Stochastic gradient descent without replacement sampling is widely used in practice for model training. However, the vast majority of SGD analyses assumes data is sampled with replacement, and when the function minimized is strongly convex, an $\mathcal{O}\left(\frac{1}{T}\right)$ rate can be established when SGD is run for $T$ iterations. A recent line of breakthrough works on SGD without replacement (SGDo) established an $\mathcal{O}\left(\frac{n}{T^2}\right)$ convergence rate when the function minimized is strongly convex and is a sum of $n$ smooth functions, and an $\mathcal{O}\left(\frac{1}{T^2}+\frac{n^3}{T^3}\right)$ rate for sums of quadratics. On the other hand, the tightest known lower bound postulates an $Ω\left(\frac{1}{T^2}+\frac{n^2}{T^3}\right)$ rate, leaving open the possibility of better SGDo convergence rates in the general case. In this paper, we close this gap and show that SGD without replacement achieves a rate of $\mathcal{O}\left(\frac{1}{T^2}+\frac{n^2}{T^3}\right)$ when the sum of the functions is a quadratic, and offer a new lower bound of $Ω\left(\frac{n}{T^2}\right)$ for strongly convex functions that are sums of smooth functions.
Microsoft Research（微软研究院）；Massachusetts Institute of Technology（麻省理工学院）
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, strong…
For solving finite-sum optimization problems, SGD without replacement sampling is empirically shown to outperform SGD. Denoting by $n$ the number of components in the cost and $K$ the number of epochs of the algorithm , several recent works have shown convergence rates of without-replacement SGD that have better dependency on $n$ and $K$ than the baseline rate of $O(1/(nK))$ for SGD. However, there are two main limitations shared among those works: the rates have extra poly-logarithmic factors on $nK$, and denoting by $κ$ the condition number of the problem, the rates hold after $κ^c\log(nK)$ …
In distributed learning, local SGD (also known as federated averaging) and its simple baseline minibatch SGD are widely studied optimization methods. Most existing analyses of these methods assume independent and unbiased gradient estimates obtained via with-replacement sampling. In contrast, we study shuffling-based variants: minibatch and local Random Reshuffling, which draw stochastic gradients without replacement and are thus closer to practice. For smooth functions satisfying the Polyak-Łojasiewicz condition, we obtain convergence bounds (in the large epoch regime) which show that these s…
We propose matrix norm inequalities that extend the Recht-Ré (2012) conjecture on a noncommutative AM-GM inequality by supplementing it with another inequality that accounts for single-shuffle, which is a widely used without-replacement sampling scheme that shuffles only once in the beginning and is overlooked in the Recht-Ré conjecture. Instead of general positive semidefinite matrices, we restrict our attention to positive definite matrices with small enough condition numbers, which are more relevant to matrices that arise in the analysis of SGD. For such matrices, we conjecture that the mea…
Princeton University（普林斯顿大学）；Weizmann Institute of Science（魏茨曼科学研究所）
Recently, there has been much interest in studying the convergence rates of without-replacement SGD, and proving that it is faster than with-replacement SGD in the worst case. However, known lower bounds ignore the problem's geometry, including its condition number, whereas the upper bounds explicitly depend on it. Perhaps surprisingly, we prove that when the condition number is taken into account, without-replacement SGD \emph{does not} significantly improve on with-replacement SGD in terms of worst-case bounds, unless the number of epochs (passes over the data) is larger than the condition n…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We study to what extent may stochastic gradient descent (SGD) be understood as a "conventional" learning rule that achieves generalization performance by obtaining a good fit to training data. We consider the fundamental stochastic convex optimization framework, where (one pass, without-replacement) SGD is classically known to minimize the population risk at rate $O(1/\sqrt n)$, and prove that, surprisingly, there exist problem instances where the SGD solution exhibits both empirical risk and generalization gap of $Ω(1)$. Consequently, it turns out that SGD is not algorithmically stable in any…
Stochastic gradient descent-ascent (SGDA) is one of the main workhorses for solving finite-sum minimax optimization problems. Most practical implementations of SGDA randomly reshuffle components and sequentially use them (i.e., without-replacement sampling); however, there are few theoretical results on this approach for minimax algorithms, especially outside the easier-to-analyze (strongly-)monotone setups. To narrow this gap, we study the convergence bounds of SGDA with random reshuffling (SGDA-RR) for smooth nonconvex-nonconcave objectives with Polyak-Łojasiewicz (PŁ) geometry. We analyze b…
