Approximating Stochastic Gradient Descent (SGD) as a Stochastic Differential Equation (SDE) has allowed researchers to enjoy the benefits of studying a continuous optimization trajectory while carefully preserving the stochasticity of SGD. Analogous study of adaptive gradient methods, such as RMSprop and Adam, has been challenging because there were no rigorously proven SDE approximations for these methods. This paper derives the SDE approximations for RMSprop and Adam, giving theoretical guarantees of their correctness as well as experimental validation of their applicability to common large-scaling vision and language settings. A key practical result is the derivation of a square root scaling rule to adjust the optimization hyperparameters of RMSprop and Adam when changing batch size, and its empirical validation in deep learning settings.

We take a Bayesian perspective to illustrate a connection between training speed and the marginal likelihood in linear models. This provides two major insights: first, that a measure of a model's training speed can be used to estimate its marginal likelihood. Second, that this measure, under certain conditions, predicts the relative weighting of models in linear model combinations trained to minimize a regression loss. We verify our results in model selection tasks for linear models and for the infinite-width limit of deep neural networks. We further provide encouraging empirical evidence that the intuition developed in these settings also holds for deep neural networks trained with stochastic gradient descent.

As in standard linear regression, in truncated linear regression, we are given access to observations (Ai, yi)i whose dependent variable equals yi Ai {\rm T} \cdot x * \etai, where x * is some fixed unknown vector of interest and \etai is independent noise; except we are only given an observation if its dependent variable yi lies in some "truncation set" S \subset \mathbb{R}. The goal is to recover x * under some favorable conditions on the Ai's and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering k-sparse n-dimensional vectors x * from m truncated samples, which attains an optimal \ell2 reconstruction error of O(\sqrt{(k \log n)/m}). As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, such as that which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaption of the LASSO optimization problem that accommodates truncation.

The vast majority of successful deep neural networks are trained using variants of stochastic gradient descent (SGD) algorithms. Recent attempts to improve SGD can be broadly categorized into two approaches: (1) adaptive learning rate schemes, such as AdaGrad and Adam and (2) accelerated schemes, such as heavy-ball and Nesterov momentum. In this paper, we propose a new optimization algorithm, Lookahead, that is orthogonal to these previous approaches and iteratively updates two sets of weights. Intuitively, the algorithm chooses a search direction by looking ahead at the sequence of fast weights" generated by another optimizer. We show that Lookahead improves the learning stability and lowers the variance of its inner optimizer with negligible computation and memory cost.

Metric learning aims to learn a distance measure that can benefit distance-based methods such as the nearest neighbour (NN) classifier. While considerable efforts have been made to improve its empirical performance and analyze its generalization ability by focusing on the data structure and model complexity, an unresolved question is how choices of algorithmic parameters, such as the number of training iterations, affect metric learning as it is typically formulated as an optimization problem and nowadays more often as a non-convex problem. In this paper, we theoretically address this question and prove the agnostic Probably Approximately Correct (PAC) learnability for metric learning algorithms with non-convex objective functions optimized via gradient descent (GD); in particular, our theoretical guarantee takes the iteration number into account. We first show that the generalization PAC bound is a sufficient condition for agnostic PAC learnability and this bound can be obtained by ensuring the uniform convergence on a densely concentrated subset of the parameter space. We then show that, for classifiers optimized via GD, their generalizability can be guaranteed if the classifier and loss function are both Lipschitz smooth, and further improved by using fewer iterations.

As the use of large embedding models in recommendation systems and language applications increases, concerns over user data privacy have also risen. DP-SGD, a training algorithm that combines differential privacy with stochastic gradient descent, has been the workhorse in protecting user privacy without compromising model accuracy by much. However, applying DP-SGD naively to embedding models can destroy gradient sparsity, leading to reduced training efficiency. To address this issue, we present two new algorithms, DP-FEST and DP-AdaFEST, that preserve gradient sparsity during the private training of large embedding models. Our algorithms achieve substantial reductions ( 10 6 \times) in gradient size, while maintaining comparable levels of accuracy, on benchmark real-world datasets.

