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Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization

Lesi Chen, Luo Luo

TL;DR

This work addresses finding near-stationary points in stochastic minimax optimization by introducing Recursive Anchored Iteration (RAIN), a framework that progressively anchors subproblems to reduce gradient norms efficiently. The core idea combines anchored regularization with a stochastic extragradient subroutine (Epoch-SEG), yielding near-optimal SFO complexity in convex-concave and strongly-convex-strongly-concave settings. The authors extend the approach to nonconvex-nonconcave cases via saddle envelopes and introduce RAIN$^{++}$ with MLMC-based debiasing to achieve similar near-optimal guarantees under comonotone and intersection-dominant conditions. Theoretical results are complemented by numerical experiments showing superior performance over established baselines across CC and NC regimes, underscoring the practical impact of near-optimal stochastic minimax optimization. Overall, the paper advances both methodological and complexity-theoretic understanding of stochastic minimax optimization and provides practical algorithms with strong guarantees.

Abstract

We study the problem of finding a near-stationary point for smooth minimax optimization. The recently proposed extra anchored gradient (EAG) methods achieve the optimal convergence rate for the convex-concave minimax problem in the deterministic setting. However, the direct extension of EAG to stochastic optimization is not efficient. In this paper, we design a novel stochastic algorithm called Recursive Anchored IteratioN (RAIN). We show that the RAIN achieves near-optimal stochastic first-order oracle (SFO) complexity for stochastic minimax optimization in both convex-concave and strongly-convex-strongly-concave cases. In addition, we extend the idea of RAIN to solve structured nonconvex-nonconcave minimax problem and it also achieves near-optimal SFO complexity.

Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization

TL;DR

This work addresses finding near-stationary points in stochastic minimax optimization by introducing Recursive Anchored Iteration (RAIN), a framework that progressively anchors subproblems to reduce gradient norms efficiently. The core idea combines anchored regularization with a stochastic extragradient subroutine (Epoch-SEG), yielding near-optimal SFO complexity in convex-concave and strongly-convex-strongly-concave settings. The authors extend the approach to nonconvex-nonconcave cases via saddle envelopes and introduce RAIN with MLMC-based debiasing to achieve similar near-optimal guarantees under comonotone and intersection-dominant conditions. Theoretical results are complemented by numerical experiments showing superior performance over established baselines across CC and NC regimes, underscoring the practical impact of near-optimal stochastic minimax optimization. Overall, the paper advances both methodological and complexity-theoretic understanding of stochastic minimax optimization and provides practical algorithms with strong guarantees.

Abstract

We study the problem of finding a near-stationary point for smooth minimax optimization. The recently proposed extra anchored gradient (EAG) methods achieve the optimal convergence rate for the convex-concave minimax problem in the deterministic setting. However, the direct extension of EAG to stochastic optimization is not efficient. In this paper, we design a novel stochastic algorithm called Recursive Anchored IteratioN (RAIN). We show that the RAIN achieves near-optimal stochastic first-order oracle (SFO) complexity for stochastic minimax optimization in both convex-concave and strongly-convex-strongly-concave cases. In addition, we extend the idea of RAIN to solve structured nonconvex-nonconcave minimax problem and it also achieves near-optimal SFO complexity.
Paper Structure (41 sections, 23 theorems, 132 equations, 4 figures, 2 tables)

This paper contains 41 sections, 23 theorems, 132 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. The Recursive Anchored Iteration
  4. Connection to Related Work
  5. Complexity Analysis for RAIN
  6. Extension to Nonconvex-Nonconcave Settings
  7. The Lower Complexity Bounds
  8. Numerical Experiments
  9. The Convex-Concave Case
  10. The Nonconvex-Nonconcave Case
  11. Conclusion
  12. The Proofs in Section \ref{['sec:RAIN']}
  13. The Proof of Lemma \ref{['lem:RAL']}
  14. The Proof of Lemma \ref{['lem:anchoring']}
  15. The Proofs in Section \ref{['sec:convex-concave']}
  16. ...and 26 more sections

Key Result

Lemma 2.1

Under Assumption asm:smooth and asm:cc, it holds that for all $z=(x,y)$ and $z'=(x',y')$.

Figures (4)

  • Figure 1: The results of the number of SFO calls against gradient norm on problem (\ref{['func-xy']}).
  • Figure 2: The results of the number of SFO calls against gradient norm on problem (\ref{['func-delta']}). SEAG diverges in (c), which does not contradict its convergence guarantee as the condition $\sigma_k^2 \le \epsilon / (k+1)$ in Theorem 6.1 lee2021fast is unsatisfied.
  • Figure 3: The results of the number of SFO calls against gradient norm on problem (\ref{['func-rho']}) with $L=1$ and $\rho = - {1}/{(8 \sqrt{2})}$.
  • Figure 4: The results of the number of SFO calls against gradient norm on problem (\ref{['func-rho']}) with $L=1$ and $\rho = - {1}/3$.

Theorems & Definitions (31)

  • Lemma 2.1: monotonicity
  • Lemma 2.2: strong monotonicity
  • Definition 2.1: nearly-stationary point
  • Lemma 3.1: recursively anchoring lemma
  • Remark 3.1
  • Lemma 3.2: anchoring lemma
  • Lemma 4.1: SEG
  • Lemma 4.2: Epoch-SEG
  • Theorem 4.1: RAIN, SCSC
  • Theorem 4.2: RAIN, CC
  • ...and 21 more