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. We present additional results for the cases when the compressor is arbitrary and when the stochastic gradient is Lipschitz.

Structural equation models (SEMs) are widely used in sciences, ranging from economics to psychology, to uncover causal relationships underlying a complex system under consideration and estimate structural parameters of interest. We study estimation in a class of generalized SEMs where the object of interest is defined as the solution to a linear operator equation. We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using the stochastic gradient descent. We consider both 2-layer and multi-layer NNs with ReLU activation functions and prove global convergence in an overparametrized regime, where the number of neurons is diverging. The results are established using techniques from online learning and local linearization of NNs, and improve in several aspects the current state-of-the-art. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.

Several works have aimed to explain why overparameterized neural networks generalize well when trained by Stochastic Gradient Descent (SGD). The consensus explanation that has emerged credits the randomized nature of SGD for the bias of the training process towards low-complexity models and, thus, for implicit regularization. We take a careful look at this explanation in the context of image classification with common deep neural network architectures. We find that if we do not regularize \emph{explicitly}, then SGD can be easily made to converge to poorly-generalizing, high-complexity models: all it takes is to first train on a random labeling on the data, before switching to properly training with the correct labels. In contrast, we find that in the presence of explicit regularization, pretraining with random labels has no detrimental effect on SGD. We believe that our results give evidence that explicit regularization plays a far more important role in the success of overparameterized neural networks than what has been understood until now. Specifically, in suppressing complicated models that got lucky with the training data, regularization not only makes simple models that fit the data well the global optima, but it also clears the way to make them discoverable by local methods, such as SGD.

We consider the problem of stochastic optimization with delayed gradients in which, at each time step $t$, the algorithm makes an update using a stale stochastic gradient from step $t - d_t$ for some arbitrary delay $d_t$. This setting abstracts asynchronous distributed optimization where a central server receives gradient updates computed by worker machines. These machines can experience computation and communication loads that might vary significantly over time. In the general non-convex smooth optimization setting, we give a simple and efficient algorithm that requires $O( \sigma^2/\epsilon^4 + \tau/\epsilon^2)$ steps for finding an $\epsilon$-stationary point $x$. Here, $\tau$ is the \emph{average} delay $\frac{1}{T}\sum_{t=1}^T d_t$ and $\sigma^2$ is the variance of the stochastic gradients. This improves over previous work, which showed that stochastic gradient decent achieves the same rate but with respect to the \emph{maximal} delay $\max_{t} d_t$, that can be significantly larger than the average delay especially in heterogeneous distributed systems. Our experiments demonstrate the efficacy and robustness of our algorithm in cases where the delay distribution is skewed or heavy-tailed.

Inverse problems are paramount in Science and Engineering. In this paper, we consider the setup of Statistical Inverse Problem (SIP) and demonstrate how Stochastic Gradient Descent (SGD) algorithms can be used to solve linear SIP. We provide consistency and finite sample bounds for the excess risk. We also propose a modification for the SGD algorithm where we leverage machine learning methods to smooth the stochastic gradients and improve empirical performance.

We prove the Fast Equilibrium Conjecture proposed by Li et al., (2020), i.e., stochastic gradient descent (SGD) on a scale-invariant loss (e.g., using networks with various normalization schemes) with learning rate $\eta$ and weight decay factor $\lambda$ mixes in function space in $\mathcal{\tilde{O}}(\frac{1}{\lambda\eta})$ steps, under two standard assumptions: (1) the noise covariance matrix is non-degenerate and (2) the minimizers of the loss form a connected, compact and analytic manifold. The analysis uses the framework of Li et al., (2021) and shows that for every $T> 0$, the iterates of SGD with learning rate $\eta$ and weight decay factor $\lambda$ on the scale-invariant loss converge in distribution in $\Theta\left(\eta^{-1}\lambda^{-1}(T+\ln(\lambda/\eta))\right)$ iterations as $\eta\lambda\to 0$ while satisfying $\eta \le O(\lambda)\le O(1)$. Moreover, the evolution of the limiting distribution can be described by a stochastic differential equation that mixes to the same equilibrium distribution for every initialization around the manifold of minimizers as $T\to\infty$.

The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of Stochastic Convex Optimization.

In this paper we propose a general framework to perform statistical online inference in a class of constant step size stochastic approximation (SA) problems, including the well-known stochastic gradient descent (SGD) and Q-learning. Regarding a constant step size SA procedure as a time-homogeneous Markov chain, we establish a functional central limit theorem (FCLT) for it under weaker conditions, and then construct confidence intervals for parameters via random scaling. To leverage the FCLT results in the Markov chain setting, an alternative condition that is more applicable for SA problems is established. We conduct experiments to perform inference with both random scaling and other traditional inference methods, and finds that the former has a more accurate and robust performance.

Although the optimization objectives for learning neural networks are highly non-convex, gradient-based methods have been wildly successful at learning neural networks in practice. This juxtaposition has led to a number of recent studies on provable guarantees for neural networks trained by gradient descent. Unfortunately, the techniques in these works are often highly specific to the particular setup in each problem, making it difficult to generalize across different settings. To address this drawback in the literature, we propose a unified non-convex optimization framework for the analysis of neural network training. We introduce the notions of proxy convexity and proxy Polyak-Lojasiewicz (PL) inequalities, which are satisfied if the original objective function induces a proxy objective function that is implicitly minimized when using gradient methods. We show that stochastic gradient descent (SGD) on objectives satisfying proxy convexity or the proxy PL inequality leads to efficient guarantees for proxy objective functions. We further show that many existing guarantees for neural networks trained by gradient descent can be unified through proxy convexity and proxy PL inequalities.

We consider a standard distributed optimisation setting where $N$ machines, each holding a $d$-dimensional function $f_i$, aim to jointly minimise the sum of the functions $\sum_{i = 1}^N f_i (x)$. This problem arises naturally in large-scale distributed optimisation, where a standard solution is to apply variants of (stochastic) gradient descent. We focus on the communication complexity of this problem: our main result provides the first fully unconditional bounds on total number of bits which need to be sent and received by the $N$ machines to solve this problem under point-to-point communication, within a given error-tolerance. Specifically, we show that $\Omega( Nd \log d / N\varepsilon)$ total bits need to be communicated between the machines to find an additive $\epsilon$-approximation to the minimum of $\sum_{i = 1}^N f_i (x)$. The result holds for both deterministic and randomised algorithms, and, importantly, requires no assumptions on the algorithm structure. The lower bound is tight under certain restrictions on parameter values, and is matched within constant factors for quadratic objectives by a new variant of quantised gradient descent, which we describe and analyse. Our results bring over tools from communication complexity to distributed optimisation, which has potential for further applications.