Electronic Information School, Wuhan University（武汉大学）；Department of Electrical and Computer Engineering, North Carolina State University（北卡罗来纳州立大学）
While machine learning has achieved remarkable results in a wide variety of domains, the training of models often requires large datasets that may need to be collected from different individuals. As sensitive information may be contained in the individual's dataset, sharing training data may lead to severe privacy concerns. Therefore, there is a compelling need to develop privacy-aware machine learning methods, for which one effective approach is to leverage the generic framework of differential privacy. Considering that stochastic gradient descent (SGD) is one of the most commonly adopted met…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Ohio State University（俄亥俄州立大学）；University of Virginia（弗吉尼亚大学）
Adaptive gradient methods, which adopt historical gradient information to automatically adjust the learning rate, despite the nice property of fast convergence, have been observed to generalize worse than stochastic gradient descent (SGD) with momentum in training deep neural networks. This leaves how to close the generalization gap of adaptive gradient methods an open problem. In this work, we show that adaptive gradient methods such as Adam, Amsgrad, are sometimes "over adapted". We design a new algorithm, called Partially adaptive momentum estimation method, which unifies the Adam/Amsgrad w…
Indian Institute of Technology, Roorkee（印度理工学院鲁尔基分校）
This work is a part of ICLR Reproducibility Challenge 2019, we try to reproduce the results in the conference submission PADAM: Closing The Generalization Gap of Adaptive Gradient Methods In Training Deep Neural Networks. Adaptive gradient methods proposed in past demonstrate a degraded generalization performance than the stochastic gradient descent (SGD) with momentum. The authors try to address this problem by designing a new optimization algorithm that bridges the gap between the space of Adaptive Gradient algorithms and SGD with momentum. With this method a new tunable hyperparameter calle…
A long-standing problem in the theory of stochastic gradient descent (SGD) is to prove that its without-replacement version RandomShuffle converges faster than the usual with-replacement version. We present the first (to our knowledge) non-asymptotic solution to this problem, which shows that after a "reasonable" number of epochs RandomShuffle indeed converges faster than SGD. Specifically, we prove that under strong convexity and second-order smoothness, the sequence generated by RandomShuffle converges to the optimal solution at the rate O(1/T^2 + n^3/T^3), where n is the number of component…
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
When solving finite-sum minimization problems, two common alternatives to stochastic gradient descent (SGD) with theoretical benefits are random reshuffling (SGD-RR) and shuffle-once (SGD-SO), in which functions are sampled in cycles without replacement. Under a convenient stochastic noise approximation which holds experimentally, we study the stationary variances of the iterates of SGD, SGD-RR and SGD-SO, whose leading terms decrease in this order, and obtain simple approximations. To obtain our results, we study the power spectral density of the stochastic gradient noise sequences. Our analy…
A recent line of ground-breaking results for permutation-based SGD has corroborated a widely observed phenomenon: random permutations offer faster convergence than with-replacement sampling. However, is random optimal? We show that this depends heavily on what functions we are optimizing, and the convergence gap between optimal and random permutations can vary from exponential to nonexistent. We first show that for 1-dimensional strongly convex functions, with smooth second derivatives, there exist permutations that offer exponentially faster convergence compared to random. However, for genera…
Stochastic Gradient Descent (SGD) is a central tool in machine learning. We prove that SGD converges to zero loss, even with a fixed (non-vanishing) learning rate - in the special case of homogeneous linear classifiers with smooth monotone loss functions, optimized on linearly separable data. Previous works assumed either a vanishing learning rate, iterate averaging, or loss assumptions that do not hold for monotone loss functions used for classification, such as the logistic loss. We prove our result on a fixed dataset, both for sampling with or without replacement. Furthermore, for logistic …
