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Stochastic Approximation Proximal Subgradient Method for Stochastic Convex-Concave Minimax Optimization

Yu-Hong Dai, Jiani Wang, Liwei Zhang

TL;DR

A linearized stochastic approximation augmented Lagrange (LSAAL) method is proposed and it is proved that this algorithm exhibits the expected convergence rate for the minimax optimality measure and the minimax optimality measure bound with high probability as well.

Abstract

This paper presents a stochastic approximation proximal subgradient (SAPS) method for stochastic convex-concave minimax optimization. By accessing unbiased and variance bounded approximate subgradients, we show that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate of the minimax optimality measure if the parameters in the algorithm are properly chosen, where $N$ denotes the number of iterations. Moreover, we show that the algorithm has ${\rm O}(\log(N)N^{-1/2})$ minimax optimality measure bound with high probability. Further we study a specific stochastic convex-concave minimax optimization problems arising from stochastic convex conic optimization problems, which the the bounded subgradient condition is fail. To overcome the lack of the bounded subgradient conditions in convex-concave minimax problems, we propose a linearized stochastic approximation augmented Lagrange (LSAAL) method and prove that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate for the minimax optimality measure and ${\rm O}(\log^2(N)N^{-1/2})$ minimax optimality measure bound with high probability as well. Preliminary numerical results demonstrate the effect of the SAPS and LSAAL methods.

Stochastic Approximation Proximal Subgradient Method for Stochastic Convex-Concave Minimax Optimization

TL;DR

A linearized stochastic approximation augmented Lagrange (LSAAL) method is proposed and it is proved that this algorithm exhibits the expected convergence rate for the minimax optimality measure and the minimax optimality measure bound with high probability as well.

Abstract

This paper presents a stochastic approximation proximal subgradient (SAPS) method for stochastic convex-concave minimax optimization. By accessing unbiased and variance bounded approximate subgradients, we show that this algorithm exhibits expected convergence rate of the minimax optimality measure if the parameters in the algorithm are properly chosen, where denotes the number of iterations. Moreover, we show that the algorithm has minimax optimality measure bound with high probability. Further we study a specific stochastic convex-concave minimax optimization problems arising from stochastic convex conic optimization problems, which the the bounded subgradient condition is fail. To overcome the lack of the bounded subgradient conditions in convex-concave minimax problems, we propose a linearized stochastic approximation augmented Lagrange (LSAAL) method and prove that this algorithm exhibits expected convergence rate for the minimax optimality measure and minimax optimality measure bound with high probability as well. Preliminary numerical results demonstrate the effect of the SAPS and LSAAL methods.
Paper Structure (14 sections, 18 theorems, 130 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 18 theorems, 130 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Special Cases
  3. Motivation and Contributions
  4. Related Work
  5. Organization and notations
  6. Stochastic Approximation Proximal Subgradient Method
  7. Expected convergence rate analysis
  8. High probability performance analysis
  9. Linearized Stochastic Approximation Augmented Lagrange Method
  10. Preliminary Numerical Experiments
  11. Stochastic strongly convex-concave minimax problems
  12. General stochastic convex-concave minimax problems
  13. Multi-class Neyman-Pearson classification
  14. Some dicussion

Key Result

Proposition 2.1

Let $\{z^k\}$ be generated by Algorithm ASPGalg, for integer $k>1$, and $\widetilde{z}^N$ be computed by formula (s2.16). Assume that conditions (A1) and (A2) are satisfied. Denote $\Delta_k=G(z^k,\xi_k)-g(z^k)$. Then for integer $N>0$, for any $z=(x,y)\in \mathbb Z$, the following inequality holds for any $v(z^k)=(v_x(x^k),v_y(y^k))$ with $v_x(x^k) \in \partial \vartheta (x^k)$ and $v_y(y^k) \

Figures (3)

  • Figure 1: The trend of the error for \ref{['SSccm1']} with respect to iteration.
  • Figure 2: The trend of the relative error for solving (\ref{['SSccm2']}) with respect to iteration
  • Figure 3: The trend of the relative error for solving (\ref{['MCNPC1']}) with respect to CPU time

Theorems & Definitions (23)

  • Proposition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.2
  • Corollary 2.1
  • Remark 2.5
  • Lemma 3.1
  • ...and 13 more