Entropic Mirror Descent for Linear Systems: Polyak's Stepsize and Implicit Bias
This work addresses the problem of solving linear systems efficiently for researchers and practitioners in the field of optimization and machine learning.
The authors tackled the challenge of applying entropic mirror descent to solve linear systems with unbounded domains, achieving sublinear and linear convergence results. They obtained a strengthened bound for $ell_1$-norm implicit bias.
This paper focuses on applying entropic mirror descent to solve linear systems, where the main challenge for the convergence analysis stems from the unboundedness of the domain. To overcome this without imposing restrictive assumptions, we introduce a variant of Polyak-type stepsizes. Along the way, we strengthen the bound for $\ell_1$-norm implicit bias, obtain sublinear and linear convergence results, and generalize the convergence result to arbitrary convex $L$-smooth functions. We also propose an alternative method that avoids exponentiation, resembling the original Hadamard descent, but with provable convergence.