Online Non-Stationary Stochastic Quasar-Convex Optimization

TL;DR

This paper investigates online stochastic quasar-convex optimization in dynamic environments, providing regret bounds for gradient descent methods and applying them to time-varying generalized linear models with various activation functions.

Contribution

It establishes regret bounds for online gradient descent in quasar-convex settings and extends these results to generalized linear models with time-varying parameters.

Findings

Regret bounds depend on path variation and gradient variance.

Results apply to GLMs with leaky ReLU, logistic, and ReLU activations.

Numerical experiments support theoretical findings.

Abstract

Recent research has shown that quasar-convexity can be found in applications such as identification of linear dynamical systems and generalized linear models. Such observations have in turn spurred exciting developments in design and analysis algorithms that exploit quasar-convexity. In this work, we study the online stochastic quasar-convex optimization problems in a dynamic environment. We establish regret bounds of online gradient descent in terms of cumulative path variation and cumulative gradient variance for losses satisfying quasar-convexity and strong quasar-convexity. We then apply the results to generalized linear models (GLM) when the underlying parameter is time-varying. We establish regret bounds of online gradient descent when applying to GLMs with leaky ReLU activation function, logistic activation function, and ReLU activation function. Numerical results are presented to corroborate our findings.