In this paper, we propose a simple, fast and easy to implement algorithm LOSSGRAD (locally optimal step-size in gradient descent), which automatically modifies the step-size in gradient descent during neural networks training. Given a function $f$, a point $x$, and the gradient $\nabla_x f$ of $f$, we aim to find the step-size $h$ which is (locally) optimal, i.e. satisfies: $$ h=arg\,min_{t \geq 0} f(x-t \nabla_x f). $$ Making use of quadratic approximation, we show that the algorithm satisfies the above assumption. We experimentally show that our method is insensitive to the choice of initial learning rate while achieving results comparable to other methods.
Huawei Technologies Canada（加拿大华为技术有限公司）；University of Alberta（阿尔伯塔大学）
Gradient descent is slow to converge for ill-conditioned problems and non-convex problems. An important technique for acceleration is step-size adaptation. The first part of this paper contains a detailed review of step-size adaptation methods, including Polyak step-size, L4, LossGrad, Adam, IDBD, and Hypergradient descent, and the relation of step-size adaptation to meta-gradient methods. In the second part of this paper, we propose a new class of methods of accelerating gradient descent that have some distinctiveness from existing techniques. The new methods, which we call {\em step-size pla…
Many machine learning models require a training procedure based on running stochastic gradient descent. A key element for the efficiency of those algorithms is the choice of the learning rate schedule. While finding good learning rates schedules using Bayesian optimisation has been tackled by several authors, adapting it dynamically in a data-driven way is an open question. This is of high practical importance to users that need to train a single, expensive model. To tackle this problem, we introduce an original probabilistic model for traces of optimisers, based on latent Gaussian processes a…
Stochastic Gradient Descent (SGD) methods are prominent for training machine learning and deep learning models. The performance of these techniques depends on their hyperparameter tuning over time and varies for different models and problems. Manual adjustment of hyperparameters is very costly and time-consuming, and even if done correctly, it lacks theoretical justification which inevitably leads to "rule of thumb" settings. In this paper, we propose a generic approach that utilizes the statistics of an unbiased gradient estimator to automatically and simultaneously adjust two paramount hyper…
Inspired by dynamic programming, we propose Stochastic Virtual Gradient Descent (SVGD) algorithm where the Virtual Gradient is defined by computational graph and automatic differentiation. The method is computationally efficient and has little memory requirements. We also analyze the theoretical convergence properties and implementation of the algorithm. Experimental results on multiple datasets and network models show that SVGD has advantages over other stochastic optimization methods.
The objective of this research is to enhance performance of Stochastic Gradient Descent (SGD) algorithm in text classification. In our research, we proposed using SGD learning with Grid-Search approach to fine-tuning hyper-parameters in order to enhance the performance of SGD classification. We explored different settings for representation, transformation and weighting features from the summary description of terrorist attacks incidents obtained from the Global Terrorism Database as a pre-classification step, and validated SGD learning on Support Vector Machine (SVM), Logistic Regression and …
IBM Research AI（IBM研究院人工智能部门）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Evolutionary stochastic gradient descent (ESGD) was proposed as a population-based approach that combines the merits of gradient-aware and gradient-free optimization algorithms for superior overall optimization performance. In this paper we investigate a variant of ESGD for optimization of acoustic models for automatic speech recognition (ASR). In this variant, we assume the existence of a well-trained acoustic model and use it as an anchor in the parent population whose good "gene" will propagate in the evolution to the offsprings. We propose an ESGD algorithm leveraging the anchor models suc…
In min-min optimization or max-min optimization, one has to compute the gradient of a function defined as a minimum. In most cases, the minimum has no closed-form, and an approximation is obtained via an iterative algorithm. There are two usual ways of estimating the gradient of the function: using either an analytic formula obtained by assuming exactness of the approximation, or automatic differentiation through the algorithm. In this paper, we study the asymptotic error made by these estimators as a function of the optimization error. We find that the error of the automatic estimator is clos…
Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze in the convex and non-convex settings a generalized version of the AdaGrad stepsizes. We show sufficient conditions for these st…
The Cartesian reverse derivative is a categorical generalization of reverse-mode automatic differentiation. We use this operator to generalize several optimization algorithms, including a straightforward generalization of gradient descent and a novel generalization of Newton's method. We then explore which properties of these algorithms are preserved in this generalized setting. First, we show that the transformation invariances of these algorithms are preserved: while generalized Newton's method is invariant to all invertible linear transformations, generalized gradient descent is invariant o…
INRIA Lille（法国国家信息与自动化研究所里尔中心）；Politecnico di Milano（米兰理工大学）
In this paper, we propose a novel approach to automatically determine the batch size in stochastic gradient descent methods. The choice of the batch size induces a trade-off between the accuracy of the gradient estimate and the cost in terms of samples of each update. We propose to determine the batch size by optimizing the ratio between a lower bound to a linear or quadratic Taylor approximation of the expected improvement and the number of samples used to estimate the gradient. The performance of the proposed approach is empirically compared with related methods on popular classification tas…
Deep learning's successes are often attributed to its ability to automatically discover new representations of the data, rather than relying on handcrafted features like other learning methods. We show, however, that deep networks learned by the standard gradient descent algorithm are in fact mathematically approximately equivalent to kernel machines, a learning method that simply memorizes the data and uses it directly for prediction via a similarity function (the kernel). This greatly enhances the interpretability of deep network weights, by elucidating that they are effectively a superposit…
Empirical risk minimization is the main tool for prediction problems, but its extension to relational data remains unsolved. We solve this problem using recent ideas from graph sampling theory to (i) define an empirical risk for relational data and (ii) obtain stochastic gradients for this empirical risk that are automatically unbiased. This is achieved by considering the method by which data is sampled from a graph as an explicit component of model design. By integrating fast implementations of graph sampling schemes with standard automatic differentiation tools, we provide an efficient turnk…
Convergence detection of iterative stochastic optimization methods is of great practical interest. This paper considers stochastic gradient descent (SGD) with a constant learning rate and momentum. We show that there exists a transient phase in which iterates move towards a region of interest, and a stationary phase in which iterates remain bounded in that region around a minimum point. We construct a statistical diagnostic test for convergence to the stationary phase using the inner product between successive gradients and demonstrate that the proposed diagnostic works well. We theoretically …
Working with any gradient-based machine learning algorithm involves the tedious task of tuning the optimizer's hyperparameters, such as its step size. Recent work has shown how the step size can itself be optimized alongside the model parameters by manually deriving expressions for "hypergradients" ahead of time. We show how to automatically compute hypergradients with a simple and elegant modification to backpropagation. This allows us to easily apply the method to other optimizers and hyperparameters (e.g. momentum coefficients). We can even recursively apply the method to its own hyper-hype…
Google DeepMind（谷歌DeepMind）；University of Oxford（牛津大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
The move from hand-designed features to learned features in machine learning has been wildly successful. In spite of this, optimization algorithms are still designed by hand. In this paper we show how the design of an optimization algorithm can be cast as a learning problem, allowing the algorithm to learn to exploit structure in the problems of interest in an automatic way. Our learned algorithms, implemented by LSTMs, outperform generic, hand-designed competitors on the tasks for which they are trained, and also generalize well to new tasks with similar structure. We demonstrate this on a nu…
University of Oxford（牛津大学）；University of British Columbia（不列颠哥伦比亚大学）
