Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping

Recently, the study of heavy-tailed noises in first-order nonconvex stochastic optimization has gotten a lot of attention since it was recognized as a more realistic condition as suggested by many empirical observations. Specifically, the stochastic noise (the difference between the stochastic and true gradient) is considered only to have a finite $\mathfrak{p}$-th moment where $\mathfrak{p}\in\left(1,2\right]$ instead of assuming it always satisfies the classical finite variance assumption. To deal with this more challenging setting, people have proposed different algorithms and proved them to converge at an optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate for smooth objectives after $T$ iterations. Notably, all these new-designed algorithms are based on the same technique - gradient clipping. Naturally, one may want to know whether the clipping method is a necessary ingredient and the only way to guarantee convergence under heavy-tailed noises. In this work, by revisiting the existing Batched Normalized Stochastic Gradient Descent with Momentum (Batched NSGDM) algorithm, we provide the first convergence result under heavy-tailed noises but without gradient clipping. Concretely, we prove that Batched NSGDM can achieve the optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate even under the relaxed smooth condition. More interestingly, we also establish the first $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{2\mathfrak{p}}})$ convergence rate in the case where the tail index $\mathfrak{p}$ is unknown in advance, which is arguably the common scenario in practice.

On the Last-Iterate Convergence of Shuffling Gradient Methods

Shuffling gradient methods, which are also known as stochastic gradient descent (SGD) without replacement, are widely implemented in practice, particularly including three popular algorithms: Random Reshuffle (RR), Shuffle Once (SO), and Incremental Gradient (IG). Compared to the empirical success, the theoretical guarantee of shuffling gradient methods was not well-understanding for a long time. Until recently, the convergence rates had just been established for the average iterate for convex functions and the last iterate for strongly convex problems (using squared distance as the metric). However, when using the function value gap as the convergence criterion, existing theories cannot interpret the good performance of the last iterate in different settings (e.g., constrained optimization). To bridge this gap between practice and theory, we prove last-iterate convergence rates for shuffling gradient methods with respect to the objective value even without strong convexity. Our new results either (nearly) match the existing last-iterate lower bounds or are as fast as the previous best upper bounds for the average iterate.

Revisiting the Last-Iterate Convergence of Stochastic Gradient Methods

In the past several years, the convergence of the last iterate of the Stochastic Gradient Descent (SGD) algorithm has triggered people's interest due to its good performance in practice but lack of theoretical understanding. For Lipschitz and convex functions, different works have established the optimal $O(\log(1/\delta)\log T/\sqrt{T})$ or $O(\sqrt{\log(1/\delta)/T})$ high-probability convergence rates for the final iterate, where $T$ is the time horizon and $\delta$ is the failure probability. However, to prove these bounds, all the existing works are limited to compact domains or require almost surely bounded noises. It is natural to ask whether the last iterate of SGD can still guarantee the optimal convergence rate but without these two restrictive assumptions. Besides this important question, there are still lots of theoretical problems lacking an answer. For example, compared with the last iterate convergence of SGD for non-smooth problems, only few results for smooth optimization have yet been developed. Additionally, the existing results are all limited to a non-composite objective and the standard Euclidean norm. It still remains unclear whether the last-iterate convergence can be provably extended to wider composite optimization and non-Euclidean norms. In this work, to address the issues mentioned above, we revisit the last-iterate convergence of stochastic gradient methods and provide the first unified way to prove the convergence rates both in expectation and in high probability to accommodate general domains, composite objectives, non-Euclidean norms, Lipschitz conditions, smoothness and (strong) convexity simultaneously. Additionally, we extend our analysis to obtain the last-iterate convergence under heavy-tailed noises.

