## 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., kF − fk_{∞} ≤ ν). Our objective is to find the -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 ≤ O(^{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.

Original language | English (US) |
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Pages (from-to) | 4896-4905 |

Number of pages | 10 |

Journal | Advances in Neural Information Processing Systems |

Volume | 2018-December |

State | Published - Jan 1 2018 |

Externally published | Yes |

Event | 32nd Conference on Neural Information Processing Systems, NeurIPS 2018 - Montreal, Canada Duration: Dec 2 2018 → Dec 8 2018 |

## All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Information Systems
- Signal Processing