We introduce a general method for improving the convergence rate of gradient-based optimizers that is easy to implement and works well in practice. We demonstrate the effectiveness of the method in a range of optimization problems by applying it to stochastic gradient descent, stochastic gradient descent with Nesterov momentum, and Adam, showing that it significantly reduces the need for the manual tuning of the initial learning rate for these commonly used algorithms. Our method works by dynamically updating the learning rate during optimization using the gradient with respect to the learning…
CEREMADE, Université Paris Dauphine - PSL Research University（CEREMADE，巴黎多芬大学-巴黎文理研究大学）
The minimization of the loss function is of paramount importance in deep neural networks. On the other hand, many popular optimization algorithms have been shown to correspond to some evolution equation of gradient flow type. Inspired by the numerical schemes used for general evolution equations we introduce a second order stochastic Runge Kutta method and show that it yields a consistent procedure for the minimization of the loss function. In addition it can be coupled, in an adaptive framework, with a Stochastic Gradient Descent (SGD) to adjust automatically the learning rate of the SGD, wit…
The Chinese University of Hong Kong（香港中文大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；The University of Hong Kong（香港大学）
One of the major issues in stochastic gradient descent (SGD) methods is how to choose an appropriate step size while running the algorithm. Since the traditional line search technique does not apply for stochastic optimization algorithms, the common practice in SGD is either to use a diminishing step size, or to tune a fixed step size by hand, which can be time consuming in practice. In this paper, we propose to use the Barzilai-Borwein (BB) method to automatically compute step sizes for SGD and its variant: stochastic variance reduced gradient (SVRG) method, which leads to two algorithms: SGD…
Training deep neural networks is a highly nontrivial task, involving carefully selecting appropriate training algorithms, scheduling step sizes and tuning other hyperparameters. Trying different combinations can be quite labor-intensive and time consuming. Recently, researchers have tried to use deep learning algorithms to exploit the landscape of the loss function of the training problem of interest, and learn how to optimize over it in an automatic way. In this paper, we propose a new learning-to-learn model and some useful and practical tricks. Our optimizer outperforms generic, hand-crafte…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；MIT（麻省理工学院）
This paper considers the problem of implementing large-scale gradient descent algorithms in a distributed computing setting in the presence of {\em straggling} processors. To mitigate the effect of the stragglers, it has been previously proposed to encode the data with an erasure-correcting code and decode at the master server at the end of the computation. We, instead, propose to encode the second-moment of the data with a low density parity-check (LDPC) code. The iterative decoding algorithms for LDPC codes have very low computational overhead and the number of decoding iterations can be mad…
IBM Research（IBM研究院）；George Mason University（乔治梅森大学）；Duke Unversity（杜克大学）
Distributed Deep Learning (DDL) is essential for large-scale Deep Learning (DL) training. Synchronous Stochastic Gradient Descent (SSGD) 1 is the de facto DDL optimization method. Using a sufficiently large batch size is critical to achieving DDL runtime speedup. In a large batch setting, the learning rate must be increased to compensate for the reduced number of parameter updates. However, a large learning rate may harm convergence in SSGD and training could easily diverge. Recently, Decentralized Parallel SGD (DPSGD) has been proposed to improve distributed training speed. In this paper, we …
University of Electronic Science and Technology of China（电子科技大学）；Harbin Institute of Technology（哈尔滨工业大学）；Peng Cheng Lab（鹏城实验室）
Stein variational gradient descent (SVGD) and its variants have shown promising successes in approximate inference for complex distributions. In practice, we notice that the kernel used in SVGD-based methods has a decisive effect on the empirical performance. Radial basis function (RBF) kernel with median heuristics is a common choice in previous approaches, but unfortunately this has proven to be sub-optimal. Inspired by the paradigm of Multiple Kernel Learning (MKL), our solution to this flaw is using a combination of multiple kernels to approximate the optimal kernel, rather than a single o…
Recent advances show that deep neural networks are not robust to deliberately crafted adversarial examples which many are generated by adding human imperceptible perturbation to clear input. Consider $l_2$ norms attacks, Project Gradient Descent (PGD) and the Carlini and Wagner (C\&W) attacks are the two main methods, where PGD control max perturbation for adversarial examples while C\&W approach treats perturbation as a regularization term optimized it with loss function together. If we carefully set parameters for any individual input, both methods become similar. In general, PGD attacks per…
