gradient-based optimization. This paper presents end-to-end learning for SPENs, where the energy function is discriminatively trained by back-propagating through gradient-based prediction. In our experience, the approach is substantially more accurate than the structured SVM method of Belanger and McCallum (2016), as it allows us to use more sophisticated non-convex energies. We provide a collection of techniques for improving the speed, accuracy, and memory requirements of end-to-end SPENs, and demonstrate the power of our method on 7-Scenes image denoising and CoNLL-2005 semantic role labeling tasks. In both, inexact minimization of non-convex SPEN energies is superior to baseline methods that use simplistic energy functions that can be minimized exactly.
summary: These methods can be used for tasks like (structured) prediction where the training method shapes the energy function to be low around the observed data manifold and high elsewhere @cite_6 . In recent years, there has been a strong push to further incorporate structured prediction methods like conditional random fields as the last layer'' of a deep network architecture @cite_19 @cite_18 @cite_23 as well as in deeper energy-based architectures @cite_8 @cite_13 @cite_14 . Learning in this context requires observed data, which isn't present in some of the contexts we consider in this paper, and also may suffer from instability issues when combined with deep energy-based architectures as observed in .