Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Diffusion Stochastic Optimization for Min-Max Problems

Haoyuan Cai, Sulaiman A. Alghunaim, Ali H. Sayed

TL;DR

This work proposes Diffusion Stochastic Same-Sample Optimistic Gradient (DSS-OG) to tackle minimax optimization with nonconvex-PL objectives in distributed settings, addressing the large-batch bottleneck of traditional stochastic OG methods. It develops a centralized SS-OG variant and a distributed diffusion implementation over left-stochastic networks, establishing convergence with a primal rate of $\mathcal{O}(1/\sqrt{T})$ and a dual rate of $\mathcal{O}(1/T)$; the overall complexity to obtain an $\varepsilon$-stationary point scales as $T_0+T_1=\mathcal{O}(\varepsilon^{-4})+\mathcal{O}(\varepsilon^{-2})$. The analysis relies on relaxed Lipschitz and PL assumptions, without requiring gradient-smoothness of the stochastic losses, and demonstrates that gradient queries can be executed in parallel with modest memory overhead. Empirical validation on Wasserstein GAN and DCGAN tasks shows that DSS-OG and its centralized variant outperform several baselines in gradient norms, MSE, and FID scores, confirming practical effectiveness for distributed minimax learning. Overall, the paper advances distributed minimax optimization by delivering a batch-flexible, diffusion-based algorithm with provable convergence and real-world GAN demonstrations.

Abstract

The optimistic gradient method is useful in addressing minimax optimization problems. Motivated by the observation that the conventional stochastic version suffers from the need for a large batch size on the order of $\mathcal{O}(\varepsilon^{-2})$ to achieve an $\varepsilon$-stationary solution, we introduce and analyze a new formulation termed Diffusion Stochastic Same-Sample Optimistic Gradient (DSS-OG). We prove its convergence and resolve the large batch issue by establishing a tighter upper bound, under the more general setting of nonconvex Polyak-Lojasiewicz (PL) risk functions. We also extend the applicability of the proposed method to the distributed scenario, where agents communicate with their neighbors via a left-stochastic protocol. To implement DSS-OG, we can query the stochastic gradient oracles in parallel with some extra memory overhead, resulting in a complexity comparable to its conventional counterpart. To demonstrate the efficacy of the proposed algorithm, we conduct tests by training generative adversarial networks.

Diffusion Stochastic Optimization for Min-Max Problems

TL;DR

This work proposes Diffusion Stochastic Same-Sample Optimistic Gradient (DSS-OG) to tackle minimax optimization with nonconvex-PL objectives in distributed settings, addressing the large-batch bottleneck of traditional stochastic OG methods. It develops a centralized SS-OG variant and a distributed diffusion implementation over left-stochastic networks, establishing convergence with a primal rate of and a dual rate of ; the overall complexity to obtain an -stationary point scales as . The analysis relies on relaxed Lipschitz and PL assumptions, without requiring gradient-smoothness of the stochastic losses, and demonstrates that gradient queries can be executed in parallel with modest memory overhead. Empirical validation on Wasserstein GAN and DCGAN tasks shows that DSS-OG and its centralized variant outperform several baselines in gradient norms, MSE, and FID scores, confirming practical effectiveness for distributed minimax learning. Overall, the paper advances distributed minimax optimization by delivering a batch-flexible, diffusion-based algorithm with provable convergence and real-world GAN demonstrations.

Abstract

The optimistic gradient method is useful in addressing minimax optimization problems. Motivated by the observation that the conventional stochastic version suffers from the need for a large batch size on the order of to achieve an -stationary solution, we introduce and analyze a new formulation termed Diffusion Stochastic Same-Sample Optimistic Gradient (DSS-OG). We prove its convergence and resolve the large batch issue by establishing a tighter upper bound, under the more general setting of nonconvex Polyak-Lojasiewicz (PL) risk functions. We also extend the applicability of the proposed method to the distributed scenario, where agents communicate with their neighbors via a left-stochastic protocol. To implement DSS-OG, we can query the stochastic gradient oracles in parallel with some extra memory overhead, resulting in a complexity comparable to its conventional counterpart. To demonstrate the efficacy of the proposed algorithm, we conduct tests by training generative adversarial networks.
Paper Structure (25 sections, 11 theorems, 103 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 25 sections, 11 theorems, 103 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Main Contributions
  4. Algorithm Description
  5. Centralized Scenario
  6. Distributed scenario
  7. Convergence Analysis
  8. Assumptions
  9. Main Results
  10. Comparison between S-OG and SS-OG
  11. Computer simulation
  12. Wasserstein GAN
  13. Deep Convolutional GANs
  14. Conclusion
  15. Basic Lemmas
  16. ...and 10 more sections

Key Result

Theorem 1

Under Assumptions muPL--LeftStochastic, choosing step sizes the non-asymptotic convergence rate of the primal objective is given by: where $L =L_f + \frac{L^2_f}{\nu}$ is the Lipschitz constant associated with $P(x)$ and are constants. Furthermore, DSS-OG outputs a primal $\varepsilon$-stationary point after $T_0 = \mathcal{O}(\varepsilon^{-4})$ iterations and gradient evaluation complexity, i.

Figures (2)

  • Figure 1: Evoluation of gradient norm and mean square error distance between the true and estimated ones: In (a) and (b), the true model is given by $\pi_k =0, \sigma^2_k = 0.001$ for all $k$; In (c) and (d), the true model is given by $\pi_k =0, \sigma^2_k = 0.1$ for all $k$.
  • Figure 2: Simulation results for DCGANs. In (a), the FID score versus different hyperparameters is demonstrated. In (b), we show a sample image generated by a single node of Adam-DSS-OG ($\mu = 0.001, \beta_1 = 0.2, \beta_2 = 0.998$).

Theorems & Definitions (12)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1: Quadratic Growth karimi2016linear
  • Lemma 2: Danskin-type Lemma nouiehed2019solving
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 2 more