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PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization
TL;DR: The results demonstrate that PAGE not only converges much faster than SGD in training but also achieves the higher test accuracy, validating the theoretical results and confirming the practical superiority of PAGE.
Abstract: In this paper, we propose a novel stochastic gradient estimator -- ProbAbilistic Gradient Estimator (PAGE) -- for nonconvex optimization. PAGE is easy to implement as it is designed via a small adjustment to vanilla SGD: in each iteration, PAGE uses the vanilla minibatch SGD update with probability $p_t$ or reuses the previous gradient with a small adjustment, at a much lower computational cost, with probability $1-p_t$. We give a simple formula for the optimal choice of $p_t$. Moreover, we prove the first tight lower bound $\Omega(n+\frac{\sqrt{n}}{\epsilon^2})$ for nonconvex finite-sum problems, which also leads to a tight lower bound $\Omega(b+\frac{\sqrt{b}}{\epsilon^2})$ for nonconvex online problems, where $b:= \min\{\frac{\sigma^2}{\epsilon^2}, n\}$. Then, we show that PAGE obtains the optimal convergence results $O(n+\frac{\sqrt{n}}{\epsilon^2})$ (finite-sum) and $O(b+\frac{\sqrt{b}}{\epsilon^2})$ (online) matching our lower bounds for both nonconvex finite-sum and online problems. Besides, we also show that for nonconvex functions satisfying the Polyak-Łojasiewicz (PL) condition, PAGE can automatically switch to a faster linear convergence rate $O(\cdot\log \frac{1}{\epsilon})$. Finally, we conduct several deep learning experiments (e.g., LeNet, VGG, ResNet) on real datasets in PyTorch showing that PAGE not only converges much faster than SGD in training but also achieves the higher test accuracy, validating the optimal theoretical results and confirming the practical superiority of PAGE.
Citations
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TL;DR: An extensive, standardized benchmark of more than a dozen particularly popular deep learning optimizers is performed, identifying a significantly reduced subset of specific algorithms and parameter choices that generally provided competitive results in the authors' experiments.
Abstract: Choosing the optimizer is considered to be among the most crucial design decisions in deep learning, and it is not an easy one. The growing literature now lists hundreds of optimization methods. In the absence of clear theoretical guidance and conclusive empirical evidence, the decision is often made based on anecdotes. In this work, we aim to replace these anecdotes, if not with a conclusive ranking, then at least with evidence-backed heuristics. To do so, we perform an extensive, standardized benchmark of fifteen particularly popular deep learning optimizers while giving a concise overview of the wide range of possible choices. Analyzing more than $50,000$ individual runs, we contribute the following three points: (i) Optimizer performance varies greatly across tasks. (ii) We observe that evaluating multiple optimizers with default parameters works approximately as well as tuning the hyperparameters of a single, fixed optimizer. (iii) While we cannot discern an optimization method clearly dominating across all tested tasks, we identify a significantly reduced subset of specific optimizers and parameter choices that generally lead to competitive results in our experiments: Adam remains a strong contender, with newer methods failing to significantly and consistently outperform it. Our open-sourced results are available as challenging and well-tuned baselines for more meaningful evaluations of novel optimization methods without requiring any further computational efforts.
85 citations
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TL;DR: An overview of recent theoretical results on global performance guarantees of optimization algorithms for non-convex optimization and a list of problems that can be solved efficiently to find the global minimizer by exploiting the structure of the problem as much as it is possible.
Abstract: Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical results on global performance guarantees of optimization algorithms for non-convex optimization. We start with classical arguments showing that general non-convex problems could not be solved efficiently in a reasonable time. Then we give a list of problems that can be solved efficiently to find the global minimizer by exploiting the structure of the problem as much as it is possible. Another way to deal with non-convexity is to relax the goal from finding the global minimum to finding a stationary point or a local minimum. For this setting, we first present known results for the convergence rates of deterministic first-order methods, which are then followed by a general theoretical analysis of optimal stochastic and randomized gradient schemes, and an overview of the stochastic first-order methods. After that, we discuss quite general classes of non-convex problems, such as minimization of $\alpha$-weakly-quasi-convex functions and functions that satisfy Polyak--Lojasiewicz condition, which still allow obtaining theoretical convergence guarantees of first-order methods. Then we consider higher-order and zeroth-order/derivative-free methods and their convergence rates for non-convex optimization problems.
