Poster
Efficient Algorithms for Sum-Of-Minimum Optimization
Lisang Ding · Ziang Chen · Xinshang Wang · Wotao Yin
Hall C 4-9
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Abstract
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Tue 23 Jul 2:30 a.m. PDT
— 4 a.m. PDT
Abstract:
In this work, we propose a novel optimization model termed ``sum-of-minimum" optimization. This model seeks to minimize the sum or average of $N$ objective functions over $k$ parameters, where each objective takes the minimum value of a predefined sub-function with respect to the $k$ parameters. This universal framework encompasses numerous clustering applications in machine learning and related fields. We develop efficient algorithms for solving sum-of-minimum optimization problems, inspired by a randomized initialization algorithm for the classic $k$-means (Arthur & Vassilvitskii, 2007) and Lloyd's algorithm (Lloyd, 1982). We establish a new tight bound for the generalized initialization algorithm and prove a gradient-descent-like convergence rate for generalized Lloyd's algorithm. The efficiency of our algorithms is numerically examined on multiple tasks, including generalized principal component analysis, mixed linear regression, and small-scale neural network training. Our approach compares favorably to previous ones based on simpler-but-less-precise optimization reformulations.
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