Sirong Dai | Searcharxiv

arXiv2026

Newton Method for Fixed-Support Doubly Entropic Wasserstein Barycenter

We study the fixed-support doubly regularized Wasserstein barycenter problem. Using the semi-dual formulation of entropic optimal transport, we reformulate the problem as a smooth, unconstrained, convex optimization problem in the dual variables. We then derive explicit expressions for the gradient and Hessian and develop an exact Newton method for high-accuracy barycenter computation. To improve scalability, we propose a sparse Newton variant that sparsifies the transport probability matrices, thereby reducing the cost of Hessian-vector products. We establish theoretical results for the proposed methods, including Hessian approximation bounds and convergence results. Experiments on synthetic and real datasets show that the sparse Newton method converges faster than

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arXiv2026

Unified Ergodic Primal-Dual Gap Rates with Unhalved Primal Stepsizes

We study ergodic primal-dual gap rates for first-order primal-dual methods applied to \[ \min_x f(x)+g(x)+h(Ax), \] where $f$ is smooth and convex, $g$ and $h$ are proper, closed, convex functions, and $A$ is linear. Standard gap-rate proofs often impose the halved smooth-stepsize condition $τ\le 1/L$, even though the corresponding convergence theory allows the larger range $τ<2/L$. We introduce a residual-to-gap transfer principle: positive residual terms in the one-step gap inequality are controlled by the decrease of a Lyapunov function. This yields $O(1/K)$ ergodic primal-dual gap bounds with the unhalved primal stepsize $τ<2/L$ for Condat--Vũ, PD3O, AFBA/PDDY, and PAPC/PDFP$^2$O, under their algorithm-dependent product conditions. We also give a two-dimensional counterexample showing that the fully separated rectangle $τ<2/L$, $τη\|A\|^2<4/3$ cannot hold in the general three-function setting.

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