29 citations
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10 Feb 2022
TL;DR: It is proved that averaged Langevin Monte Carlo outputs a sample with ε -relative Fisher information after O ( L 2 d 2 /ε 2 ) iterations, which constitutes a first step towards the general theory of non-log-concave sampling.
Abstract: For the task of sampling from a density $\pi \propto \exp(-V)$ on $\mathbb{R}^d$, where $V$ is possibly non-convex but $L$-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with $\varepsilon$-relative Fisher information after $O( L^2 d^2/\varepsilon^2)$ iterations. This is the sampling analogue of complexity bounds for finding an $\varepsilon$-approximate first-order stationary points in non-convex optimization and therefore constitutes a first step towards the general theory of non-log-concave sampling. We discuss numerous extensions and applications of our result; in particular, it yields a new state-of-the-art guarantee for sampling from distributions which satisfy a Poincar\'e inequality.
27 citations
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02 Feb 2022
TL;DR: In this paper , a new class of gradient communication mechanisms for communicationefficient training, three point compressors (3PC), is proposed and studied, which allows the compressors to evolve throughout the training process with the aim of improving the theoretical communication complexity and practical efficiency of underlying methods.
Abstract: We propose and study a new class of gradient communication mechanisms for communicationefficient training—three point compressors (3PC)—as well as efficient distributed nonconvex optimization algorithms that can take advantage of them. Unlike most established approaches, which rely on a static compressor choice (e.g., Top-K), our class allows the compressors to evolve throughout the training process, with the aim of improving the theoretical communication complexity and practical efficiency of the underlying methods. We show that our general approach can recover the recently proposed state-of-the-art error feedback mechanism EF21 (Richtárik et al., 2021) and its theoretical properties as a special case, but also leads to a number of new efficient methods. Notably, our approach allows us to improve upon the state of the art in the algorithmic and theoretical foundations of the lazy aggregation literature (Chen et al., 2018). As a by-product that may be of independent interest, we provide a new and fundamental link between the lazy aggregation and error feedback literature. A special feature of our work is that we do not require the compressors to be unbiased.
15 citations
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24 Feb 2022
TL;DR: A comprehensive convergence analysis of a simple stochastic algorithm for both non-convex and convex objectives is provided, which results in the improved oracle complexity with the parallel speed-up by using the moving-average based estimator with mini-batching.
Abstract: This paper studies stochastic optimization for a sum of compositional functions, where the inner-level function of each summand is coupled with the corresponding summation index. We refer to this family of problems as finite-sum coupled compositional optimization (FCCO). It has broad applications in machine learning for optimizing non-convex or convex compositional measures/objectives such as average precision (AP), p-norm push, listwise ranking losses, neighborhood component analysis (NCA), deep survival analysis, deep latent variable models, etc., which deserves finer analysis. Yet, existing algorithms and analyses are restricted in one or other aspects. The contribution of this paper is to provide a comprehensive convergence analysis of a simple stochastic algorithm for both non-convex and convex objectives. Our key result is the improved oracle complexity with the parallel speed-up by using the moving-average based estimator with mini-batching. Our theoretical analysis also exhibits new insights for improving the practical implementation by sampling the batches of equal size for the outer and inner levels. Numerical experiments on AP maximization, NCA, and p-norm push corroborate some aspects of the theory.
11 citations
References
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27 Jun 2016TL;DR: In this article, the authors proposed a residual learning framework to ease the training of networks that are substantially deeper than those used previously, which won the 1st place on the ILSVRC 2015 classification task.
