Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize
This work addresses optimization challenges in machine learning by improving convergence guarantees and adaptive stepsizing for SMD, offering incremental advancements for researchers and practitioners in convex optimization.
The paper tackles convergence analysis of stochastic mirror descent (SMD) under interpolation in convex optimization, providing new guarantees without bounded gradient or variance assumptions and proposing the mirror stochastic Polyak stepsize (mSPS) for adaptive stepsizing, with experiments showing effectiveness in supervised learning tasks.
We investigate the convergence of stochastic mirror descent (SMD) under interpolation in relatively smooth and smooth convex optimization. In relatively smooth convex optimization we provide new convergence guarantees for SMD with a constant stepsize. For smooth convex optimization we propose a new adaptive stepsize scheme -- the mirror stochastic Polyak stepsize (mSPS). Notably, our convergence results in both settings do not make bounded gradient assumptions or bounded variance assumptions, and we show convergence to a neighborhood that vanishes under interpolation. Consequently, these results correspond to the first convergence guarantees under interpolation for the exponentiated gradient algorithm for fixed or adaptive stepsizes. mSPS generalizes the recently proposed stochastic Polyak stepsize (SPS) (Loizou et al. 2021) to mirror descent and remains both practical and efficient for modern machine learning applications while inheriting the benefits of mirror descent. We complement our results with experiments across various supervised learning tasks and different instances of SMD, demonstrating the effectiveness of mSPS.