Shuffling the Data, Stretching the Step-size: Sharper Bias in constant step-size SGD

From adversarial robustness to multi-agent learning, many machine learning tasks can be cast as finite-sum min-max optimization or, more generally, as variational inequality problems (VIPs). Owing to their simplicity and scalability, stochastic gradient methods with constant step size are widely used, despite the fact that they converge only up to a constant term. Among the many heuristics adopted in practice, two classical techniques have recently attracted attention to mitigate this issue: \emph{Random Reshuffling} of data and \emph{Richardson--Romberg extrapolation} across iterates. Random Reshuffling sharpens the mean-squared error (MSE) of the estimated solution, while Richardson-Romberg extrapolation acts orthogonally, providing a second-order reduction in its bias. In this work, we show that their composition is strictly better than both, not only maintaining the enhanced MSE guarantees but also yielding an even greater cubic refinement in the bias. To the best of our knowledge, our work provides the first theoretical guarantees for such a synergy in structured non-monotone VIPs. Our analysis proceeds in two steps: (i) we smooth the discrete noise induced by reshuffling and leverage tools from continuous-state Markov chain theory to establish a novel law of large numbers and a central limit theorem for its iterates; and (ii) we employ spectral tensor techniques to prove that extrapolation debiases and sharpens the asymptotic behavior even under the biased gradient oracle induced by reshuffling. Finally, extensive experiments validate our theory, consistently demonstrating substantial speedups in practice.

On the Exploitability of FTRL Dynamics

In this paper we investigate the exploitability of a Follow-the-Regularized-Leader (FTRL) learner with constant step size $η$ in $n\times m$ two-player zero-sum games played over $T$ rounds against a clairvoyant optimizer. In contrast with prior analysis, we show that exploitability is an inherent feature of the FTRL family, rather than an artifact of specific instantiations. First, for fixed optimizer, we establish a sweeping law of order $Ω(N/η)$, proving that exploitation scales to the number of the learner's suboptimal actions $N$ and vanishes in their absence. Second, for alternating optimizer, a surplus of $Ω(ηT/\mathrm{poly}(n,m))$ can be guaranteed regardless of the equilibrium structure, with high probability, in random games. Our analysis uncovers once more the sharp geometric dichotomy: non-steep regularizers allow the optimizer to extract maximum surplus via finite-time elimination of suboptimal actions, whereas steep ones introduce a vanishing correction that may delay exploitation. Finally, we discuss whether this leverage persists under bilateral payoff uncertainty and we propose susceptibility measure to quantify which regularizers are most vulnerable to strategic manipulation.

Armada: Memory-Efficient Distributed Training of Large-Scale Graph Neural Networks

We study distributed training of Graph Neural Networks (GNNs) on billion-scale graphs that are partitioned across machines. Efficient training in this setting relies on min-edge-cut partitioning algorithms, which minimize cross-machine communication due to GNN neighborhood sampling. Yet, min-edge-cut partitioning over large graphs remains a challenge: State-of-the-art (SoTA) offline methods (e.g., METIS) are effective, but they require orders of magnitude more memory and runtime than GNN training itself, while computationally efficient algorithms (e.g., streaming greedy approaches) suffer from increased edge cuts. Thus, in this work we introduce Armada, a new end-to-end system for distributed GNN training whose key contribution is GREM, a novel min-edge-cut partitioning algorithm that can efficiently scale to large graphs. GREM builds on streaming greedy approaches with one key addition: prior vertex assignments are continuously refined during streaming, rather than frozen after an initial greedy selection. Our theoretical analysis and experimental results show that this refinement is critical to minimizing edge cuts and enables GREM to reach partition quality comparable to METIS but with 8-65x less memory and 8-46x faster. Given a partitioned graph, Armada leverages a new disaggregated architecture for distributed GNN training to further improve efficiency; we find that on common cloud machines, even with zero communication, GNN neighborhood sampling and feature loading bottleneck training. Disaggregation allows Armada to independently allocate resources for these operations and ensure that expensive GPUs remain saturated with computation. We evaluate Armada against SoTA systems for distributed GNN training and find that the disaggregated architecture leads to runtime improvements up to 4.5x and cost reductions up to 3.1x.

Contracting with a Learning Agent

Many real-life contractual relations differ completely from the clean, static model at the heart of principal-agent theory. Typically, they involve repeated strategic interactions of the principal and agent, taking place under uncertainty and over time. While appealing in theory, players seldom use complex dynamic strategies in practice, often preferring to circumvent complexity and approach uncertainty through learning. We initiate the study of repeated contracts with a learning agent, focusing on agents who achieve no-regret outcomes. Optimizing against a no-regret agent is a known open problem in general games; we achieve an optimal solution to this problem for a canonical contract setting, in which the agent's choice among multiple actions leads to success/failure. The solution has a surprisingly simple structure: for some $\alpha > 0$, initially offer the agent a linear contract with scalar $\alpha$, then switch to offering a linear contract with scalar $0$. This switch causes the agent to ``free-fall'' through their action space and during this time provides the principal with non-zero reward at zero cost. Despite apparent exploitation of the agent, this dynamic contract can leave \emph{both} players better off compared to the best static contract. Our results generalize beyond success/failure, to arbitrary non-linear contracts which the principal rescales dynamically. Finally, we quantify the dependence of our results on knowledge of the time horizon, and are the first to address this consideration in the study of strategizing against learning agents.

