Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal Algorithms for Stochastic Multi-Level Compositional Optimization   In this paper, we investigate the problem of stochastic multi-level
compositional optimization, where the objective function is a composition of
multiple smooth but possibly non-convex functions. Existing methods for solving
this problem either suffer from sub-optimal sample complexities or need a huge
batch size. To address this limitation, we propose a Stochastic Multi-level
Variance Reduction method (SMVR), which achieves the optimal sample complexity
of $\mathcal{O}\left(1 / \epsilon^{3}\right)$ to find an $\epsilon$-stationary
point for non-convex objectives. Furthermore, when the objective function
satisfies the convexity or Polyak-{\L}ojasiewicz (PL) condition, we propose a
stage-wise variant of SMVR and improve the sample complexity to
$\mathcal{O}\left(1 / \epsilon^{2}\right)$ for convex functions or
$\mathcal{O}\left(1 /(\mu\epsilon)\right)$ for non-convex functions satisfying
the $\mu$-PL condition. The latter result implies the same complexity for
$\mu$-strongly convex functions. To make use of adaptive learning rates, we
also develop Adaptive SMVR, which achieves the same optimal complexities but
converges faster in practice. All our complexities match the lower bounds not
only in terms of $\epsilon$ but also in terms of $\mu$ (for PL or strongly
convex functions), without using a large batch size in each iteration.

33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom