Babak Hassibi

Danil Akhtiamov, Reza Ghane, Babak Hassibi

We investigate Stochastic Mirror Descent (SMD) with matrix parameters and vector-valued predictions, a framework relevant to multi-class classification and matrix completion problems. Focusing on the overparameterized regime, where the total number of parameters exceeds the number of training samples, we prove that SMD with matrix mirror functions $ψ(\cdot)$ converges exponentially to a global interpolator. Furthermore, we generalize classical implicit bias results of vector SMD by demonstrating that the matrix SMD algorithm converges to the unique solution minimizing the Bregman divergence induced by $ψ(\cdot)$ from initialization subject to interpolating the data. These findings reveal how matrix mirror maps dictate inductive bias in high-dimensional, multi-output problems.

Gautam Goel, Babak Hassibi

We consider control from the perspective of competitive analysis. Unlike much prior work on learning-based control, which focuses on minimizing regret against the best controller selected in hindsight from some specific class, we focus on designing an online controller which competes against a clairvoyant offline optimal controller. A natural performance metric in this setting is competitive ratio, which is the ratio between the cost incurred by the online controller and the cost incurred by the offline optimal controller. Using operator-theoretic techniques from robust control, we derive a computationally efficient state-space description of the the controller with optimal competitive ratio in both finite-horizon and infinite-horizon settings. We extend competitive control to nonlinear systems using Model Predictive Control (MPC) and present numerical experiments which show that our competitive controller can significantly outperform standard $H_2$ and $H_{\infty}$ controllers in the MPC setting.