Importance Sampling for Minibatches

Minibatching is a very well studied and highly popular technique in supervised learning, used by practitioners due to its ability to accelerate training through better utilization of parallel processing power and reduction of stochastic variance. Another popular technique is importance sampling -- a strategy for preferential sampling of more important examples also capable of accelerating the training process. However, despite considerable effort by the community in these areas, and due to the inherent technical difficulty of the problem, there is no existing work combining the power of importance sampling with the strength of minibatching. In this paper we propose the first {\em importance sampling for minibatches} and give simple and rigorous complexity analysis of its performance. We illustrate on synthetic problems that for training data of certain properties, our sampling can lead to several orders of magnitude improvement in training time. We then test the new sampling on several popular datasets, and show that the improvement can reach an order of magnitude

Convergence analysis of stochastic higher-order majorization-minimization algorithms

Majorization-minimization schemes are a broad class of iterative methods targeting general optimization problems, including nonconvex, nonsmooth and stochastic. These algorithms minimize successively a sequence of upper bounds of the objective function so that along the iterations the objective value decreases. We present a stochastic higher-order algorithmic framework for minimizing the average of a very large number of sufficiently smooth functions. Our stochastic framework is based on the notion of stochastic higher-order upper bound approximations of the finite-sum objective function and minibatching. We derive convergence results for nonconvex and convex optimization problems when the higher-order approximation of the objective function yields an error that is p times differentiable and has Lipschitz continuous p derivative. More precisely, for general nonconvex problems we present asymptotic stationary point guarantees and under Kurdyka-Lojasiewicz property we derive local convergence rates ranging from sublinear to linear. For convex problems with uniformly convex objective function we derive local (super)linear convergence results for our algorithm. Compared to existing stochastic (first-order) methods, our algorithm adapts to the problem's curvature and allows using any batch size. Preliminary numerical tests support the effectiveness of our algorithmic framework.Comment: 28 page

Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization

Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t.~the Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We further prove the faster, linear convergence of our methods when a Polyak-{\L}ojasiewicz (P{\L}) condition holds for the objective function. To the best of our knowledge, our work is the first to provide variance-reduced convergence guarantees for a cyclic block coordinate method. Our experimental results demonstrate the efficacy of the proposed variance-reduced cyclic scheme in training deep neural nets