We analyze the convergence rates of stochastic gradient algorithms for smooth finite-sum minimax optimization and show that, for many such algorithms, sampling the data points \emph{without replacement} leads to faster convergence compared to sampling with replacement. For the smooth and strongly convex-strongly concave setting, we consider gradient descent ascent and the proximal point method, and present a unified analysis of two popular without-replacement sampling strategies, namely \emph{Random Reshuffling} (RR), which shuffles the data every epoch, and \emph{Single Shuffling} or \emph{Shuffle Once} (SO), which shuffles only at the beginning. We obtain tight convergence rates for RR and SO and demonstrate that these strategies lead to faster convergence than uniform sampling. Moving beyond convexity, we obtain similar results for smooth nonconvex-nonconcave objectives satisfying a two-sided Polyak-\L{}ojasiewicz inequality. Finally, we demonstrate that our techniques are general enough to analyze the effect of \emph{data-ordering attacks}, where an adversary manipulates the order in which data points are supplied to the optimizer.

Several works have proposed Simplicity Bias (SB)---the tendency of standard training procedures such as Stochastic Gradient Descent (SGD) to find simple models---to justify why neural networks generalize well [Arpit et al. 2017, Nakkiran et al. 2019, Valle-Perez et al. 2019]. However, the precise notion of simplicity remains vague. Furthermore, previous settings [Soudry et al. 2018, Gunasekar et al. 2018] that use SB to theoretically justify why neural networks generalize well do not simultaneously capture the non-robustness of neural networks---a widely observed phenomenon in practice [Goodfellow et al. 2014, Jo and Bengio 2017]. We attempt to reconcile SB and the superior standard generalization of neural networks with the non-robustness observed in practice by introducing piecewise-linear and image-based datasets, which (a) incorporate a precise notion of simplicity, (b) comprise multiple predictive features with varying levels of simplicity, and (c) capture the non-robustness of neural networks trained on real data. Using theory and empirics on these datasets, we make four observations: (i) SB of SGD and variants can be extreme: neural networks can exclusively rely on the simplest feature and remain invariant to all predictive complex features. Given the role of SB in training neural networks, we hope that the proposed datasets and methods serve as an effective testbed to evaluate novel algorithmic approaches aimed at avoiding the pitfalls of SB.

We analyze in a closed form the learning dynamics of stochastic gradient descent (SGD) for a single layer neural network classifying a high-dimensional Gaussian mixture where each cluster is assigned one of two labels. This problem provides a prototype of a non-convex loss landscape with interpolating regimes and a large generalization gap. We define a particular stochastic process for which SGD can be extended to a continuous-time limit that we call stochastic gradient flow. In the full-batch limit we recover the standard gradient flow. We apply dynamical mean-field theory from statistical physics to track the dynamics of the algorithm in the high-dimensional limit via a self-consistent stochastic process.

Stochastic gradient descent (SGD) is a prevalent optimization technique for large-scale distributed machine learning. While SGD computation can be efficiently divided between multiple machines, communication typically becomes a bottleneck in the distributed setting. Gradient compression methods can be used to alleviate this problem, and a recent line of work shows that SGD augmented with gradient compression converges to an \varepsilon -first-order stationary point. In this paper we extend these results to convergence to an \varepsilon -second-order stationary point ( \varepsilon -SOSP), which is to the best of our knowledge the first result of this type. In addition, we show that, when the stochastic gradient is not Lipschitz, compressed SGD with RandomK compressor converges to an \varepsilon -SOSP with the same number of iterations as uncompressed SGD [Jin et al.,2021] (JACM), while improving the total communication by a factor of \tilde \Theta(\sqrt{d} \varepsilon {-3/4}), where d is the dimension of the optimization problem.