Despite superior training outcomes, adaptive optimization methods such as Adam, Adagrad or RMSprop have been found to generalize poorly compared to Stochastic gradient descent (SGD). These methods tend to perform well in the initial portion of training but are outperformed by SGD at later stages of training. We investigate a hybrid strategy that begins training with an adaptive method and switches to SGD when appropriate. Concretely, we propose SWATS, a simple strategy which switches from Adam to SGD when a triggering condition is satisfied. The condition we propose relates to the projection o…
We study the performance of stochastic gradient descent (SGD) on smooth and strongly-convex finite-sum optimization problems. In contrast to the majority of existing theoretical works, which assume that individual functions are sampled with replacement, we focus here on popular but poorly-understood heuristics, which involve going over random permutations of the individual functions. This setting has been investigated in several recent works, but the optimal error rates remain unclear. In this paper, we provide lower bounds on the expected optimization error with these heuristics (using SGD wi…
Télécom Paris Tech（巴黎电信学院）；University of Edinburgh（爱丁堡大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We propose a general yet simple theorem describing the convergence of SGD under the arbitrary sampling paradigm. Our theorem describes the convergence of an infinite array of variants of SGD, each of which is associated with a specific probability law governing the data selection rule used to form mini-batches. This is the first time such an analysis is performed, and most of our variants of SGD were never explicitly considered in the literature before. Our analysis relies on the recently introduced notion of expected smoothness and does not rely on a uniform bound on the variance of the stoch…
We study without-replacement SGD for solving finite-sum optimization problems. Specifically, depending on how the indices of the finite-sum are shuffled, we consider the RandomShuffle (shuffle at the beginning of each epoch) and SingleShuffle (shuffle only once) algorithms. First, we establish minimax optimal convergence rates of these algorithms up to poly-log factors. Notably, our analysis is general enough to cover gradient dominated nonconvex costs, and does not rely on the convexity of individual component functions unlike existing optimal convergence results. Secondly, assuming convexity…
From Continual Learning to SGD and Back: Better Rates for Continual Linear Models
Meta；Technion（技术ion学院）；Department of Computer Science（计算机科学系）；Tel Aviv University（特拉维夫大学）；Blavatnik School of Computer Science and AI（Blavatnik计算机科学与人工智能学院）；Google Research（谷歌研究）；Department of Electrical and Computing Engineering（电气与计算工程系）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
Statistical Guarantees for High-Dimensional Stochastic Gradient Descent
Department of Statistics University of Chicago（统计学系芝加哥大学）；Department of Mathematics University of California, San Diego（数学系加州大学圣地亚哥分校）；Department of Applied Mathematics University of Twente（应用数学系埃因霍温大学）
University of Pennsylvania（宾夕法尼亚大学）；DAMO Academy, Alibaba Group（阿里巴巴集团达摩院）；Tsinghua University（清华大学）
When applying a stochastic algorithm, one must choose an order to draw samples. The practical choices are without-replacement sampling orders, which are empirically faster and more cache-friendly than uniform-iid-sampling but often have inferior theoretical guarantees. Without-replacement sampling is well understood only for SGD without variance reduction. In this paper, we will improve the convergence analysis and rates of variance reduction under without-replacement sampling orders for composite finite-sum minimization. Our results are in two-folds. First, we develop a damped variant of Fini…
Rapid Overfitting of Multi-Pass Stochastic Gradient Descent in Stochastic Convex Optimization
Blavatnik School of Computer Science and AI, Tel Aviv University（塔尔塔尔大学Blavatnik计算机科学与人工智能学院）；Blavatnik School of Computer Science and AI, Tel Aviv University, Google Research（塔尔塔尔大学Blavatnik计算机科学与人工智能学院，Google研究）；School of Electrical & Computer Engineering, Tel Aviv University（塔尔塔尔大学电气与计算机工程学院）
We prove local convergence of several notable gradient descent algorithms used in machine learning, for which standard stochastic gradient descent theory does not apply directly. This includes, first, online algorithms for recurrent models and dynamical systems, such as \emph{Real-time recurrent learning} (RTRL) and its computationally lighter approximations NoBackTrack and UORO; second, several adaptive algorithms such as RMSProp, online natural gradient, and Adam with $β^2\to 1$.Despite local convergence being a relatively weak requirement for a new optimization algorithm, no local analysis …