We introduce a technique that can automatically tune the parameters of a rule-based computer vision system comprised of thresholds, combinational logic, and time constants. This lets us retain the flexibility and perspicacity of a conventionally structured system while allowing us to perform approximate gradient descent using labeled data. While this is only a heuristic procedure, as far as we are aware there is no other efficient technique for tuning such systems. We describe the components of the system and the associated supervised learning mechanism. We also demonstrate the utility of the …
Randomized Forward Mode of Automatic Differentiation For Optimization Algorithms
Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Mathematics, North Carolina State University（北卡罗来纳州立大学数学系）；Mathematical Institute for Data Science, Pohang University of Science and Technology (POSTECH)（浦项科技大学数据科学数学研究所）
Provable and Practical Online Learning Rate Adaptation with Hypergradient Descent
Stanford University（斯坦福大学）；ICME, Stanford University（斯坦福大学ICME）；Department of Mathematics, Stanford University（斯坦福大学数学系）；Department of Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程学院）
Donders Centre for Cognition, Radboud University（拉德堡德大学唐德斯认知中心）；One Planet（一个星球（研究机构））
The automation of probabilistic reasoning is one of the primary aims of machine learning. Recently, the confluence of variational inference and deep learning has led to powerful and flexible automatic inference methods that can be trained by stochastic gradient descent. In particular, normalizing flows are highly parameterized deep models that can fit arbitrarily complex posterior densities. However, normalizing flows struggle in highly structured probabilistic programs as they need to relearn the forward-pass of the program. Automatic structured variational inference (ASVI) remedies this prob…
Paradigm Inc.（第四范式公司）；Hong Kong University of Science and Technology（香港科技大学）
Feature crossing captures interactions among categorical features and is useful to enhance learning from tabular data in real-world businesses. In this paper, we present AutoCross, an automatic feature crossing tool provided by 4Paradigm to its customers, ranging from banks, hospitals, to Internet corporations. By performing beam search in a tree-structured space, AutoCross enables efficient generation of high-order cross features, which is not yet visited by existing works. Additionally, we propose successive mini-batch gradient descent and multi-granularity discretization to further improve …
We used convolutional neural networks (CNNs) for automatic sleep stage scoring based on single-channel electroencephalography (EEG) to learn task-specific filters for classification without using prior domain knowledge. We used an openly available dataset from 20 healthy young adults for evaluation and applied 20-fold cross-validation. We used class-balanced random sampling within the stochastic gradient descent (SGD) optimization of the CNN to avoid skewed performance in favor of the most represented sleep stages. We achieved high mean F1-score (81%, range 79-83%), mean accuracy across indivi…
The successes of deep learning, variational inference, and many other fields have been aided by specialized implementations of reverse-mode automatic differentiation (AD) to compute gradients of mega-dimensional objectives. The AD techniques underlying these tools were designed to compute exact gradients to numerical precision, but modern machine learning models are almost always trained with stochastic gradient descent. Why spend computation and memory on exact (minibatch) gradients only to use them for stochastic optimization? We develop a general framework and approach for randomized automa…
Private Gradient Descent for Linear Regression: Tighter Error Bounds and Instance-Specific Uncertainty Estimation
Paul G Allen School of Computer Science and Engineering, University of Washington（华盛顿大学保罗·G·艾伦计算机科学与工程学院）；Boston University（波士顿大学）；Google DeepMind（谷歌DeepMind）
We study two practically important cases of model based clustering using Gaussian Mixture Models: (1) when there is misspecification and (2) on high dimensional data, in the light of recent advances in Gradient Descent (GD) based optimization using Automatic Differentiation (AD). Our simulation studies show that EM has better clustering performance, measured by Adjusted Rand Index, compared to GD in cases of misspecification, whereas on high dimensional data GD outperforms EM. We observe that both with EM and GD there are many solutions with high likelihood but poor cluster interpretation. To …