Abstract: Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers—8× deeper than VGG nets [40] but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.
123,388 citations
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04 Sep 2014TL;DR: This work investigates the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting using an architecture with very small convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers.
Abstract: In this work we investigate the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting. Our main contribution is a thorough evaluation of networks of increasing depth using an architecture with very small (3x3) convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers. These findings were the basis of our ImageNet Challenge 2014 submission, where our team secured the first and the second places in the localisation and classification tracks respectively. We also show that our representations generalise well to other datasets, where they achieve state-of-the-art results. We have made our two best-performing ConvNet models publicly available to facilitate further research on the use of deep visual representations in computer vision.
55,235 citations
"PAGE: A Simple and Optimal Probabil..." refers methods in this paper
...Concretely, we compare our PAGE algorithm with vanilla SGD by running standard LeNet [22], VGG [42] and ResNet [10] models on MNIST [22] and CIFAR-10 [17] datasets....
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...We conduct several deep learning experiments (e.g., LeNet, VGG, ResNet) on real datasets in PyTorch showing that PAGE indeed not only converges much faster than SGD in training but also achieves higher test accuracy....
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01 Jan 2015TL;DR: In this paper, the authors investigated the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting and showed that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 layers.
Abstract: In this work we investigate the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting. Our main contribution is a thorough evaluation of networks of increasing depth using an architecture with very small (3x3) convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers. These findings were the basis of our ImageNet Challenge 2014 submission, where our team secured the first and the second places in the localisation and classification tracks respectively. We also show that our representations generalise well to other datasets, where they achieve state-of-the-art results. We have made our two best-performing ConvNet models publicly available to facilitate further research on the use of deep visual representations in computer vision.
49,914 citations
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TL;DR: Deep learning is making major advances in solving problems that have resisted the best attempts of the artificial intelligence community for many years, and will have many more successes in the near future because it requires very little engineering by hand and can easily take advantage of increases in the amount of available computation and data.
Abstract: Deep learning allows computational models that are composed of multiple processing layers to learn representations of data with multiple levels of abstraction. These methods have dramatically improved the state-of-the-art in speech recognition, visual object recognition, object detection and many other domains such as drug discovery and genomics. Deep learning discovers intricate structure in large data sets by using the backpropagation algorithm to indicate how a machine should change its internal parameters that are used to compute the representation in each layer from the representation in the previous layer. Deep convolutional nets have brought about breakthroughs in processing images, video, speech and audio, whereas recurrent nets have shone light on sequential data such as text and speech.
46,982 citations
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01 Jan 1998TL;DR: In this article, a graph transformer network (GTN) is proposed for handwritten character recognition, which can be used to synthesize a complex decision surface that can classify high-dimensional patterns, such as handwritten characters.
Abstract: Multilayer neural networks trained with the back-propagation algorithm constitute the best example of a successful gradient based learning technique. Given an appropriate network architecture, gradient-based learning algorithms can be used to synthesize a complex decision surface that can classify high-dimensional patterns, such as handwritten characters, with minimal preprocessing. This paper reviews various methods applied to handwritten character recognition and compares them on a standard handwritten digit recognition task. Convolutional neural networks, which are specifically designed to deal with the variability of 2D shapes, are shown to outperform all other techniques. Real-life document recognition systems are composed of multiple modules including field extraction, segmentation recognition, and language modeling. A new learning paradigm, called graph transformer networks (GTN), allows such multimodule systems to be trained globally using gradient-based methods so as to minimize an overall performance measure. Two systems for online handwriting recognition are described. Experiments demonstrate the advantage of global training, and the flexibility of graph transformer networks. A graph transformer network for reading a bank cheque is also described. It uses convolutional neural network character recognizers combined with global training techniques to provide record accuracy on business and personal cheques. It is deployed commercially and reads several million cheques per day.
42,067 citations