While SGD, which samples from the data with replacement is widely studied in theory, a variant called Random Reshuffling (RR) is more common in practice. RR iterates through random permutations of the dataset and has been shown to converge faster than SGD. When the order is chosen deterministically, a variant called incremental gradient descent (IG), the existing convergence bounds show improvement over SGD but are worse than RR. However, these bounds do not differentiate between a good and a bad ordering and hold for the worst choice of order. Meanwhile, in some cases, choosing the right orde…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Random Reshuffling (RR), also known as Stochastic Gradient Descent (SGD) without replacement, is a popular and theoretically grounded method for finite-sum minimization. We propose two new algorithms: Proximal and Federated Random Reshuffing (ProxRR and FedRR). The first algorithm, ProxRR, solves composite convex finite-sum minimization problems in which the objective is the sum of a (potentially non-smooth) convex regularizer and an average of $n$ smooth objectives. We obtain the second algorithm, FedRR, as a special case of ProxRR applied to a reformulation of distributed problems with eithe…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Random Reshuffling (RR), which is a variant of Stochastic Gradient Descent (SGD) employing sampling without replacement, is an immensely popular method for training supervised machine learning models via empirical risk minimization. Due to its superior practical performance, it is embedded and often set as default in standard machine learning software. Under the name FedRR, this method was recently shown to be applicable to federated learning (Mishchenko et al.,2021), with superior performance when compared to common baselines such as Local SGD. Inspired by this development, we design three ne…
In modern supervised learning, many deep neural networks are able to interpolate the data: the empirical loss can be driven to near zero on all samples simultaneously. In this work, we explicitly exploit this interpolation property for the design of a new optimization algorithm for deep learning, which we term Adaptive Learning-rates for Interpolation with Gradients (ALI-G). ALI-G retains the two main advantages of Stochastic Gradient Descent (SGD), which are (i) a low computational cost per iteration and (ii) good generalization performance in practice. At each iteration, ALI-G exploits the i…
Peking University（北京大学）；Microsoft Research（微软研究院）；Beijing Jiaotong University（北京交通大学）；Chinese Academy of Mathematics and Systems Science（中国科学院数学与系统科学研究院）
When using stochastic gradient descent to solve large-scale machine learning problems, a common practice of data processing is to shuffle the training data, partition the data across multiple machines if needed, and then perform several epochs of training on the re-shuffled (either locally or globally) data. The above procedure makes the instances used to compute the gradients no longer independently sampled from the training data set. Then does the distributed SGD method have desirable convergence properties in this practical situation? In this paper, we give answers to this question. First, …
Differential privacy is a useful tool to build machine learning models which do not release too much information about the training data. We study the Rényi differential privacy of stochastic gradient descent when each training example is sampled without replacement (also known as cyclic SGD). Cyclic SGD is typically faster than traditional SGD and is the algorithm of choice in large-scale implementations. We recover privacy guarantees for cyclic SGD which are competitive with those known for sampling with replacement. Our proof techniques make no assumptions on the model or on the data and ar…
Case Western Reserve University（凯斯西储大学）；Harvard Medical School（哈佛医学院）；Dana-Farber Cancer Institute（丹娜-法伯癌症研究所）；Koç University（科奇大学）；University of Chicago（芝加哥大学）
The stochastic proximal point (SPP) methods have gained recent attention for stochastic optimization, with strong convergence guarantees and superior robustness to the classic stochastic gradient descent (SGD) methods showcased at little to no cost of computational overhead added. In this article, we study a minibatch variant of SPP, namely M-SPP, for solving convex composite risk minimization problems. The core contribution is a set of novel excess risk bounds of M-SPP derived through the lens of algorithmic stability theory. Particularly under smoothness and quadratic growth conditions, we s…
Embedding learning has found widespread applications in recommendation systems and natural language modeling, among other domains. To learn quality embeddings efficiently, adaptive learning rate algorithms have demonstrated superior empirical performance over SGD, largely accredited to their token-dependent learning rate. However, the underlying mechanism for the efficiency of token-dependent learning rate remains underexplored. We show that incorporating frequency information of tokens in the embedding learning problems leads to provably efficient algorithms, and demonstrate that common adapt…
