Online Non-Stationary Stochastic Quasar-Convex Optimization
This work addresses optimization problems in non-stationary settings for applications like system identification and GLMs, but it is incremental as it extends existing quasar-convexity methods to online stochastic cases.
The paper tackles online stochastic quasar-convex optimization in dynamic environments by establishing regret bounds for online gradient descent in terms of cumulative path variation and gradient variance, and applies these results to generalized linear models with various activation functions, showing numerical corroboration.
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.