Block majorization-minimization with diminishing radius for constrained nonconvex optimization

• URL: http://arxiv.org/abs/2012.03503v5
• Date: Tue, 7 Nov 2023 04:45:38 GMT
• Title: Block majorization-minimization with diminishing radius for constrained nonconvex optimization
• Authors: Hanbaek Lyu and Yuchen Li
• Abstract summary: Block tensor regularization-minimization (BMM) is a simple iterative algorithm for non constrained optimization that minimizes major surrogates in each block. We show that BMM can produce a gradient $O(epsilon-2(logepsilon-1)2)$ when convex surrogates are used.
• Score: 9.907540661545328
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Block majorization-minimization (BMM) is a simple iterative algorithm for nonconvex constrained optimization that sequentially minimizes majorizing surrogates of the objective function in each block coordinate while the other coordinates are held fixed. BMM entails a large class of optimization algorithms such as block coordinate descent and its proximal-point variant, expectation-minimization, and block projected gradient descent. We establish that for general constrained nonconvex optimization, BMM with strongly convex surrogates can produce an $\epsilon$-stationary point within $O(\epsilon^{-2}(\log \epsilon^{-1})^{2})$ iterations and asymptotically converges to the set of stationary points. Furthermore, we propose a trust-region variant of BMM that can handle surrogates that are only convex and still obtain the same iteration complexity and asymptotic stationarity. These results hold robustly even when the convex sub-problems are inexactly solved as long as the optimality gaps are summable. As an application, we show that a regularized version of the celebrated multiplicative update algorithm for nonnegative matrix factorization by Lee and Seung has iteration complexity of $O(\epsilon^{-2}(\log \epsilon^{-1})^{2})$. The same result holds for a wide class of regularized nonnegative tensor decomposition algorithms as well as the classical block projected gradient descent algorithm. These theoretical results are validated through various numerical experiments.

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