Implicit Bias of Mirror Flow on Separable Data
This work addresses the problem of understanding algorithmic bias in optimization for machine learning practitioners, though it is incremental as it extends known results from mirror descent to its continuous-time counterpart.
The paper investigates the implicit bias of mirror flow, the continuous-time version of mirror descent, on linearly separable classification problems, showing that the algorithm converges to a maximum margin classifier defined by the horizon function of the mirror potential, with numerical experiments supporting the findings.
We examine the continuous-time counterpart of mirror descent, namely mirror flow, on classification problems which are linearly separable. Such problems are minimised `at infinity' and have many possible solutions; we study which solution is preferred by the algorithm depending on the mirror potential. For exponential tailed losses and under mild assumptions on the potential, we show that the iterates converge in direction towards a $φ_\infty$-maximum margin classifier. The function $φ_\infty$ is the \textit{horizon function} of the mirror potential and characterises its shape `at infinity'. When the potential is separable, a simple formula allows to compute this function. We analyse several examples of potentials and provide numerical experiments highlighting our results.