Stochastic convex optimization algorithms are the most popular way to train machine learning models on large-scale data. Scaling up the training process of these models is crucial, but the most popular algorithm, Stochastic Gradient Descent (SGD), is a serial method that is surprisingly hard to parallelize. In this paper, we propose an efficient distributed stochastic optimization method by combining adaptivity with variance reduction techniques. Our analysis yields a linear speedup in the number of machines, constant memory footprint, and only a logarithmic number of communication rounds. Critically, our approach is a black-box reduction that parallelizes any serial online learning algorithm, streamlining prior analysis and allowing us to leverage the significant progress that has been made in designing adaptive algorithms. In particular, we achieve optimal convergence rates without any prior knowledge of smoothness parameters, yielding a more robust algorithm that reduces the need for hyperparameter tuning. We implement our algorithm in the Spark distributed framework and exhibit dramatic performance gains on large-scale logistic regression problems.

Uncertainty sampling, a popular active learning algorithm, is used to reduce the amount of data required to learn a classifier, but it has been observed in practice to converge to different parameters depending on the initialization and sometimes to even better parameters than standard training on all the data. In this work, we give a theoretical explanation of this phenomenon, showing that uncertainty sampling on a convex (e.g., logistic) loss can be interpreted as performing a preconditioned stochastic gradient step on the population zero-one loss. Experiments on synthetic and real datasets support this connection.

Neural networks have many successful applications, while much less theoretical understanding has been gained. Towards bridging this gap, we study the problem of learning a two-layer overparameterized ReLU neural network for multi-class classification via stochastic gradient descent (SGD) from random initialization. In the overparameterized setting, when the data comes from mixtures of well-separated distributions, we prove that SGD learns a network with a small generalization error, albeit the network has enough capacity to fit arbitrary labels. Furthermore, the analysis provides interesting insights into several aspects of learning neural networks and can be verified based on empirical studies on synthetic data and on the MNIST dataset.

In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization. Specifically, we propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. Furthermore, under a structural sparsity assumption, we first illustrate an implicit regularization phenomenon where the standard stochastic gradient algorithm with zeroth-order information adapts to the sparsity of the problem at hand by just varying the step-size. Next, we propose a truncated stochastic gradient algorithm with zeroth-order information, whose rate of convergence depends only poly-logarithmically on the dimensionality.

Distributed model training suffers from communication overheads due to frequent gradient updates transmitted between compute nodes. To mitigate these overheads, several studies propose the use of sparsified stochastic gradients. We argue that these are facets of a general sparsification method that can operate on any possible atomic decomposition. Notable examples include element-wise, singular value, and Fourier decompositions.

Stochastic Gradient Langevin Dynamics (SGLD) has emerged as a key MCMC algorithm for Bayesian learning from large scale datasets. While SGLD with decreasing step sizes converges weakly to the posterior distribution, the algorithm is often used with a constant step size in practice and has demonstrated spectacular successes in machine learning tasks. The current practice is to set the step size inversely proportional to N where N is the number of training samples. As N becomes large, we show that the SGLD algorithm has an invariant probability measure which significantly departs from the target posterior and behaves like as Stochastic Gradient Descent (SGD). This difference is inherently due to the high variance of the stochastic gradients. Several strategies have been suggested to reduce this effect; among them, SGLD Fixed Point (SGLDFP) uses carefully designed control variates to reduce the variance of the stochastic gradients. We show that SGLDFP gives approximate samples from the posterior distribution, with an accuracy comparable to the Langevin Monte Carlo (LMC) algorithm for a computational cost sublinear in the number of data points. We provide a detailed analysis of the Wasserstein distances between LMC, SGLD, SGLDFP and SGD and explicit expressions of the means and covariance matrices of their invariant distributions. Our findings are supported by limited numerical experiments.

Distributed training of massive machine learning models, in particular deep neural networks, via Stochastic Gradient Descent (SGD) is becoming commonplace. Several families of communication-reduction methods, such as quantization, large-batch methods, and gradient sparsification, have been proposed. To date, gradient sparsification methods--where each node sorts gradients by magnitude, and only communicates a subset of the components, accumulating the rest locally--are known to yield some of the largest practical gains. Such methods can reduce the amount of communication per step by up to \emph{three orders of magnitude}, while preserving model accuracy. Yet, this family of methods currently has no theoretical justification. This is the question we address in this paper. We prove that, under analytic assumptions, sparsifying gradients by magnitude with local error correction provides convergence guarantees, for both convex and non-convex smooth objectives, for data-parallel SGD. The main insight is that sparsification methods implicitly maintain bounds on the maximum impact of stale updates, thanks to selection by magnitude. Our analysis and empirical validation also reveal that these methods do require analytical conditions to converge well, justifying existing heuristics.

A residual network (or ResNet) is a standard deep neural net architecture, with state-of-the-art performance across numerous applications. The main premise of ResNets is that they allow the training of each layer to focus on fitting just the residual of the previous layer's output and the target output. Thus, we should expect that the trained network is no worse than what we can obtain if we remove the residual layers and train a shallower network instead. However, due to the non-convexity of the optimization problem, it is not at all clear that ResNets indeed achieve this behavior, rather than getting stuck at some arbitrarily poor local minimum. In this paper, we rigorously prove that arbitrarily deep, nonlinear residual units indeed exhibit this behavior, in the sense that the optimization landscape contains no local minima with value above what can be obtained with a linear predictor (namely a 1-layer network). Notably, we show this under minimal or no assumptions on the precise network architecture, data distribution, or loss function used. We also provide a quantitative analysis of approximate stationary points for this problem. Finally, we show that with a certain tweak to the architecture, training the network with standard stochastic gradient descent achieves an objective value close or better than any linear predictor.

Distributed stochastic gradient descent is an important subroutine in distributed learning. A setting of particular interest is when the clients are mobile devices, where two important concerns are communication efficiency and the privacy of the clients. Several recent works have focused on reducing the communication cost or introducing privacy guarantees, but none of the proposed communication efficient methods are known to be privacy preserving and none of the known privacy mechanisms are known to be communication efficient. To this end, we study algorithms that achieve both communication efficiency and differential privacy. For $d$ variables and $n \approx d$ clients, the proposed method uses $\cO(\log \log(nd))$ bits of communication per client per coordinate and ensures constant privacy. We also improve previous analysis of the \emph{Binomial mechanism} showing that it achieves nearly the same utility as the Gaussian mechanism, while requiring fewer representation bits, which can be of independent interest.

The performance of neural networks on high-dimensional data distributions suggests that it may be possible to parameterize a representation of a given high-dimensional function with controllably small errors, potentially outperforming standard interpolation methods. We demonstrate, both theoretically and numerically, that this is indeed the case. We map the parameters of a neural network to a system of particles relaxing with an interaction potential determined by the loss function. We show that in the limit that the number of parameters $n$ is large, the landscape of the mean-squared error becomes convex and the representation error in the function scales as $O(n^{-1})$. In this limit, we prove a dynamical variant of the universal approximation theorem showing that the optimal representation can be attained by stochastic gradient descent, the algorithm ubiquitously used for parameter optimization in machine learning. In the asymptotic regime, we study the fluctuations around the optimal representation and show that they arise at a scale $O(n^{-1})$. These fluctuations in the landscape identify the natural scale for the noise in stochastic gradient descent. Our results apply to both single and multi-layer neural networks, as well as standard kernel methods like radial basis functions.