Second-Order Optimization for Non-Convex Machine Learning: An Empirical Study

While first-order optimization methods such as stochastic gradient descent (SGD) are popular in machine learning (ML), they come with well-known deficiencies, including relatively-slow convergence, sensitivity to the settings of hyper-parameters such as learning rate, stagnation at high training errors, and difficulty in escaping flat regions and saddle points. These issues are particularly acute in highly non-convex settings such as those arising in neural networks. Motivated by this, there has been recent interest in second-order methods that aim to alleviate these shortcomings by capturing curvature information. In this paper, we report detailed empirical evaluations of a class of Newton-type methods, namely sub-sampled variants of trust region (TR) and adaptive regularization with cubics (ARC) algorithms, for non-convex ML problems. In doing so, we demonstrate that these methods not only can be computationally competitive with hand-tuned SGD with momentum, obtaining comparable or better generalization performance, but also they are highly robust to hyper-parameter settings. Further, in contrast to SGD with momentum, we show that the manner in which these Newton-type methods employ curvature information allows them to seamlessly escape flat regions and saddle points.Comment: 21 pages, 11 figures. Restructure the paper and add experiment

Proximal boosting and its acceleration

Gradient boosting is a prediction method that iteratively combines weak learners to produce a complex and accurate model. From an optimization point of view, the learning procedure of gradient boosting mimics a gradient descent on a functional variable. This paper proposes to build upon the proximal point algorithm when the empirical risk to minimize is not differentiable to introduce a novel boosting approach, called proximal boosting. Besides being motivated by non-differentiable optimization, the proposed algorithm benefits from Nesterov's acceleration in the same way as gradient boosting [Biau et al., 2018]. This leads to a variant, called accelerated proximal boosting. Advantages of leveraging proximal methods for boosting are illustrated by numerical experiments on simulated and real-world data. In particular, we exhibit a favorable comparison over gradient boosting regarding convergence rate and prediction accuracy