Exploiting hidden structures in non-convex games for convergence to Nash equilibrium

A wide array of modern machine learning applications - from adversarial models to multi-agent reinforcement learning - can be formulated as non-cooperative games whose Nash equilibria represent the system's desired operational states. Despite having a highly non-convex loss landscape, many cases of interest possess a latent convex structure that could potentially be leveraged to yield convergence to equilibrium. Driven by this observation, our paper proposes a flexible first-order method that successfully exploits such "hidden structures" and achieves convergence under minimal assumptions for the transformation connecting the players' control variables to the game's latent, convex-structured layer. The proposed method - which we call preconditioned hidden gradient descent (PHGD) - hinges on a judiciously chosen gradient preconditioning scheme related to natural gradient methods. Importantly, we make no separability assumptions for the game's hidden structure, and we provide explicit convergence rate guarantees for both deterministic and stochastic environments.

A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games

Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $\mathcal{O}(d/\epsilon^2)$ iterations to $\epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $\mathcal{O}(d/\epsilon)$ iterations to $\epsilon$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $\epsilon$-Nash equilibria in quantum zero-sum games.

Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds

Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before.

Stochastic Methods in Variational Inequalities: Ergodicity, Bias and Refinements

For min-max optimization and variational inequalities problems (VIP) encountered in diverse machine learning tasks, Stochastic Extragradient (SEG) and Stochastic Gradient Descent Ascent (SGDA) have emerged as preeminent algorithms. Constant step-size variants of SEG/SGDA have gained popularity, with appealing benefits such as easy tuning and rapid forgiveness of initial conditions, but their convergence behaviors are more complicated even in rudimentary bilinear models. Our work endeavors to elucidate and quantify the probabilistic structures intrinsic to these algorithms. By recasting the constant step-size SEG/SGDA as time-homogeneous Markov Chains, we establish a first-of-its-kind Law of Large Numbers and a Central Limit Theorem, demonstrating that the average iterate is asymptotically normal with a unique invariant distribution for an extensive range of monotone and non-monotone VIPs. Specializing to convex-concave min-max optimization, we characterize the relationship between the step-size and the induced bias with respect to the Von-Neumann's value. Finally, we establish that Richardson-Romberg extrapolation can improve proximity of the average iterate to the global solution for VIPs. Our probabilistic analysis, underpinned by experiments corroborating our theoretical discoveries, harnesses techniques from optimization, Markov chains, and operator theory.

Chaos persists in large-scale multi-agent learning despite adaptive learning rates

Multi-agent learning is intrinsically harder, more unstable and unpredictable than single agent optimization. For this reason, numerous specialized heuristics and techniques have been designed towards the goal of achieving convergence to equilibria in self-play. One such celebrated approach is the use of dynamically adaptive learning rates. Although such techniques are known to allow for improved convergence guarantees in small games, it has been much harder to analyze them in more relevant settings with large populations of agents. These settings are particularly hard as recent work has established that learning with fixed rates will become chaotic given large enough populations.In this work, we show that chaos persists in large population congestion games despite using adaptive learning rates even for the ubiquitous Multiplicative Weight Updates algorithm, even in the presence of only two strategies. At a technical level, due to the non-autonomous nature of the system, our approach goes beyond conventional period-three techniques Li-Yorke by studying fundamental properties of the dynamics including invariant sets, volume expansion and turbulent sets. We complement our theoretical insights with experiments showcasing that slight variations to system parameters lead to a wide variety of unpredictable behaviors.

On the convergence of policy gradient methods to Nash equilibria in general stochastic games

Learning in stochastic games is a notoriously difficult problem because, in addition to each other's strategic decisions, the players must also contend with the fact that the game itself evolves over time, possibly in a very complicated manner. Because of this, the convergence properties of popular learning algorithms - like policy gradient and its variants - are poorly understood, except in specific classes of games (such as potential or two-player, zero-sum games). In view of this, we examine the long-run behavior of policy gradient methods with respect to Nash equilibrium policies that are second-order stationary (SOS) in a sense similar to the type of sufficiency conditions used in optimization. Our first result is that SOS policies are locally attracting with high probability, and we show that policy gradient trajectories with gradient estimates provided by the REINFORCE algorithm achieve an $\mathcal{O}(1/\sqrt{n})$ distance-squared convergence rate if the method's step-size is chosen appropriately. Subsequently, specializing to the class of deterministic Nash policies, we show that this rate can be improved dramatically and, in fact, policy gradient methods converge within a finite number of iterations in that case.