mirror descent analysis

Existing stochastic optimization theory is confined to Hilbert spaces, limiting its applicability to non-Euclidean geometries—such as simplices and probability manifolds—arising in mirror descent, natural gradient methods, and KL-regularized language model training. Method: We propose the first unified stochastic optimization framework grounded in Banach spaces and Bregman geometry, eliminating reliance on inner products and supporting arbitrary smooth convex norms. Crucially, we introduce a novel over-relaxation parameter λ > 2, enabling the first derivation of Bregman–Fejér monotonicity and convergence guarantees in general Banach spaces. Contribution/Results: Our framework unifies stochastic mirror descent, adaptive learning, and large-model training dynamics. Empirical evaluation on UCI datasets, Transformer, Actor-Critic, and distilGPT-2 demonstrates up to 20% faster convergence, significantly reduced variance, and improved accuracy—validating both theoretical generality and practical efficacy.

This work addresses the instability of classical mirror descent in monotone variational inequality problems, where the method may diverge or cycle due to the absence of a stabilizing convergence mechanism. To overcome this limitation, the paper introduces the Targeted Mirror Descent (TMD) framework, which incorporates a target-point correction mechanism into the dual update to effectively stabilize the optimization dynamics. TMD unifies classical algorithms such as the proximal point method and extragradient method, revealing their underlying convergence principles. It also rectifies the equilibrium misalignment issue present in discounted mirror descent and enables geometric ensembling, allowing multiple heterogeneous mirror maps to operate collaboratively in parallel. The framework guarantees convergence for monotone variational inequalities and facilitates the construction of novel mirror maps, substantially enhancing algorithmic robustness and applicability.

This work studies the *last-iterate convergence* of Optimistic Multiplicative Weights Update (OMWU) for *constrained convex-concave minimax optimization* within the no-regret online learning framework. Prior theoretical guarantees are limited to unconstrained or bilinear settings; local last-iterate convergence of OMWU in general constrained convex-concave games remains unaddressed. We establish, for the first time, a local last-iterate convergence guarantee for OMWU over *general convex-concave constraint sets*. Our proof integrates tools from no-regret analysis, saddle-point dynamics, and convex optimization. The result significantly relaxes structural assumptions required by existing theory—removing reliance on bilinearity or absence of constraints—while preserving sharp convergence rates. Numerical experiments corroborate the predicted fast convergence behavior. This work closes a fundamental theoretical gap in the convergence analysis of no-regret learning algorithms for non-bilinear, constrained convex-concave games.

This work systematically uncovers the decisive role of problem geometry—specifically, the curvature of the constraint set and the structure of gradients—in governing the statistical-computational trade-offs of stochastic and online optimization algorithms. We introduce the first geometric measure quantifying the deviation of a constraint set from quadratic convexity, rigorously identifying the geometric origins of suboptimality in subgradient methods. We prove that diagonal-preconditioned SGD achieves minimax-optimal convergence rates under quadratic convex constraints. For non-Euclidean, non-quadratically-convex domains—such as ℓₚ-balls with p < 2—we establish tight convergence bounds for mirror descent and adaptive gradient methods, and uncover, for the first time, a precise correspondence between their convergence rates and the accuracy-computation trade-off in Gaussian sequence estimation. Our results provide geometric criteria for algorithm selection and unify the understanding of when nonlinear updates—e.g., via mirror descent—are necessary to attain statistical optimality.

This work addresses the open problem posed by Syrgkanis et al. concerning last-iterate convergence of Optimistic Multiplicative Weights Update (OMWU) in constrained convex-concave minimax optimization—particularly relevant to zero-sum games and GANs. We establish, for the first time, global convergence of OMWU to exact saddle points under general convex constraints, without requiring unconstrained domains or strong regularization assumptions. Our analysis introduces a novel framework combining monotonic KL-divergence descent with local contraction mapping properties, integrating fixed-point theory and contraction mapping techniques. This approach overcomes key limitations of prior analyses reliant on either unbounded domains or stringent regularization. The result provides the first rigorous theoretical guarantee for OMWU’s last-iterate convergence in constrained saddle-point optimization and furnishes a principled foundation for termination criteria in practical iterative implementations.

This work investigates the convergence properties and implicit bias of matrix-form stochastic mirror descent (SMD) in over-parameterized, multi-output high-dimensional problems such as multiclass classification and matrix completion. By extending the implicit bias theory of vector-valued SMD to the matrix setting, it reveals for the first time how the Bregman divergence induced by the mirror map governs the uniqueness of interpolating solutions and the associated inductive bias. The theoretical analysis demonstrates that, under over-parameterization, matrix SMD converges exponentially to the unique solution that both interpolates the data and minimizes the Bregman divergence from the initialization. This result elucidates the pivotal role of the mirror map in shaping the generalization behavior of the learned model.

Optimization over large-scale Riemannian manifolds poses significant challenges for the direct application of classical mirror descent methods. This work proposes a Riemannian Mirror Descent (RMD) framework that generalizes mirror descent to arbitrary Riemannian manifolds through reparameterization and develops its stochastic variant. For the first time, non-asymptotic convergence guarantees are established for mirror descent on general Riemannian manifolds. When specialized to the Stiefel manifold, the framework naturally yields Riemannian gradient descent and its stochastic extension. The theoretical analysis integrates tools from Riemannian geometry, stochastic gradient estimation, and non-convex optimization. Empirical evaluations demonstrate the efficiency and scalability of the proposed methods across diverse tasks, including image processing, policy optimization, and neural network training.

This work addresses the poor sample efficiency and weak policy convergence of value-iteration–based methods in high-precision and online reinforcement learning settings. To overcome these limitations, the authors introduce mirror descent—a technique from convex optimization—into value function optimization, proposing Value Mirror Descent (VMD) and its stochastic variant SVMD, enhanced with variance reduction. By explicitly controlling the Bregman divergence between successive policies, the approach bridges a theoretical gap in existing value iteration methods for continual learning. Under general convex regularization, SVMD achieves a near-optimal sample complexity of Õ(|S||A|(1−γ)⁻³ε⁻²), which further improves to Õ(|S||A|(1−γ)⁻⁵ε⁻¹) under strong convexity, while guaranteeing convergence of the policy to the optimal solution.

This work addresses the challenge that mirror descent methods in non-convex optimization often fail to converge to Karush–Kuhn–Tucker (KKT) points when iterates approach the boundary of the feasible domain. To overcome this, the authors propose a definable boundary-extension reparameterization scheme grounded in metric flattening. Convergence of the reparameterized objective is established via the Kurdyka–Łojasiewicz (KL) inequality, and continuity of the inverse mapping is leveraged to recover convergence of the original iterates to KKT points. This study provides the first theoretical guarantee for mirror descent converging to KKT points without excluding boundary-limiting behavior, offering verifiable conditions that jointly account for the objective function, Legendre kernel, and feasible set geometry. The framework is validated on canonical examples including Shannon entropy, Fermi–Dirac entropy, and power kernels, laying foundational groundwork for convergence analysis of generalized Bregman-type algorithms.