DAMA: A Unified Accelerated Approach for Decentralized Nonconvex Minimax Optimization-Part II: Convergence and Performance Analyses

Abstract: In Part I of this work [1], we developed an accelerated algorithmic framework, DAMA (Decentralized Accelerated Minimax Approach), for nonconvex Polyak-Lojasiewicz (PL) minimax optimization over decentralized multi-agent networks. To further enhance convergence in online and offline scenarios, Part I of this work [1] also proposed a novel accelerated gradient estimator, namely, GRACE (GRadient ACceleration Estimator), which unifies several momentum-based methods (e.g., STORM) and loopless variance-reduction techniques (e.g., PAGE, Loopless SARAH), thereby enabling accelerated gradient updates within DAMA. Part I reported a unified performance bound for DAMA and refined guarantees for specific algorithmic instances, demonstrating the superior performance of several new variants on sparsely connected networks. In this Part II, we focus on the convergence and performance bounds that substantiate the main results presented in Part I [1]. In particular, we establish a unified performance bound for DAMA using the transformed recursion derived in Part I and subsequently refine this bound for its various special cases.

• The paper presents DAMA, a unified accelerated approach that integrates momentum-based variance reduction with gradient tracking for decentralized nonconvex minimax optimization.
• It derives explicit convergence bounds that quantify consensus and gradient-tracking errors while linking iteration complexity to network topology, batch sizes, and stochastic noise.
• The analysis extends to popular estimators like STORM, PAGE, and SARAH, providing practical guidelines for achieving communication efficiency and scalability in distributed settings.

Convergence and Performance Analysis of DAMA for Decentralized Nonconvex Minimax Optimization

Overview of the DAMA Framework

The DAMA (Decentralized Accelerated Minimax Approach) framework addresses decentralized nonconvex minimax problems over networks, where the objective is to find an $\epsilon$-stationary point for

$$\min_{x \in \mathbb{R}^{d_1}} \max_{y \in \mathbb{R}^{d_2}} J(x,y) = \frac{1}{K}\sum_{k=1}^K J_k(x,y)$$

with stochastic or finite-sum objectives $J_k$ handled by agents in a peer-to-peer network. The framework imposes $L_f$-smoothness and a Polyak-Łojasiewicz (PL) condition for the $y$-variable, covering nonconvexity in $x$ and possible nonconcavity in $y$, extending applicability to nonconvex-PL problems.

DAMA's key innovation is the integration of an accelerated gradient estimator, GRACE (Gradient ACceleration Estimator), which encapsulates and unifies momentum-based variance reduction (e.g., STORM) and loopless stochastic estimators (e.g., PAGE, SARAH). DAMA subsumes existing decentralized minimax methods and gradient tracking mechanisms, incorporating momentum and bias-correction strategies (e.g., exact diffusion, EXTRA, GT variants) in a unified primal-dual recursion.

Theoretical Contributions and Main Results

The central contribution of DAMA Part II is a rigorous convergence analysis of the generic and instantiated forms of DAMA. The analysis proceeds via transformed recursions at the network centroid and auxiliary error coordinates, quantifying consensus and gradient-tracking errors. The technical approach opens a direct path to bounding the following:

• Consensus error (deviation from the centroid) is tightly upper bounded by the coupled error norm on the transformed error system, with constants determined by the spectral properties of the network mixing matrix.
• The deviation between estimated and true stochastic gradients (both local and centroidal) is bounded in expectation, with rate expressions coupling batch size, estimator parameters, network algebra, and noise variance.
• The iteration complexity to reach an $\epsilon$-stationary solution in (\texttt{primal}, \texttt{dual}) is quantified with explicit constants for initial error, variance, and step-size, yielding a unified upper bound.

The main theorem provides an upper bound

$$\frac{1}{T}\sum_{i=0}^{T-1} \mathbb{E}\|\nabla_x J(x_{c,i}, y_{c,i})\|^2 + \mathbb{E}\|\nabla_y J(x_{c,i},y_{c,i})\|^2 \le \mathcal{O}\left( \frac{E G_{p,0}}{T\mu_x} + \frac{\kappa^2 E\Delta_{c,0}}{T\mu_y} + \cdots \right)$$

where $E G_{p,0}$ and $E\Delta_{c,0}$ encapsulate the initial primal/dual suboptimality, with explicit scaling dependence on $T$, batch sizes, mixing properties, and instability sources. Separate terms are isolated for deterministic and stochastic error contributions and initial network/model discrepancy.

The bound systematically specializes to popular cases: variance-reduced and loopless estimators (STORM, PAGE, SARAH), different algorithmic architectures (ED, EXTRA, GT variants), and both online and offline regimes. The technical treatment includes explicit dependence (and matching lower bounds) on network topology (via spectral gap $1-\lambda$ of $W$), batch/sample size, and noise.

Technical Insights

Several layers of technical depth underpin the convergence and performance characterizations:

• Recursion Transformation: The iterative process is analyzed under centroid (consensus) and disagreement (auxiliary) coordinates, leading to coupled linear systems whose spectral norms directly impact convergence rates.
• Variance Reduction: The GRACE estimator, when instantiated as (e.g.) STORM or PAGE, yields complexity upper bounds that scale as $\mathcal{O}((TK)^{-2/3})$ or better with respect to $\epsilon$, $K$, and the spectral gap.
• Communication-Sample Tradeoff: Sample and communication complexity analyses are detailed for both offline and online learning, highlighting that network sparsity (small spectral gap) increases transient time, but variance reduction can dramatically reduce sample complexity relative to non-accelerated alternatives.
• Parameter Specification and Layered Minimax Problems: Lemmas provide guidelines on step-size, smoothing/momentum factor, and batch size choices (with interlinked constraints ensuring stability across estimator switchings and network coupling). The results subsume decentralized minimax methods with and without momentum and stochastic variance reduction.

Implications and Future Directions

Practically, DAMA enables communication-efficient, scalable stochastic optimization for distributed multi-agent settings in adversarial, robust, or distributionally-robust applications—including GANs, robust RL, and federated adversarial training. The theory provides guidelines for selecting batch, step-size, and estimator tuning in heterogeneous or time-varying topologies. Modular instantiation further enables adoption in large-scale settings and across alternative network architectures.

Theoretically, Part II demonstrates that gradient-tracking, momentum, and variance reduction can be unified under a common recursion with matching finite-sample and communication guarantees, and that the PL condition for the dual variable is sufficient for convergence matching state-of-the-art rates in both convex and nonconvex settings. The framework is robust to network connectivity, initial model discrepancy, and stochasticity under mild assumptions.

Future research can extend DAMA to handle time-varying directed graphs, adapt to non-stationary data, or integrate adaptive or federated regularization. Further exploration into breaking the $\mathcal{O}((TK)^{-2/3})$ barrier for nonconvex-PL minimax under weaker conditions, or in asynchronous communication, remains compelling.

Conclusion

This work provides a comprehensive convergence and complexity characterization of the DAMA framework for decentralized nonconvex minimax optimization, showing that a unified accelerated estimator approach can systematize the design space of decentralized minimax learning. The abstract performance bounds and their instantiations accurately quantify the network, variance, and stochasticity bottlenecks, and point toward principled distributed learning protocols and networked game-theoretic learning (2512.13923).