We study the problem of minimizing the sum of a smooth function and a nonsmooth convex regularizer over a compact Riemannian submanifold embedded in Euclidean space. By introducing an auxiliary splitting variable, we propose an adaptive Riemannian alternating direction method of multipliers (ARADMM), which, for the first time, achieves convergence without requiring smoothing of the nonsmooth term. Our approach involves only one Riemannian gradient evaluation and one proximal update per iteration. Through careful and adaptive coordination of the stepsizes and penalty parameters, we establish an optimal iteration complexity of order O(ϵ 3) for finding an ϵ-approximate KKT point, matching the complexity of existing smoothing technique-based Riemannian ADMM methods. Extensive numerical experiments on sparse PCA and robust subspace recovery demonstrate that our ARADMM consistently outperforms state-of-the-art Riemannian ADMM variants in convergence speed and solution quality.

We explore the matrix sensing problem from near-isotropic linear measurements, distributed across a network of agents modeled as an undirected graph, with no server. We provide the first study of statistical, computational/communication guarantees for a decentralized gradient algorithm that solves the (nonconvex) Burer-Monteiro type decomposition associated to the low-rank matrix estimation. With small random initialization, the algorithm displays an approximate two-phase convergence: (i) a spectral phase that aligns the iterates' column space with the underlying low-rank matrix, mimicking centralized spectral initialization (not directly implementable over networks); and (ii) a local refinement phase that diverts the iterates from certain degenerate saddle points, while ensuring swift convergence to the underlying low-rank matrix. Central to our analysis is a novel "in-network" Restricted Isometry Property which accommodates for the decentralized nature of the optimization, revealing an intriguing interplay between sample complexity, network connectivity & topology, and communication complexity.

Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function f: Rn!R, it outputs an -approximate second-order stationary point in O(logn/ 1.75)iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with O(log4 n/ 2) or O(log6 n/ 1.75) iterations, our algorithm is polynomially better in terms of logn and matches their complexities in terms of 1/ .

Federated learning (FL) has emerged as an enabling framework for communicationefficient decentralized training. We study three broadly applicable problem classes in FL: (i) Nondifferentiable nonconvex federated optimization; (ii) Federated bilevel optimization; (iii) Federated minimax problems. Notably, in an implicit sense, both (ii) and (iii) are instances of (i). However, the hierarchical problems in (ii) and (iii) are often complicated by the absence of a closed-form expression for the implicit objective function. Unfortunately, research on these problems has been limited and afflicted by reliance on strong assumptions, including the need for differentiability and L-smoothness of the implicit function. We address this shortcoming by making the following contributions. In (i), by leveraging convolution-based smoothing and Clarke's subdifferential calculus, we devise a randomized smoothing-enabled zeroth-order FL method and derive communication and iteration complexity guarantees for computing an approximate Clarke stationary point. To contend with (ii) and (iii), we devise a unified randomized implicit zeroth-order FL framework, equipped with explicit communication and iteration complexities. Importantly, our method utilizes delays during local steps to skip making calls to the inexact lower-level FL oracle.

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Cubic-regularized Newton's method (CR) is a popular algorithm that guarantees

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