On the Regret Minimization of Nonconvex Online Gradient Ascent for Online PCA
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Non-convex optimization with global convergence guarantees is gaining significant interest in machine learning research in recent years. However, while most works consider either offline settings in which all data is given beforehand, or simple online stochastic i.i.d. settings, very little is known about non-convex optimization for adversarial online learning settings. In this paper we focus on the problem of Online Principal Component Analysis in the regret minimization framework. For this problem, all existing regret minimization algorithms are based on a positive semidefinite convex relaxation, and hence require quadratic memory and SVD computation (either thin of full) on each iteration, which amounts to at least quadratic runtime per iteration. This is in stark contrast to a corresponding stochastic i.i.d. variant of the problem which admits very efficient gradient ascent algorithms that work directly on the natural non-convex formulation of the problem, and hence require only linear memory and linear runtime per iteration. This raises the question: can non-convex online gradient ascent algorithms be shown to minimize regret in online adversarial settings? In this paper we take a step forward towards answering this question. We introduce an adversarially-perturbed spiked-covariance model in which, each data point is assumed to follow a fixed stochastic distribution, but is then perturbed by adversarial noise. We show that in a certain regime of parameters, when the non-convex online gradient ascent algorithm is initialized with a "warm-start" vector, it provably minimizes the regret with high probability. We further discuss the possibility of computing such a "warm-start" vector. Our theoretical findings are supported by empirical experiments on both synthetic and real-world data.
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