We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e.

We propose a randomized first order optimization method-- SEGA (SkEtched GrAdient)--which progressively throughout its iterations builds a variance-reduced estimate of the gradient from random linear measurements (sketches) of the gradient. In each iteration, SEGA updates the current estimate of the gradient through a sketch-and-project operation using the information provided by the latest sketch, and this is subsequently used to compute an unbiased estimate of the true gradient through a random relaxation procedure. This unbiased estimate is then used to perform a gradient step. Unlike standard subspace descent methods, such as coordinate descent, SEGA can be used for optimization problems with a non-separable proximal term. We provide a general convergence analysis and prove linear convergence for strongly convex objectives. In the special case of coordinate sketches, SEGA can be enhanced with various techniques such as importance sampling, minibatching and acceleration, and its rate is up to a small constant factor identical to the best-known rate of coordinate descent.

Neural Information Processing Systems http://nips.cc/

This paper proposes a stochastic variant of a classic algorithm--the cubic-regularized Newton method [Nesterov and Polyak, 2006].

We propose a reduction for non-convex optimization that can (1) turn an stationary-point finding algorithm into an local-minimum finding one, and (2) replace the Hessian-vector product computations with only gradient computations. It works both in the stochastic and the deterministic settings, without hurting the algorithm's performance. As applications, our reduction turns Natasha2 into a first-order method without hurting its theoretical performance. It also converts SGD, GD, SCSG, and SVRG into algorithms finding approximate local minima, outperforming some best known results.

The loss functions of deep neural networks are complex and their geometric properties are not well understood.

We show that most local optima of PNN optimizations are global while we have a characterization of regions where bad local optimizers may exist.

More complex data consisting of measurements across multiple modalities can be organized as higher dimensional data arrays, or higher order tensors.

Neural Information Processing Systems http://nips.cc/

Neural network training relies on our ability to find "good" minimizers of highly