Efficient Algorithms for Smooth Minimax Optimization

This paper studies first order methods for solving smooth minimax optimization problems $\min_x \max_y g(x,y)$ where $g(\cdot,\cdot)$ is smooth and $g(x,\cdot)$ is concave for each $x$. In terms of $g(\cdot,y)$, we consider two settings -- strongly convex and nonconvex -- and improve upon the best known rates in both. For strongly-convex $g(\cdot, y),\ \forall y$, we propose a new algorithm combining Mirror-Prox and Nesterov's AGD, and show that it can find global optimum in $\tilde{O}(1/k^2)$ iterations, improving over current state-of-the-art rate of $O(1/k)$. We use this result along with an inexact proximal point method to provide $\tilde{O}(1/k^{1/3})$ rate for finding stationary points in the nonconvex setting where $g(\cdot, y)$ can be nonconvex. This improves over current best-known rate of $O(1/k^{1/5})$. Finally, we instantiate our result for finite nonconvex minimax problems, i.e., $\min_x \max_{1\leq i\leq m} f_i(x)$, with nonconvex $f_i(\cdot)$, to obtain convergence rate of $O(m(\log m)^{3/2}/k^{1/3})$ total gradient evaluations for finding a stationary point.

Making the Last Iterate of SGD Information Theoretically Optimal

Stochastic gradient descent (SGD) is one of the most widely used algorithms for large scale optimization problems. While classical theoretical analysis of SGD for convex problems studies (suffix) \emph{averages} of iterates and obtains information theoretically optimal bounds on suboptimality, the \emph{last point} of SGD is, by far, the most preferred choice in practice. The best known results for last point of SGD \cite{shamir2013stochastic} however, are suboptimal compared to information theoretic lower bounds by a $\log T$ factor, where $T$ is the number of iterations. \cite{harvey2018tight} shows that in fact, this additional $\log T$ factor is tight for standard step size sequences of $\OTheta{\frac{1}{\sqrt{t}}}$ and $\OTheta{\frac{1}{t}}$ for non-strongly convex and strongly convex settings, respectively. Similarly, even for subgradient descent (GD) when applied to non-smooth, convex functions, the best known step-size sequences still lead to $O(\log T)$-suboptimal convergence rates (on the final iterate). The main contribution of this work is to design new step size sequences that enjoy information theoretically optimal bounds on the suboptimality of \emph{last point} of SGD as well as GD. We achieve this by designing a modification scheme, that converts one sequence of step sizes to another so that the last point of SGD/GD with modified sequence has the same suboptimality guarantees as the average of SGD/GD with original sequence. We also show that our result holds with high-probability. We validate our results through simulations which demonstrate that the new step size sequence indeed improves the final iterate significantly compared to the standard step size sequences.

Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression

We study the problem of robust linear regression with response variable corruptions. We consider the oblivious adversary model, where the adversary corrupts a fraction of the responses in complete ignorance of the data. We provide a nearly linear time estimator which consistently estimates the true regression vector, even with $1-o(1)$ fraction of corruptions. Existing results in this setting either don't guarantee consistent estimates or can only handle a small fraction of corruptions. We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting. Finally, we apply our estimator to the problem of linear regression with heavy-tailed noise and show that our estimator consistently estimates the regression vector even when the noise has unbounded variance (e.g., Cauchy distribution), for which most existing results don't even apply. Our estimator is based on a novel variant of outlier removal via hard thresholding in which the threshold is chosen adaptively and crucially relies on randomness to escape bad fixed points of the non-convex hard thresholding operation.

SGD without Replacement: Sharper Rates for General Smooth Convex Functions

We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement}~\cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence properties are not well understood as sampling without replacement leads to coupling between iterates and gradients. By using method of exchangeable pairs to bound Wasserstein distance, we provide the first non-asymptotic results for \sgdwor when applied to {\em general smooth, strongly-convex} functions. In particular, we show that \sgdwor converges at a rate of $O(1/K^2)$ while \sgd~is known to converge at $O(1/K)$ rate, where $K$ denotes the number of passes over data and is required to be {\em large enough}. Existing results for \sgdwor in this setting require additional {\em Hessian Lipschitz assumption}~\cite{gurbuzbalaban2015random,haochen2018random}. For {\em small} $K$, we show \sgdwor can achieve same convergence rate as \sgd for {\em general smooth strongly-convex} functions. Existing results in this setting require $K=1$ and hold only for generalized linear models \cite{shamir2016without}. In addition, by careful analysis of the coupling, for both large and small $K$, we obtain better dependence on problem dependent parameters like condition number.

FastGRNN: A Fast, Accurate, Stable and Tiny Kilobyte Sized Gated Recurrent Neural Network

This paper develops the FastRNN and FastGRNN algorithms to address the twin RNN limitations of inaccurate training and inefficient prediction. Previous approaches have improved accuracy at the expense of prediction costs making them infeasible for resource-constrained and real-time applications. Unitary RNNs have increased accuracy somewhat by restricting the range of the state transition matrix's singular values but have also increased the model size as they require a larger number of hidden units to make up for the loss in expressive power. Gated RNNs have obtained state-of-the-art accuracies by adding extra parameters thereby resulting in even larger models. FastRNN addresses these limitations by adding a residual connection that does not constrain the range of the singular values explicitly and has only two extra scalar parameters. FastGRNN then extends the residual connection to a gate by reusing the RNN matrices to match state-of-the-art gated RNN accuracies but with a 2-4x smaller model. Enforcing FastGRNN's matrices to be low-rank, sparse and quantized resulted in accurate models that could be up to 35x smaller than leading gated and unitary RNNs. This allowed FastGRNN to accurately recognize the "Hey Cortana" wakeword with a 1 KB model and to be deployed on severely resource-constrained IoT microcontrollers too tiny to store other RNN models. FastGRNN's code is available at https://github.com/Microsoft/EdgeML/.