STDA-Meta: A Meta-Learning Framework for Few-Shot Traffic Prediction

As the development of cities, traffic congestion becomes an increasingly pressing issue, and traffic prediction is a classic method to relieve that issue. Traffic prediction is one specific application of spatio-temporal prediction learning, like taxi scheduling, weather prediction, and ship trajectory prediction. Against these problems, classical spatio-temporal prediction learning methods including deep learning, require large amounts of training data. In reality, some newly developed cities with insufficient sensors would not hold that assumption, and the data scarcity makes predictive performance worse. In such situation, the learning method on insufficient data is known as few-shot learning (FSL), and the FSL of traffic prediction remains challenges. On the one hand, graph structures' irregularity and dynamic nature of graphs cannot hold the performance of spatio-temporal learning method. On the other hand, conventional domain adaptation methods cannot work well on insufficient training data, when transferring knowledge from different domains to the intended target domain.To address these challenges, we propose a novel spatio-temporal domain adaptation (STDA) method that learns transferable spatio-temporal meta-knowledge from data-sufficient cities in an adversarial manner. This learned meta-knowledge can improve the prediction performance of data-scarce cities. Specifically, we train the STDA model using a Model-Agnostic Meta-Learning (MAML) based episode learning process, which is a model-agnostic meta-learning framework that enables the model to solve new learning tasks using only a small number of training samples. We conduct numerous experiments on four traffic prediction datasets, and our results show that the prediction performance of our model has improved by 7\% compared to baseline models on the two metrics of MAE and RMSE.

Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises

Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper bounded by $\sigma^{p}$ for some $\sigma\geq0$) where $p\in(1,2]$, which not only generalizes the traditional finite variance assumption ($p=2$) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon $T$ even when $T$ is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to $\sigma$, in other words, the algorithm can still guarantee convergence without any prior knowledge of $\sigma$.

High Probability Convergence of Stochastic Gradient Methods

In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. This method can be applied to the non-convex case. We demonstrate an $O((1+\sigma^{2}\log(1/\delta))/T+\sigma/\sqrt{T})$ convergence rate when the number of iterations $T$ is known and an $O((1+\sigma^{2}\log(T/\delta))/\sqrt{T})$ convergence rate when $T$ is unknown for SGD, where $1-\delta$ is the desired success probability. These bounds improve over existing bounds in the literature. Additionally, we demonstrate that our techniques can be used to obtain high probability bound for AdaGrad-Norm (Ward et al., 2019) that removes the bounded gradients assumption from previous works. Furthermore, our technique for AdaGrad-Norm extends to the standard per-coordinate AdaGrad algorithm (Duchi et al., 2011), providing the first noise-adapted high probability convergence for AdaGrad.

Breaking the Lower Bound with (Little) Structure: Acceleration in Non-Convex Stochastic Optimization with Heavy-Tailed Noise

We consider the stochastic optimization problem with smooth but not necessarily convex objectives in the heavy-tailed noise regime, where the stochastic gradient's noise is assumed to have bounded $p$th moment ($p\in(1,2]$). Zhang et al. (2020) is the first to prove the $\Omega(T^{\frac{1-p}{3p-2}})$ lower bound for convergence (in expectation) and provides a simple clipping algorithm that matches this optimal rate. Cutkosky and Mehta (2021) proposes another algorithm, which is shown to achieve the nearly optimal high-probability convergence guarantee $O(\log(T/\delta)T^{\frac{1-p}{3p-2}})$, where $\delta$ is the probability of failure. However, this desirable guarantee is only established under the additional assumption that the stochastic gradient itself is bounded in $p$th moment, which fails to hold even for quadratic objectives and centered Gaussian noise. In this work, we first improve the analysis of the algorithm in Cutkosky and Mehta (2021) to obtain the same nearly optimal high-probability convergence rate $O(\log(T/\delta)T^{\frac{1-p}{3p-2}})$, without the above-mentioned restrictive assumption. Next, and curiously, we show that one can achieve a faster rate than that dictated by the lower bound $\Omega(T^{\frac{1-p}{3p-2}})$ with only a tiny bit of structure, i.e., when the objective function $F(x)$ is assumed to be in the form of $\mathbb{E}_{\Xi\sim\mathcal{D}}[f(x,\Xi)]$, arguably the most widely applicable class of stochastic optimization problems. For this class of problems, we propose the first variance-reduced accelerated algorithm and establish that it guarantees a high-probability convergence rate of $O(\log(T/\delta)T^{\frac{1-p}{2p-1}})$ under a mild condition, which is faster than $\Omega(T^{\frac{1-p}{3p-2}})$. Notably, even when specialized to the finite-variance case, our result yields the (near-)optimal high-probability rate $O(\log(T/\delta)T^{-1/3})$.

