Convergence analysis of stochastic higher-order majorization-minimization algorithms

Majorization-minimization schemes are a broad class of iterative methods targeting general optimization problems, including nonconvex, nonsmooth and stochastic. These algorithms minimize successively a sequence of upper bounds of the objective function so that along the iterations the objective valu...

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Published in:Optimization methods & software Vol. 39; no. 2; pp. 384 - 413
Main Authors: Lupu, Daniela, Necoara, Ion
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 03.03.2024
Taylor & Francis Ltd
Subjects:
Algorithms
Approximation
Convergence
convergence rates
Convexity
Finite-sum optimization
Iterative methods
majorization-minimization
minibatch
Optimization
stochastic higher-order algorithms
Upper bounds
ISSN:1055-6788, 1029-4937
Online Access:Get full text
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Summary:Majorization-minimization schemes are a broad class of iterative methods targeting general optimization problems, including nonconvex, nonsmooth and stochastic. These algorithms minimize successively a sequence of upper bounds of the objective function so that along the iterations the objective value decreases. We present a stochastic higher-order algorithmic framework for minimizing the average of a very large number of sufficiently smooth functions. Our stochastic framework is based on the notion of stochastic higher-order upper bound approximations of the finite-sum objective function and minibatching. We derive convergence results for nonconvex and convex optimization problems when the higher-order approximation of the objective function yields an error that is p times differentiable and has Lipschitz continuous p derivative. More precisely, for general nonconvex problems we present asymptotic stationary point guarantees and under Kurdyka-Lojasiewicz property we derive local convergence rates ranging from sublinear to linear. For convex problems with uniformly convex objective function, we derive local (super)linear convergence results for our algorithm. Compared to existing stochastic (first-order) methods, our algorithm adapts to the problem's curvature and allows using any batch size. Preliminary numerical tests support the effectiveness of our algorithmic framework.
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ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2023.2256447