Neural-Guided Deductive Search for Real-Time Program Synthesis from Examples

Synthesizing user-intended programs from a small number of input-output examples is a challenging problem with several important applications like spreadsheet manipulation, data wrangling and code refactoring. Existing synthesis systems either completely rely on deductive logic techniques that are extensively hand-engineered or on purely statistical models that need massive amounts of data, and in general fail to provide real-time synthesis on challenging benchmarks. In this work, we propose Neural Guided Deductive Search (NGDS), a hybrid synthesis technique that combines the best of both symbolic logic techniques and statistical models. Thus, it produces programs that satisfy the provided specifications by construction and generalize well on unseen examples, similar to data-driven systems. Our technique effectively utilizes the deductive search framework to reduce the learning problem of the neural component to a simple supervised learning setup. Further, this allows us to both train on sparingly available real-world data and still leverage powerful recurrent neural network encoders. We demonstrate the effectiveness of our method by evaluating on real-world customer scenarios by synthesizing accurate programs with up to 12x speed-up compared to state-of-the-art systems.

On the insufficiency of existing momentum schemes for Stochastic Optimization

Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov's accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). Rigorously speaking, "fast gradient" methods have provable improvements over gradient descent only for the deterministic case, where the gradients are exact. In the stochastic case, the popular explanations for their wide applicability is that when these fast gradient methods are applied in the stochastic case, they partially mimic their exact gradient counterparts, resulting in some practical gain. This work provides a counterpoint to this belief by proving that there exist simple problem instances where these methods cannot outperform SGD despite the best setting of its parameters. These negative problem instances are, in an informal sense, generic; they do not look like carefully constructed pathological instances. These results suggest (along with empirical evidence) that HB or NAG's practical performance gains are a by-product of mini-batching. Furthermore, this work provides a viable (and provable) alternative, which, on the same set of problem instances, significantly improves over HB, NAG, and SGD's performance. This algorithm, referred to as Accelerated Stochastic Gradient Descent (ASGD), is a simple to implement stochastic algorithm, based on a relatively less popular variant of Nesterov's Acceleration. Extensive empirical results in this paper show that ASGD has performance gains over HB, NAG, and SGD.

Accelerating Stochastic Gradient Descent For Least Squares Regression

There is widespread sentiment that it is not possible to effectively utilize fast gradient methods (e.g. Nesterov's acceleration, conjugate gradient, heavy ball) for the purposes of stochastic optimization due to their instability and error accumulation, a notion made precise in d'Aspremont 2008 and Devolder, Glineur, and Nesterov 2014. This work considers these issues for the special case of stochastic approximation for the least squares regression problem, and our main result refutes the conventional wisdom by showing that acceleration can be made robust to statistical errors. In particular, this work introduces an accelerated stochastic gradient method that provably achieves the minimax optimal statistical risk faster than stochastic gradient descent. Critical to the analysis is a sharp characterization of accelerated stochastic gradient descent as a stochastic process. We hope this characterization gives insights towards the broader question of designing simple and effective accelerated stochastic methods for more general convex and non-convex optimization problems.

Parallelizing Stochastic Gradient Descent for Least Squares Regression: mini-batching, averaging, and model misspecification

This work characterizes the benefits of averaging schemes widely used in conjunction with stochastic gradient descent (SGD). In particular, this work provides a sharp analysis of: (1) mini-batching, a method of averaging many samples of a stochastic gradient to both reduce the variance of the stochastic gradient estimate and for parallelizing SGD and (2) tail-averaging, a method involving averaging the final few iterates of SGD to decrease the variance in SGD's final iterate. This work presents non-asymptotic excess risk bounds for these schemes for the stochastic approximation problem of least squares regression. Furthermore, this work establishes a precise problem-dependent extent to which mini-batch SGD yields provable near-linear parallelization speedups over SGD with batch size one. This allows for understanding learning rate versus batch size tradeoffs for the final iterate of an SGD method. These results are then utilized in providing a highly parallelizable SGD method that obtains the minimax risk with nearly the same number of serial updates as batch gradient descent, improving significantly over existing SGD methods. A non-asymptotic analysis of communication efficient parallelization schemes such as model-averaging/parameter mixing methods is then provided. Finally, this work sheds light on some fundamental differences in SGD's behavior when dealing with agnostic noise in the (non-realizable) least squares regression problem. In particular, the work shows that the stepsizes that ensure minimax risk for the agnostic case must be a function of the noise properties. This paper builds on the operator view of analyzing SGD methods, introduced by Defossez and Bach (2015), followed by developing a novel analysis in bounding these operators to characterize the excess risk. These techniques are of broader interest in analyzing computational aspects of stochastic approximation.