A new gradient-based optimization approach by automatically scheduling the learning rate has been proposed recently, which is called Binary Forward Exploration (BFE). The Adaptive version of BFE has also been discussed thereafter. In this paper, the improved algorithms based on them will be investigated, in order to optimize the efficiency and robustness of the new methodology. This improved approach provides a new perspective to scheduling the update of learning rate and will be compared with the stochastic gradient descent, aka SGD algorithm with momentum or Nesterov momentum and the most su…
The University of Oslo（奥斯陆大学）；Axon research（Axon研究机构）
While Standard gradient descent is one very popular optimisation method, its convergence cannot be proven beyond the class of functions whose gradient is globally Lipschitz continuous. As such, it is not actually applicable to realistic applications such as Deep Neural Networks. In this paper, we prove that its backtracking variant behaves very nicely, in particular convergence can be shown for all Morse functions. The main theoretical result of this paper is as follows. Theorem. Let $f:\mathbb{R}^k\rightarrow \mathbb{R}$ be a $C^1$ function, and $\{z_n\}$ a sequence constructed from the Backt…
This paper investigates the stochastic optimization problem with a focus on developing scalable parallel algorithms for deep learning tasks. Our solution involves a reformation of the objective function for stochastic optimization in neural network models, along with a novel parallel strategy, coined weighted aggregating stochastic gradient descent (WASGD). Following a theoretical analysis on the characteristics of the new objective function, WASGD introduces a decentralized weighted aggregating scheme based on the performance of local workers. Without any center variable, the new method autom…
University of Oxford（牛津大学）；National University of Ireland Maynooth（爱尔兰国立梅努斯大学）；Microsoft（微软）；University of British Columbia（不列颠哥伦比亚大学）
Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic differentiation algorithms that also includes the forward mode. We present a method to compute gradients based solely on the directional derivative that one can compute exactly and efficiently via the forward mode. We call this formulation the forward gradient, an unbiased estimate of the gradient that can be evaluated in a single forward run of the function, ent…
Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）
We reconsider the stochastic (sub)gradient approach to the unconstrained primal L1-SVM optimization. We observe that if the learning rate is inversely proportional to the number of steps, i.e., the number of times any training pattern is presented to the algorithm, the update rule may be transformed into the one of the classical perceptron with margin in which the margin threshold increases linearly with the number of steps. Moreover, if we cycle repeatedly through the possibly randomly permuted training set the dual variables defined naturally via the expansion of the weight vector as a linea…
University of Electronic Science and Technology of China（电子科技大学）
In this paper, we propose a physically inspired graph-theoretical clustering method, which first makes the data points organized into an attractive graph, called In-Tree, via a physically inspired rule, called Nearest Descent (ND). In particular, the rule of ND works to select the nearest node in the descending direction of potential as the parent node of each node, which is in essence different from the classical Gradient Descent or Steepest Descent. The constructed In-Tree proves a very good candidate for clustering due to its particular features and properties. In the In-Tree, the original …
Solving Inverse Problems with Deep Linear Neural Networks: Global Convergence Guarantees for Gradient Descent with Weight Decay
Department of Mathematics, Technical University of Munich, Munich Center for Machine Learning (MCML), Munich, Germany（数学系，慕尼黑技术大学，慕尼黑机器学习中心（MCML），慕尼黑，德国）；Committee on Computational and Applied Mathematics, University of Chicago, Chicago, IL（计算与应用数学委员会，芝加哥大学，芝加哥，伊利诺伊）；Department of Electrical & Computer Engineering, University of Washington, Seattle, WA（电气与计算机工程系，华盛顿大学，西雅图，华盛顿）；Departments of Statistics and Computer Science, University of Chicago, Chicago, IL（统计学与计算机科学系，芝加哥大学，芝加哥，伊利诺伊）
IBM Research AI（IBM研究院人工智能部）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）
The past decade has witnessed great progress in Automatic Speech Recognition (ASR) due to advances in deep learning. The improvements in performance can be attributed to both improved models and large-scale training data. Key to training such models is the employment of efficient distributed learning techniques. In this article, we provide an overview of distributed training techniques for deep neural network acoustic models for ASR. Starting with the fundamentals of data parallel stochastic gradient descent (SGD) and ASR acoustic modeling, we will investigate various distributed training stra…
