Implicit bias of any algorithm: bounding bias via margin

Consider $n$ points $x_1$,\ldots,$x_n$ in finite-dimensional euclidean space, each having one of two colors. Suppose there exists a separating hyperplane (identified with its unit normal vector $w)$ for the points, i.e a hyperplane such that points of same color lie on the same side of the hyperplane. We measure the quality of such a hyperplane by its margin $\gamma(w)$, defined as minimum distance between any of the points $x_i$ and the hyperplane. In this paper, we prove that the margin function $\gamma$ satisfies a nonsmooth Kurdyka-Lojasiewicz inequality with exponent $1/2$. This result has far-reaching consequences. For example, let $\gamma^{opt}$ be the maximum possible margin for the problem and let $w^{opt}$ be the parameter for the hyperplane which attains this value. Given any other separating hyperplane with parameter $w$, let $d(w):=\|w-w^{opt}\|$ be the euclidean distance between $w$ and $w^{opt}$, also called the bias of $w$. From the previous KL-inequality, we deduce that $(\gamma^{opt}-\gamma(w)) / R \le d(w) \le 2\sqrt{(\gamma^{opt}-\gamma(w))/\gamma^{opt}}$, where $R:=\max_i \|x_i\|$ is the maximum distance of the points $x_i$ from the origin. Consequently, for any optimization algorithm (gradient-descent or not), the bias of the iterates converges at least as fast as the square-root of the rate of their convergence of the margin. Thus, our work provides a generic tool for analyzing the implicit bias of any algorithm in terms of its margin, in situations where a specialized analysis might not be available: it is sufficient to establish a good rate for converge of the margin, a task which is usually much easier.