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We investigate the convergence of stochastic mirror descent (SMD) underinterpolation in relatively smooth and smooth convex optimization. Inrelatively smooth convex optimization we provide new convergence guarantees forSMD with a constant stepsize. For smooth convex optimization we propose a newadaptive stepsize scheme -- the mirror stochastic Polyak stepsize (mSPS).Notably, our convergence results in both settings do not make bounded gradientassumptions or bounded variance assumptions, and we show convergence to aneighborhood that vanishes under interpolation. Consequently, these resultscorrespond to the first convergence guarantees under interpolation for theexponentiated gradient algorithm for fixed or adaptive stepsizes. mSPSgeneralizes the recently proposed stochastic Polyak stepsize (SPS) (Loizou etal. 2021) to mirror descent and remains both practical and efficient for modernmachine learning applications while inheriting the benefits of mirror descent.We complement our results with experiments across various supervised learningtasks and different instances of SMD, demonstrating the effectiveness of mSPS.

Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize