On the Local Minima of the Empirical Risk

Abstract

Population risk is always of primary interest in machine learning; however, learning algorithms only have access to the empirical risk. Even for applications with nonconvex nonsmooth losses (such as modern deep networks), the population risk is generally significantly more well-behaved from an optimization point of view than the empirical risk. In particular, sampling can create many spurious local minima. We consider a general framework which aims to optimize a smooth nonconvex function $F$ (population risk) given only access to an approximation $f$ (empirical risk) that is pointwise close to $F$ (i.e., $\|F-f\|_{\infty} \le ν$). Our objective is to find the $ε$-approximate local minima of the underlying function $F$ while avoiding the shallow local minima---arising because of the tolerance $ν$---which exist only in $f$. We propose a simple algorithm based on stochastic gradient descent (SGD) on a smoothed version of $f$ that is guaranteed to achieve our goal as long as $ν\le O(ε^{1.5}/d)$. We also provide an almost matching lower bound showing that our algorithm achieves optimal error tolerance $ν$ among all algorithms making a polynomial number of queries of $f$. As a concrete example, we show that our results can be directly used to give sample complexities for learning a ReLU unit.

To appear in NIPS 2018