Near-Optimal High-Probability Convergence for Non-Convex Stochastic Optimization with Variance Reduction

Traditional analyses for non-convex stochastic optimization problems characterize convergence bounds in expectation, which is inadequate as it does not supply a useful performance guarantee on a single run. Motivated by its importance, an emerging line of literature has recently studied the high-probability convergence behavior of several algorithms, including the classic stochastic gradient descent (SGD). However, no high-probability results are established for optimization algorithms with variance reduction, which is known to accelerate the convergence process and has been the de facto algorithmic technique for stochastic optimization at large. To close this important gap, we introduce a new variance-reduced algorithm for non-convex stochastic optimization, which we call Generalized SignSTORM. We show that with probability at least $1-\delta$, our algorithm converges at the rate of $O(\log(dT/\delta)/T^{1/3})$ after $T$ iterations where $d$ is the problem dimension. This convergence guarantee matches the existing lower bound up to a log factor, and to our best knowledge, is the first high-probability minimax (near-)optimal result. Finally, we demonstrate the effectiveness of our algorithm through numerical experiments.

META-STORM: Generalized Fully-Adaptive Variance Reduced SGD for Unbounded Functions

We study the application of variance reduction (VR) techniques to general non-convex stochastic optimization problems. In this setting, the recent work STORM [Cutkosky-Orabona '19] overcomes the drawback of having to compute gradients of "mega-batches" that earlier VR methods rely on. There, STORM utilizes recursive momentum to achieve the VR effect and is then later made fully adaptive in STORM+ [Levy et al., '21], where full-adaptivity removes the requirement for obtaining certain problem-specific parameters such as the smoothness of the objective and bounds on the variance and norm of the stochastic gradients in order to set the step size. However, STORM+ crucially relies on the assumption that the function values are bounded, excluding a large class of useful functions. In this work, we propose META-STORM, a generalized framework of STORM+ that removes this bounded function values assumption while still attaining the optimal convergence rate for non-convex optimization. META-STORM not only maintains full-adaptivity, removing the need to obtain problem specific parameters, but also improves the convergence rate's dependency on the problem parameters. Furthermore, META-STORM can utilize a large range of parameter settings that subsumes previous methods allowing for more flexibility in a wider range of settings. Finally, we demonstrate the effectiveness of META-STORM through experiments across common deep learning tasks. Our algorithm improves upon the previous work STORM+ and is competitive with widely used algorithms after the addition of per-coordinate update and exponential moving average heuristics.

On the Convergence of AdaGrad on $\R^{d}$: Beyond Convexity, Non-Asymptotic Rate and Acceleration

Existing analysis of AdaGrad and other adaptive methods for smooth convex optimization is typically for functions with bounded domain diameter. In unconstrained problems, previous works guarantee an asymptotic convergence rate without an explicit constant factor that holds true for the entire function class. Furthermore, in the stochastic setting, only a modified version of AdaGrad, different from the one commonly used in practice, in which the latest gradient is not used to update the stepsize, has been analyzed. Our paper aims at bridging these gaps and developing a deeper understanding of AdaGrad and its variants in the standard setting of smooth convex functions as well as the more general setting of quasar convex functions. First, we demonstrate new techniques to explicitly bound the convergence rate of the vanilla AdaGrad for unconstrained problems in both deterministic and stochastic settings. Second, we propose a variant of AdaGrad for which we can show the convergence of the last iterate, instead of the average iterate. Finally, we give new accelerated adaptive algorithms and their convergence guarantee in the deterministic setting with explicit dependency on the problem parameters, improving upon the asymptotic rate shown in previous works.