An Adaptive Remote Stochastic Gradient Method for Training Neural Networks

We present the remote stochastic gradient (RSG) method, which computes the gradients at configurable remote observation points, in order to improve the convergence rate and suppress gradient noise at the same time for different curvatures. RSG is further combined with adaptive methods to construct ARSG for acceleration. The method is efficient in computation and memory, and is straightforward to implement. We analyze the convergence properties by modeling the training process as a dynamic system, which provides a guideline to select the configurable observation factor without grid search. ARSG yields $O(1/\sqrt{T})$ convergence rate in non-convex settings, that can be further improved to $O(\log(T)/T)$ in strongly convex settings. Numerical experiments demonstrate that ARSG achieves both faster convergence and better generalization, compared with popular adaptive methods, such as ADAM, NADAM, AMSGRAD, and RANGER for the tested problems. In particular, for training ResNet-50 on ImageNet, ARSG outperforms ADAM in convergence speed and meanwhile it surpasses SGD in generalization.Comment: The generalization is improved by modifying the preconditioner. For training ResNet-50 on ImageNet, ARSG outperforms ADAM in convergence speed and meanwhile it surpasses SGD in generalization. We also present a convergence bound in non-convex setting

On the Ineffectiveness of Variance Reduced Optimization for Deep Learning

The application of stochastic variance reduction to optimization has shown remarkable recent theoretical and practical success. The applicability of these techniques to the hard non-convex optimization problems encountered during training of modern deep neural networks is an open problem. We show that naive application of the SVRG technique and related approaches fail, and explore why

Closing the Generalization Gap of Adaptive Gradient Methods in Training Deep Neural Networks

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 with SGD by introducing a partial adaptive parameter $p$, to achieve the best from both worlds. We also prove the convergence rate of our proposed algorithm to a stationary point in the stochastic nonconvex optimization setting. Experiments on standard benchmarks show that our proposed algorithm can maintain a fast convergence rate as Adam/Amsgrad while generalizing as well as SGD in training deep neural networks. These results would suggest practitioners pick up adaptive gradient methods once again for faster training of deep neural networks.Comment: 17 pages, 4 figures, 4 tables. In IJCAI 202

Interpreting Deep Learning: The Machine Learning Rorschach Test?

Theoretical understanding of deep learning is one of the most important tasks facing the statistics and machine learning communities. While deep neural networks (DNNs) originated as engineering methods and models of biological networks in neuroscience and psychology, they have quickly become a centerpiece of the machine learning toolbox. Unfortunately, DNN adoption powered by recent successes combined with the open-source nature of the machine learning community, has outpaced our theoretical understanding. We cannot reliably identify when and why DNNs will make mistakes. In some applications like text translation these mistakes may be comical and provide for fun fodder in research talks, a single error can be very costly in tasks like medical imaging. As we utilize DNNs in increasingly sensitive applications, a better understanding of their properties is thus imperative. Recent advances in DNN theory are numerous and include many different sources of intuition, such as learning theory, sparse signal analysis, physics, chemistry, and psychology. An interesting pattern begins to emerge in the breadth of possible interpretations. The seemingly limitless approaches are mostly constrained by the lens with which the mathematical operations are viewed. Ultimately, the interpretation of DNNs appears to mimic a type of Rorschach test --- a psychological test wherein subjects interpret a series of seemingly ambiguous ink-blots. Validation for DNN theory requires a convergence of the literature. We must distinguish between universal results that are invariant to the analysis perspective and those that are specific to a particular network configuration. Simultaneously we must deal with the fact that many standard statistical tools for quantifying generalization or empirically assessing important network features are difficult to apply to DNNs.Comment: 13 pages, 1 figure. Preprint is related to an upcoming Society for Industrial and Applied Mathematics (SIAM) News articl

Adaptive Weight Decay for Deep Neural Networks

Regularization in the optimization of deep neural networks is often critical to avoid undesirable over-fitting leading to better generalization of model. One of the most popular regularization algorithms is to impose L-2 penalty on the model parameters resulting in the decay of parameters, called weight-decay, and the decay rate is generally constant to all the model parameters in the course of optimization. In contrast to the previous approach based on the constant rate of weight-decay, we propose to consider the residual that measures dissimilarity between the current state of model and observations in the determination of the weight-decay for each parameter in an adaptive way, called adaptive weight-decay (AdaDecay) where the gradient norms are normalized within each layer and the degree of regularization for each parameter is determined in proportional to the magnitude of its gradient using the sigmoid function. We empirically demonstrate the effectiveness of AdaDecay in comparison to the state-of-the-art optimization algorithms using popular benchmark datasets: MNIST, Fashion-MNIST, and CIFAR-10 with conventional neural network models ranging from shallow to deep. The quantitative evaluation of our proposed algorithm indicates that AdaDecay improves generalization leading to better accuracy across all the datasets and models

