We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that stochastic gradient descent converges to a local minimum in a polynomial number of iterations. To the best of our knowledge this is the first work that gives global convergence guarantees for stochastic gradient descent on non-convex functions with exponentially many local minima and saddle points. Our analysis can be applied to orthogonal tensor decomposition, which is widely used in learning a rich class of latent variable models. We propose a new optimization formulation for the tensor decomposition problem that has strict saddle property. As a result we get the first online algorithm for orthogonal tensor decomposition with global convergence guarantee.

The stochastic gradient (SG) method can minimize an objective function composed of a large number of differentiable functions, or solve a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, handles problems with multiple blocks of variables by updating them one at a time; when the blocks of variables are easier to update individually than together, BCD has a lower per-iteration cost. This paper introduces a method that combines the features of SG and BCD for problems with many components in the objective and with multiple (blocks of) variables. Specifically, a block stochastic gradient (BSG) method is proposed for solving both convex and nonconvex programs. At each iteration, BSG approximates the gradient of the differentiable part of the objective by randomly sampling a small set of data or sampling a few functions from the sum term in the objective, and then, using those samples, it updates all the blocks of variables in either a deterministic or a randomly shuffled order. Its convergence for both convex and nonconvex cases are established in different senses. In the convex case, the proposed method has the same order of convergence rate as the SG method. In the nonconvex case, its convergence is established in terms of the expected violation of a first-order optimality condition. The proposed method was numerically tested on problems including stochastic least squares and logistic regression, which are convex, as well as low-rank tensor recovery and bilinear logistic regression, which are nonconvex.

Stochastic gradient descent (SGD) on a low-rank factorization is commonly employed to speed up matrix problems including matrix completion, subspace tracking, and SDP relaxation. In this paper, we exhibit a step size scheme for SGD on a low-rank least-squares problem, and we prove that, under broad sampling conditions, our method converges globally from a random starting point within $O(\epsilon^{-1} n \log n)$ steps with constant probability for constant-rank problems. Our modification of SGD relates it to stochastic power iteration. We also show experiments to illustrate the runtime and convergence of the algorithm.

We study regularized estimation in high-dimensional longitudinal classification problems, using the lasso and fused lasso regularizers. The constructed coefficient estimates are piecewise constant across the time dimension in the longitudinal problem, with adaptively selected change points (break points). We present an efficient algorithm for computing such estimates, based on proximal gradient descent. We apply our proposed technique to a longitudinal data set on Alzheimer's disease from the Cardiovascular Health Study Cognition Study, and use this data set to motivate and demonstrate several practical considerations such as the selection of tuning parameters, and the assessment of model stability.

Determining the 3D structures of biological molecules is a key problem for both biology and medicine. Electron Cryomicroscopy (Cryo-EM) is a promising technique for structure estimation which relies heavily on computational methods to reconstruct 3D structures from 2D images. This paper introduces the challenging Cryo-EM density estimation problem as a novel application for stochastic optimization techniques. Structure discovery is formulated as MAP estimation in a probabilistic latent-variable model, resulting in an optimization problem to which an array of seven stochastic optimization methods are applied. The methods are tested on both real and synthetic data, with some methods recovering reasonable structures in less than one epoch from a random initialization. Complex quasi-Newton methods are found to converge more slowly than simple gradient-based methods, but all stochastic methods are found to converge to similar optima. This method represents a major improvement over existing methods as it is significantly faster and is able to converge from a random initialization.

We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning $(L/\mu)^2$ (where $L$ is a bound on the smoothness and $\mu$ on the strong convexity) to a linear dependence on $L/\mu$. Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate.

In certain situations that shall be undoubtedly more and more common in the Big Data era, the datasets available are so massive that computing statistics over the full sample is hardly feasible, if not unfeasible. A natural approach in this context consists in using survey schemes and substituting the "full data" statistics with their counterparts based on the resulting random samples, of manageable size. It is the main purpose of this paper to investigate the impact of survey sampling with unequal inclusion probabilities on stochastic gradient descent-based M-estimation methods in large-scale statistical and machine-learning problems. Precisely, we prove that, in presence of some a priori information, one may significantly increase asymptotic accuracy when choosing appropriate first order inclusion probabilities, without affecting complexity. These striking results are described here by limit theorems and are also illustrated by numerical experiments.

Uniform sampling of training data has been commonly used in traditional stochastic optimization algorithms such as Proximal Stochastic Gradient Descent (prox-SGD) and Proximal Stochastic Dual Coordinate Ascent (prox-SDCA). Although uniform sampling can guarantee that the sampled stochastic quantity is an unbiased estimate of the corresponding true quantity, the resulting estimator may have a rather high variance, which negatively affects the convergence of the underlying optimization procedure. In this paper we study stochastic optimization with importance sampling, which improves the convergence rate by reducing the stochastic variance. Specifically, we study prox-SGD (actually, stochastic mirror descent) with importance sampling and prox-SDCA with importance sampling. For prox-SGD, instead of adopting uniform sampling throughout the training process, the proposed algorithm employs importance sampling to minimize the variance of the stochastic gradient. For prox-SDCA, the proposed importance sampling scheme aims to achieve higher expected dual value at each dual coordinate ascent step. We provide extensive theoretical analysis to show that the convergence rates with the proposed importance sampling methods can be significantly improved under suitable conditions both for prox-SGD and for prox-SDCA. Experiments are provided to verify the theoretical analysis.

We provide novel theoretical results regarding local optima of regularized $M$-estimators, allowing for nonconvexity in both loss and penalty functions. Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, we prove that \emph{any stationary point} of the composite objective function will lie within statistical precision of the underlying parameter vector. Our theory covers many nonconvex objective functions of interest, including the corrected Lasso for errors-in-variables linear models; regression for generalized linear models with nonconvex penalties such as SCAD, MCP, and capped-$\ell_1$; and high-dimensional graphical model estimation. We quantify statistical accuracy by providing bounds on the $\ell_1$-, $\ell_2$-, and prediction error between stationary points and the population-level optimum. We also propose a simple modification of composite gradient descent that may be used to obtain a near-global optimum within statistical precision $\epsilon$ in $\log(1/\epsilon)$ steps, which is the fastest possible rate of any first-order method. We provide simulation studies illustrating the sharpness of our theoretical results.

We formulate the problem of metric learning for k nearest neighbor classification as a large margin structured prediction problem, with a latent variable representing the choice of neighbors and the task loss directly corresponding to classification error. We describe an efficient algorithm for exact loss augmented inference, and a fast gradient descent algorithm for learning in this model. The objective drives the metric to establish neighborhood boundaries that benefit the true class labels for the training points. Our approach, reminiscent of gerrymandering (redrawing of political boundaries to provide advantage to certain parties), is more direct in its handling of optimizing classification accuracy than those previously proposed. In experiments on a variety of data sets our method is shown to achieve excellent results compared to current state of the art in metric learning.