Stochastic Online Convex Optimization: Fast Convergence Rates via Self-Normalized Martingale Inequalities

This talk is about an extension of the framework of Online Convex Optimization to a stochastic adversarial setting. 

Under the Stochastic Directional Derivative condition, we prove that the Online Newton Step algorithm achieves fast convergence rates. 

A distinguishing feature of our approach is its applicability to non-convex loss functions, significantly broadening its scope. 

We demonstrate the framework’s utility through applications to non-convex losses, such as the likelihood of volatility models. 

The analysis hinges on a fundamental self-normalized martingale inequality, as developed by Bercu and Touati (2008).

This work builds upon Wintenberger (2024), Stochastic Online Convex Optimization; Application to Probabilistic Time Series Forecasting, published in EJS, 18, 429–464.