Predictive Local Smoothness for Stochastic Gradient Methods

Stochastic gradient methods are dominant in nonconvex optimization especially for deep models but have low asymptotical convergence due to the fixed smoothness. To address this problem, we propose a simple yet effective method for improving stochastic gradient methods named predictive local smoothness (PLS). First, we create a convergence condition to build a learning rate which varies adaptively with local smoothness. Second, the local smoothness can be predicted by the latest gradients. Third, we use the adaptive learning rate to update the stochastic gradients for exploring linear convergence rates. By applying the PLS method, we implement new variants of three popular algorithms: PLS-stochastic gradient descent (PLS-SGD), PLS-accelerated SGD (PLS-AccSGD), and PLS-AMSGrad. Moreover, we provide much simpler proofs to ensure their linear convergence. Empirical results show that the variants have better performance gains than the popular algorithms, such as, faster convergence and alleviating explosion and vanish of gradients.Comment: 14 pages, 7 figure

A Modular Analysis of Provable Acceleration via Polyak's Momentum: Training a Wide ReLU Network and a Deep Linear Network

Incorporating a so-called "momentum" dynamic in gradient descent methods is widely used in neural net training as it has been broadly observed that, at least empirically, it often leads to significantly faster convergence. At the same time, there are very few theoretical guarantees in the literature to explain this apparent acceleration effect. Even for the classical strongly convex quadratic problems, several existing results only show Polyak's momentum has an accelerated linear rate asymptotically. In this paper, we first revisit the quadratic problems and show a non-asymptotic accelerated linear rate of Polyak's momentum. Then, we provably show that Polyak's momentum achieves acceleration for training a one-layer wide ReLU network and a deep linear network, which are perhaps the two most popular canonical models for studying optimization and deep learning in the literature. Prior work Du at al. 2019 and Wu et al. 2019 showed that using vanilla gradient descent, and with an use of over-parameterization, the error decays as $(1- \Theta(\frac{1}{ \kappa'}))^t$ after $t$ iterations, where $\kappa'$ is the condition number of a Gram Matrix. Our result shows that with the appropriate choice of parameters Polyak's momentum has a rate of $(1-\Theta(\frac{1}{\sqrt{\kappa'}}))^t$. For the deep linear network, prior work Hu et al. 2020 showed that vanilla gradient descent has a rate of $(1-\Theta(\frac{1}{\kappa}))^t$, where $\kappa$ is the condition number of a data matrix. Our result shows an acceleration rate $(1- \Theta(\frac{1}{\sqrt{\kappa}}))^t$ is achievable by Polyak's momentum. All the results in this work are obtained from a modular analysis, which can be of independent interest. This work establishes that momentum does indeed speed up neural net training.Comment: Accepted at ICML 202

Aggregated Momentum: Stability Through Passive Damping

Momentum is a simple and widely used trick which allows gradient-based optimizers to pick up speed along low curvature directions. Its performance depends crucially on a damping coefficient $\beta$. Large $\beta$ values can potentially deliver much larger speedups, but are prone to oscillations and instability; hence one typically resorts to small values such as 0.5 or 0.9. We propose Aggregated Momentum (AggMo), a variant of momentum which combines multiple velocity vectors with different $\beta$ parameters. AggMo is trivial to implement, but significantly dampens oscillations, enabling it to remain stable even for aggressive $\beta$ values such as 0.999. We reinterpret Nesterov's accelerated gradient descent as a special case of AggMo and analyze rates of convergence for quadratic objectives. Empirically, we find that AggMo is a suitable drop-in replacement for other momentum methods, and frequently delivers faster convergence.Comment: 11 primary pages, 11 supplementary pages, 12 figures tota

Which Algorithmic Choices Matter at Which Batch Sizes? Insights From a Noisy Quadratic Model

Increasing the batch size is a popular way to speed up neural network training, but beyond some critical batch size, larger batch sizes yield diminishing returns. In this work, we study how the critical batch size changes based on properties of the optimization algorithm, including acceleration and preconditioning, through two different lenses: large scale experiments, and analysis of a simple noisy quadratic model (NQM). We experimentally demonstrate that optimization algorithms that employ preconditioning, specifically Adam and K-FAC, result in much larger critical batch sizes than stochastic gradient descent with momentum. We also demonstrate that the NQM captures many of the essential features of real neural network training, despite being drastically simpler to work with. The NQM predicts our results with preconditioned optimizers, previous results with accelerated gradient descent, and other results around optimal learning rates and large batch training, making it a useful tool to generate testable predictions about neural network optimization.Comment: NeurIPS 201

Demon: Improved Neural Network Training with Momentum Decay

Momentum is a widely used technique for gradient-based optimizers in deep learning. In this paper, we propose a decaying momentum (\textsc{Demon}) rule. We conduct the first large-scale empirical analysis of momentum decay methods for modern neural network optimization, in addition to the most popular learning rate decay schedules. Across 28 relevant combinations of models, epochs, datasets, and optimizers, \textsc{Demon} achieves the highest number of Top-1 and Top-3 finishes at 39\% and 85\% respectively, almost doubling the second-placed learning rate cosine schedule at 17\% and 60\%, respectively. \textsc{Demon} also outperforms other widely used schedulers including, but not limited to, the learning rate step schedule, linear schedule, OneCycle schedule, and exponential schedule. Compared with the widely used learning rate step schedule, \textsc{Demon} is observed to be less sensitive to parameter tuning, which is critical to training neural networks in practice. Results are demonstrated across a variety of settings and architectures, including image classification, generative models, and language models. \textsc{Demon} is easy to implement, requires no additional tuning, and incurs almost no extra computational overhead compared to the vanilla counterparts. Code is readily available.Comment: 12 page