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On the Convergence Rate of Stochastic Mirror Descent for Nonsmooth Nonconvex Optimization

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arxiv 1806.04781 v1 pith:M6FCWYQU submitted 2018-06-12 math.OC

classification math.OC
keywords convergencestochasticconvexnonconvexoptimizationdescentmirrornonsmooth
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abstract

In this paper, we investigate the non-asymptotic stationary convergence behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We focus on a general class of nonconvex nonsmooth stochastic optimization problems, in which the objective can be decomposed into a relatively weakly convex function (possibly non-Lipschitz) and a simple non-smooth convex regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to converge to a stationary point in a convergence rate of $ \mathcal{O}(1/\sqrt{t}) $. The efficiency estimate matches with existing results for stochastic subgradient method, but is evaluated under a stronger stationarity measure. Our convergence analysis applies to both the original SMD and its proximal version, as well as the deterministic variants, for solving relatively weakly convex problems.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization

    math.OC 2026-08 conditional novelty 7.0 of 10

    Under verifiable joint conditions on the objective, the Legendre kernel, and the feasible geometry, mirror descent converges to a boundary KKT point with explicit rates.

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    Develops infinite-horizon stationary robust mean-field games incorporating distributional uncertainty, proves equilibrium existence via fixed-point on contractive Bellman operator, gives convergent algorithm, and deri...

  3. Relaxation-Free Min-k-Partition for PCI Assignment in 5G Networks

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    A Chinese Remainder Theorem decomposition plus a penalized mirror descent solver for Min-k-Partition assigns 5G PCIs with near-zero mod-3 and mod-30 interference in experiments.

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