Stochastic Chebyshev gradient descent for spectral optimization

Insu Han, Haim Avron, Jinwoo Shin

Research output: Contribution to journalConference articlepeer-review

4 Scopus citations


A large class of machine learning techniques requires the solution of optimization problems involving spectral functions of parametric matrices, e.g. log-determinant and nuclear norm. Unfortunately, computing the gradient of a spectral function is generally of cubic complexity, as such gradient descent methods are rather expensive for optimizing objectives involving the spectral function. Thus, one naturally turns to stochastic gradient methods in hope that they will provide a way to reduce or altogether avoid the computation of full gradients. However, here a new challenge appears: there is no straightforward way to compute unbiased stochastic gradients for spectral functions. In this paper, we develop unbiased stochastic gradients for spectral-sums, an important subclass of spectral functions. Our unbiased stochastic gradients are based on combining randomized trace estimators with stochastic truncation of the Chebyshev expansions. A careful design of the truncation distribution allows us to offer distributions that are variance-optimal, which is crucial for fast and stable convergence of stochastic gradient methods. We further leverage our proposed stochastic gradients to devise stochastic methods for objective functions involving spectral-sums, and rigorously analyze their convergence rate. The utility of our methods is demonstrated in numerical experiments.

Original languageEnglish
Pages (from-to)7386-7396
Number of pages11
JournalAdvances in Neural Information Processing Systems
StatePublished - 2018
Event32nd Conference on Neural Information Processing Systems, NeurIPS 2018 - Montreal, Canada
Duration: 2 Dec 20188 Dec 2018


FundersFunder number
National Research Foundation of Korea2018R1A5A1059921, 1272/17
Israel Science Foundation
Ministry of Science and ICT, South Korea


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