DAMA: A Unified Accelerated Approach for Decentralized Nonconvex Minimax Optimization-Part I: Algorithm Development and Results

In this work and its accompanying Part II [1], we develop an accelerated algorithmic framework, DAMA (Decentralized Accelerated Minimax Approach), for nonconvex Polyak-Lojasiewicz minimax optimization over decentralized multi-agent networks. Our approach integrates online and offline stochastic minimax algorithms with various decentralized learning strategies, yielding a versatile framework with broader flexibility than existing methods. Our unification is threefold: (i) we propose a unified decentralized learning strategy for minimax optimization that subsumes existing bias-correction techniques, such as gradient tracking, while introducing new variants that achieve tighter network-dependent bounds; (ii) we introduce a probabilistic gradient estimator, GRACE (Gradient Acceleration Estimator), which unifies momentum-based methods and loopless variance-reduction techniques for constructing accelerated gradients within DAMA, and is broadly applicable to general stochastic optimization problems; and (iii) we develop a unified analytical framework that establishes a general performance bound for DAMA, achieving state-of-the-art results with the best-known sample complexity. To the best of our knowledge, DAMA is the first framework to achieve a multi-level unification of decentralized learning strategies and accelerated gradient techniques. This work focuses on algorithm development and the main results, while Part II provides the theoretical analysis that substantiates these results and presents empirical validation across diverse network topologies using synthetic and real-world datasets.

💡 Research Summary

This paper introduces DAMA (Decentralized Accelerated Minimax Approach), a unified algorithmic framework for solving non‑convex Polyak‑Łojasiewicz (PL) minimax problems over decentralized multi‑agent networks. The authors consider a stochastic minimax objective J(x, y)= (1/K)∑_{k=1}^K J_k(x, y), where each local function J_k can be either an online expectation or an offline finite‑sum. J is smooth and non‑convex in the minimization variable x, while it satisfies the PL condition in the maximization variable y. Traditional gradient‑descent‑ascent (GDA) with a single stepsize fails because the asymmetric PL structure creates unbalanced dynamics; prior work therefore relies on two‑time‑scale stepsizes and variance‑reduction or momentum techniques to control gradient noise.

DAMA unifies three previously disparate components. First, it proposes a generalized decentralized learning strategy that subsumes gradient tracking (GT) as a special case and also incorporates Exact Diffusion (ED) and EXTRA, both of which have shown tighter network‑dependent convergence bounds in pure minimization settings. By casting these bias‑correction mechanisms into a single template, DAMA can adapt to sparse communication graphs and heterogeneous data distributions without sacrificing convergence speed.

Second, the paper introduces GRACE (Gradient Acceleration Estimator), a probabilistic gradient estimator that blends momentum‑based acceleration (e.g., STORM) with loopless variance‑reduction schemes such as PAGE and Loopless SARAH. GRACE operates by randomly deciding, at each iteration, whether to refresh a full‑batch gradient or to update a momentum‑type surrogate using a single stochastic sample. This probabilistic design simultaneously reduces the correlation between the x‑ and y‑gradients (a known source of instability in minimax problems) and eliminates the need for large batch sizes or double‑loop structures.

Third, the authors develop a transformed‑recursion analytical framework. Instead of analyzing each algorithmic variant separately, they rewrite the original stochastic recursion in a transformed domain where the influence of the network spectral gap (1‑λ) appears only in higher‑order terms. This yields a generic performance bound that applies uniformly to all combinations of decentralized strategies (GT, ED, EXTRA) and gradient estimators (STORM, PAGE, Loopless SARAH). The dominant term of the sample complexity becomes O(κ³ ε⁻³ K⁻¹), independent of the network topology, while the residual dependence on (1‑λ) is limited to lower‑order additive terms.

Key theoretical results include:

• STORM + GT achieves a per‑agent sample complexity O(κ³ ε⁻³ K⁻¹ + κ² ε⁻² (1‑λ)⁻³), where the leading term no longer depends on the spectral gap.
• STORM + ED improves the gap dependence to O((1‑λ)⁻²).
• Loopless SARAH + ED attains O(κ² √N ε⁻² K⁻¹), improving upon the prior state‑of‑the‑art DREAM method by a factor of √K when the dataset size N is large.
• PAGE + ED in the offline finite‑sum setting yields O(κ² √N ε⁻² (1‑λ)⁻¹ √K), outperforming existing methods under sparse connectivity.

The paper also derives transient‑time bounds, quantifying how many iterations are needed before the linear speedup (convergence rate scaling with K) manifests for the STORM‑based variants.

Empirical validation is performed on synthetic data and real‑world datasets (including image and text benchmarks) across various network topologies (fully connected, ring, random graphs). The experiments confirm that DAMA variants consistently converge faster and achieve lower final objective values than competing decentralized minimax algorithms, especially when the communication graph is sparse or the local data distributions are highly heterogeneous. Moreover, the probabilistic parameter of GRACE automatically adapts to the noise level, demonstrating robustness without manual tuning.

In summary, DAMA delivers the first multi‑level unification of decentralized bias‑correction strategies and accelerated gradient estimators for non‑convex PL minimax optimization. It provides state‑of‑the‑art sample complexity, reduces dependence on network spectral properties, and offers practical, single‑loop algorithms suitable for both online streaming and offline finite‑sum scenarios. Part II of the work supplies the detailed proofs of the transformed‑recursion analysis and further experimental studies.