arXiv:2608. 07248v1 Announce Type: cross Abstract: We prove that mirror descent converges to a KKT point for the nonconvex problem without excluding boundary limits.
arXiv:2608. 01658v1 Announce Type: cross Abstract: For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush--Kuhn--Tucker (KKT) stationary under proper stepsizes?
arXiv:2504.09409v3 Announce Type: replace-cross Abstract: In this paper, we study nonconvex constrained stochastic zeroth-order optimization problems with exact constraints and stochastic objective e...
arXiv:2502. 00753v4 Announce Type: replace-cross Abstract: Smoothness is crucial for attaining fast rates in first-order optimization.
arXiv:2609. 03762v1 Announce Type: new Abstract: The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information.
The paper introduces computable error bounds and a certified early‑stopping criterion for regularized inverse problems by exploiting an exact Fenchel–Young duality‑gap identity. The total duality gap splits into a data‑fidelity loss and a regularizer loss, both expressed as Fenchel–Young losses that are oracle‑free and vanish exactly at Mirror Alignment. Using a constructive Brønsted–Rockafellar approach, the authors build a dual‑feasible proxy via a proximal step in the fidelity geometry, enabling an early‑stopping rule based on the regularizer loss.
arXiv:2411. 15067v2 Announce Type: replace-cross Abstract: We investigate proximal descent methods, inspired by the minimizing movement scheme introduced by Jordan, Kinderlehrer and Otto, for optimizing entropy-regularized functionals on the Wasserstein space.
The paper introduces a Projected Riemannian Gradient Descent (RGD) algorithm for computing the Bures‑Wasserstein barycenter of positive definite matrices, achieving dimension‑independent linear convergence at unit step size. It resolves a previous dichotomy by showing that clipping eigenvalues to a fixed interval yields a closed‑form, non‑expansive projection in the BW metric, allowing the algorithm to match the empirical speed of unit‑step RGD while maintaining theoretical guarantees. The method also extends to the invariant matrix projection problem, providing a unified dimension‑independent analysis.
arXiv:2606. 11431v1 Announce Type: new Abstract: Mirror Descent (MD) extends Gradient Descent (GD) beyond Euclidean geometry and has recently reappeared as a lens for KL-regularized policy optimization in reinforcement learning and LLM post-training.
The paper studies algorithms for computing the Entropic Gromov-Wasserstein (EGW) distance, a measure of discrepancy between metric measure spaces. It introduces Averaged Mirror Descent (AMD), which averages successive Mirror Descent steps and is proven to converge for any cost function, and shows that a dual gradient method with a fixed step size also converges for arbitrary costs, even when iterations are inexact. Empirical comparisons demonstrate that both AMD and the dual gradient method succeed on cases where classical Mirror Descent fails.
arXiv:2607. 16384v1 Announce Type: new Abstract: For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons.
arXiv:2602. 02877v2 Announce Type: replace Abstract: This paper studies optimization for a family of problems termed $\textbf{compositional entropic risk minimization}$, in which each data's loss is formulated as a Log-Expectation-Exponential (Log-E-Exp) function.
arXiv:2607. 07468v1 Announce Type: cross Abstract: We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning.