Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of cross-validation performance with respect to all hyperparameters by chaining derivatives backwards through the entire training procedure. These gradients allow us to optimize thousands of hyperparameters, including step-size and momentum schedules, weight initialization distributions, richly parameterized regularization schemes, and neural network architectures. We compute hyperparameter gradients by exactly reversing the dynamics of stochastic gradient descent with momentum.
Generalization is a central problem in Machine Learning. Most prediction methods require careful calibration of hyperparameters carried out on a hold-out \textit{validation} dataset to achieve generalization. The main goal of this paper is to present a novel approach based on a new measure of risk that allows us to develop novel fully automatic procedures for generalization. We illustrate the pertinence of this new framework in the regression problem. The main advantages of this new approach are: (i) it can simultaneously train the model and perform regularization in a single run of a gradient…
Cornell University（康奈尔大学）；University of Connecticut（康涅狄格大学）
It is well known that adding any skew symmetric matrix to the gradient of Langevin dynamics algorithm results in a non-reversible diffusion with improved convergence rate. This paper presents a gradient algorithm to adaptively optimize the choice of the skew symmetric matrix. The resulting algorithm involves a non-reversible diffusion algorithm cross coupled with a stochastic gradient algorithm that adapts the skew symmetric matrix. The algorithm uses the same data as the classical Langevin algorithm. A weak convergence proof is given for the optimality of the choice of the skew symmetric matr…
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…
Indian Institute of Management Ahmedabad（印度管理学院艾哈迈达巴德分校）
Hyperparameter tuning is an active area of research in machine learning, where the aim is to identify the optimal hyperparameters that provide the best performance on the validation set. Hyperparameter tuning is often achieved using naive techniques, such as random search and grid search. However, most of these methods seldom lead to an optimal set of hyperparameters and often get very expensive. In this paper, we propose a bilevel solution method for solving the hyperparameter optimization problem that does not suffer from the drawbacks of the earlier studies. The proposed method is general a…
In reinforcement learning, temporal difference (TD) is the most direct algorithm to learn the value function of a policy. For large or infinite state spaces, exact representations of the value function are usually not available, and it must be approximated by a function in some parametric family. However, with \emph{nonlinear} parametric approximations (such as neural networks), TD is not guaranteed to converge to a good approximation of the true value function within the family, and is known to diverge even in relatively simple cases. TD lacks an interpretation as a stochastic gradient descen…
Stochastic Gradient Langevin Dynamics (SGLD) is a powerful algorithm for optimizing a non-convex objective, where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. SGLD is based on the overdamped Langevin diffusion which is reversible in time. By adding an anti-symmetric matrix to the drift term of the overdamped Langevin diffusion, one gets a non-reversible diffusion that converges to the same stationary distribution with a faster convergence rate. In this paper, we study the non reversible Stochastic Gradient …
We study two procedures (reverse-mode and forward-mode) for computing the gradient of the validation error with respect to the hyperparameters of any iterative learning algorithm such as stochastic gradient descent. These procedures mirror two methods of computing gradients for recurrent neural networks and have different trade-offs in terms of running time and space requirements. Our formulation of the reverse-mode procedure is linked to previous work by Maclaurin et al. [2015] but does not require reversible dynamics. The forward-mode procedure is suitable for real-time hyperparameter update…
D.E. Shaw Research（D.E. Shaw 研究所）；Department of Biochemistry and Molecular Biophysics, Columbia University（哥伦比亚大学生物化学与分子生物物理学系）
Identifying optimal values for a high-dimensional set of hyperparameters is a problem that has received growing attention given its importance to large-scale machine learning applications such as neural architecture search. Recently developed optimization methods can be used to select thousands or even millions of hyperparameters. Such methods often yield overfit models, however, leading to poor performance on unseen data. We argue that this overfitting results from using the standard hyperparameter optimization objective function. Here we present an alternative objective that is equivalent to…
Université Paris-Saclay（巴黎-萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Institut Mathématique de Bourgogne（勃艮第数学研究所）；Université de Bourgogne（勃艮第大学）；Google Research（谷歌研究院）；CNRS（法国国家科学研究中心）；IMAG（蒙彼利埃亚历山大·格罗滕迪克研究所）；Université de Montpellier（蒙彼利埃大学）
Setting regularization parameters for Lasso-type estimators is notoriously difficult, though crucial in practice. The most popular hyperparameter optimization approach is grid-search using held-out validation data. Grid-search however requires to choose a predefined grid for each parameter, which scales exponentially in the number of parameters. Another approach is to cast hyperparameter optimization as a bi-level optimization problem, one can solve by gradient descent. The key challenge for these methods is the estimation of the gradient with respect to the hyperparameters. Computing this gra…
JD Finance America Corporation（京东金融美国公司）；University of Pittsburgh（匹兹堡大学）；Nanjing University of Information Science & Technology（南京信息工程大学）
Modern machine learning algorithms usually involve tuning multiple (from one to thousands) hyperparameters which play a pivotal role in terms of model generalizability. Black-box optimization and gradient-based algorithms are two dominant approaches to hyperparameter optimization while they have totally distinct advantages. How to design a new hyperparameter optimization technique inheriting all benefits from both approaches is still an open problem. To address this challenging problem, in this paper, we propose a new hyperparameter optimization method with zeroth-order hyper-gradients (HOZOG)…
University of Freiburg（弗赖堡大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
We propose probabilistic models that can extrapolate learning curves of iterative machine learning algorithms, such as stochastic gradient descent for training deep networks, based on training data with variable-length learning curves. We study instantiations of this framework based on random forests and Bayesian recurrent neural networks. Our experiments show that these models yield better predictions than state-of-the-art models from the hyperparameter optimization literature when extrapolating the performance of neural networks trained with different hyperparameter settings.
Google Research（谷歌研究院）；Vector Institute（向量研究所）；University of Toronto（多伦多大学）；DeepMind（深度思维）；Institute for Advanced Study（普林斯顿高等研究院）
Selecting an optimizer is a central step in the contemporary deep learning pipeline. In this paper, we demonstrate the sensitivity of optimizer comparisons to the hyperparameter tuning protocol. Our findings suggest that the hyperparameter search space may be the single most important factor explaining the rankings obtained by recent empirical comparisons in the literature. In fact, we show that these results can be contradicted when hyperparameter search spaces are changed. As tuning effort grows without bound, more general optimizers should never underperform the ones they can approximate (i…
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…
Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a variant of stochastic gradient descent where the random samples are taken on the trajectory of a Markov chain. Existing results of this method assume convex objectives and a reversible Markov chain and thus have their limitations. We establish new non-ergodic convergence under wider step sizes, for nonconvex problems, and for non-reversible finite-state Markov…
State Key Laboratory of Scientific and Engineering Computing（科学与工程计算国家重点实验室）；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）
Following the same routine as [SSJ20], we continue to present the theoretical analysis for stochastic gradient descent with momentum (SGD with momentum) in this paper. Differently, for SGD with momentum, we demonstrate it is the two hyperparameters together, the learning rate and the momentum coefficient, that play the significant role for the linear rate of convergence in non-convex optimization. Our analysis is based on the use of a hyperparameters-dependent stochastic differential equation (hp-dependent SDE) that serves as a continuous surrogate for SGD with momentum. Similarly, we establis…
An Exploration of Non-Euclidean Gradient Descent: Muon and its Many Variants
Department of Computer Science, George Mason University（乔治·马歇尔大学计算机科学系）；Center for Cosmology and Particle Physics, New York University（纽约大学宇宙与粒子物理中心）；Center for Computational Mathematics, Flatiron Institute（Flatiron研究所计算数学中心）
It has long been argued that minibatch stochastic gradient descent can generalize better than large batch gradient descent in deep neural networks. However recent papers have questioned this claim, arguing that this effect is simply a consequence of suboptimal hyperparameter tuning or insufficient compute budgets when the batch size is large. In this paper, we perform carefully designed experiments and rigorous hyperparameter sweeps on a range of popular models, which verify that small or moderately large batch sizes can substantially outperform very large batches on the test set. This occurs …
A traditional artificial neural network (ANN) is normally trained slowly by a gradient descent algorithm, such as the backpropagation algorithm, since a large number of hyperparameters of the ANN need to be fine-tuned with many training epochs. Since a large number of hyperparameters of a deep neural network, such as a convolutional neural network, occupy much memory, a memory-inefficient deep learning model is not ideal for real-time Internet of Things (IoT) applications on various devices, such as mobile phones. Thus, it is necessary to develop fast and memory-efficient Artificial Intelligen…
We propose ActiveLR, an optimization meta algorithm that localizes the learning rate, $α$, and adapts them at each epoch according to whether the gradient at each epoch changes sign or not. This sign-conscious algorithm is aware of whether from the previous step to the current one the update of each parameter has been too large or too small and adjusts the $α$ accordingly. We implement the Active version (ours) of widely used and recently published gradient descent optimizers, namely SGD with momentum, AdamW, RAdam, and AdaBelief. Our experiments on ImageNet, CIFAR-10, WikiText-103, WikiText-2…
Tuning complex machine learning systems is challenging. Machine learning typically requires to set hyperparameters, be it regularization, architecture, or optimization parameters, whose tuning is critical to achieve good predictive performance. To democratize access to machine learning systems, it is essential to automate the tuning. This paper presents Amazon SageMaker Automatic Model Tuning (AMT), a fully managed system for gradient-free optimization at scale. AMT finds the best version of a trained machine learning model by repeatedly evaluating it with different hyperparameter configuratio…
It is already reported in the literature that the performance of a machine learning algorithm is greatly impacted by performing proper Hyper-Parameter optimization. One of the ways to perform Hyper-Parameter optimization is by manual search but that is time consuming. Some of the common approaches for performing Hyper-Parameter optimization are Grid search Random search and Bayesian optimization using Hyperopt. In this paper, we propose a brand new approach for hyperparameter improvement i.e. Randomized-Hyperopt and then tune the hyperparameters of the XGBoost i.e. the Extreme Gradient Boostin…
University of Utah（犹他大学）；University of Kentucky（肯塔基大学）
Momentum plays a crucial role in stochastic gradient-based optimization algorithms for accelerating or improving training deep neural networks (DNNs). In deep learning practice, the momentum is usually weighted by a well-calibrated constant. However, tuning hyperparameters for momentum can be a significant computational burden. In this paper, we propose a novel \emph{adaptive momentum} for improving DNNs training; this adaptive momentum, with no momentum related hyperparameter required, is motivated by the nonlinear conjugate gradient (NCG) method. Stochastic gradient descent (SGD) with this n…
Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer
John Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·保罗森工程与应用科学学校）；Center for Brain Sciences（脑科学研究中心）；Kempner Institute（凯普纳研究所）
Practical results have shown that deep learning optimizers using small constant learning rates, hyperparameters close to one, and large batch sizes can find the model parameters of deep neural networks that minimize the loss functions. We first show theoretical evidence that the momentum method (Momentum) and adaptive moment estimation (Adam) perform well in the sense that the upper bound of the theoretical performance measure is small with a small constant learning rate, hyperparameters close to one, and a large batch size. Next, we show that there exists a batch size called the critical batc…
Convergence and convergence rate analyses of adaptive methods, such as Adaptive Moment Estimation (Adam) and its variants, have been widely studied for nonconvex optimization. The analyses are based on assumptions that the expected or empirical average loss function is Lipschitz smooth (i.e., its gradient is Lipschitz continuous) and the learning rates depend on the Lipschitz constant of the Lipschitz continuous gradient. Meanwhile, numerical evaluations of Adam and its variants have clarified that using small constant learning rates without depending on the Lipschitz constant and hyperparamet…
Université Paris-Saclay（巴黎萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Université de Bourgogne（勃艮第大学）；Institut Mathématique de Bourgogne（勃艮第数学研究所）；Università degli Studi di Genova（热那亚大学）；Google Research（谷歌研究院）；CNRS（法国国家科学研究中心）；Université de Montpellier（蒙彼利埃大学）
Finding the optimal hyperparameters of a model can be cast as a bilevel optimization problem, typically solved using zero-order techniques. In this work we study first-order methods when the inner optimization problem is convex but non-smooth. We show that the forward-mode differentiation of proximal gradient descent and proximal coordinate descent yield sequences of Jacobians converging toward the exact Jacobian. Using implicit differentiation, we show it is possible to leverage the non-smoothness of the inner problem to speed up the computation. Finally, we provide a bound on the error made …
Adaptive Moment Estimation (ADAM) is a very popular training algorithm for deep neural networks and belongs to the family of adaptive gradient descent optimizers. However to the best of the authors knowledge no complete convergence analysis exists for ADAM. The contribution of this paper is a method for the local convergence analysis in batch mode for a deterministic fixed training set, which gives necessary conditions for the hyperparameters of the ADAM algorithm. Due to the local nature of the arguments the objective function can be non-convex but must be at least twice continuously differen…
Bayesian optimization is popular for optimizing time-consuming black-box objectives. Nonetheless, for hyperparameter tuning in deep neural networks, the time required to evaluate the validation error for even a few hyperparameter settings remains a bottleneck. Multi-fidelity optimization promises relief using cheaper proxies to such objectives --- for example, validation error for a network trained using a subset of the training points or fewer iterations than required for convergence. We propose a highly flexible and practical approach to multi-fidelity Bayesian optimization, focused on effic…
We propose a genetic algorithm (GA) for hyperparameter optimization of artificial neural networks which includes chromosomal crossover as well as a decoupling of parameters (i.e., weights and biases) from hyperparameters (e.g., learning rate, weight decay, and dropout) during sexual reproduction. Children are produced from three parents; two contributing hyperparameters and one contributing the parameters. Our version of population-based training (PBT) combines traditional gradient-based approaches such as stochastic gradient descent (SGD) with our GA to optimize both parameters and hyperparam…
KAUST（阿卜杜拉国王科技大学）；Stanford University（斯坦福大学）；University of California, Berkeley（加州大学伯克利分校）
Adaptivity is an important yet under-studied property in modern optimization theory. The gap between the state-of-the-art theory and the current practice is striking in that algorithms with desirable theoretical guarantees typically involve drastically different settings of hyperparameters, such as step-size schemes and batch sizes, in different regimes. Despite the appealing theoretical results, such divisive strategies provide little, if any, insight to practitioners to select algorithms that work broadly without tweaking the hyperparameters. In this work, blending the "geometrization" techn…
Tensor networks have demonstrated significant value for machine learning in a myriad of different applications. However, optimizing tensor networks using standard gradient descent has proven to be difficult in practice. Tensor networks suffer from initialization problems resulting in exploding or vanishing gradients and require extensive hyperparameter tuning. Efforts to overcome these problems usually depend on specific network architectures, or ad hoc prescriptions. In this paper we address the problems of initialization and hyperparameter tuning, making it possible to train tensor networks …
The performance of policy gradient methods is sensitive to hyperparameter settings that must be tuned for any new application. Widely used grid search methods for tuning hyperparameters are sample inefficient and computationally expensive. More advanced methods like Population Based Training that learn optimal schedules for hyperparameters instead of fixed settings can yield better results, but are also sample inefficient and computationally expensive. In this paper, we propose Hyperparameter Optimisation on the Fly (HOOF), a gradient-free algorithm that requires no more than one training run …
Federated learning enables multiple, distributed participants (potentially on different clouds) to collaborate and train machine/deep learning models by sharing parameters/gradients. However, sharing gradients, instead of centralizing data, may not be as private as one would expect. Reverse engineering attacks on plaintext gradients have been demonstrated to be practically feasible. Existing solutions for differentially private federated learning, while promising, lead to less accurate models and require nontrivial hyperparameter tuning. In this paper, we examine the use of additive homomorphi…
Phase diagram and eigenvalue dynamics of stochastic gradient descent in multilayer neural networks
Centre for Quantum Fields and Gravity（量子场与引力中心）；Department of Physics, Swansea University（斯旺西大学物理系）；School of Mathematical Sciences, Queen Mary University of London（伦敦大学量子玛丽学院数学科学学院）
Bayesian Optimization is a popular tool for tuning algorithms in automatic machine learning (AutoML) systems. Current state-of-the-art methods leverage Random Forests or Gaussian processes to build a surrogate model that predicts algorithm performance given a certain set of hyperparameter settings. In this paper, we propose a new surrogate model based on gradient boosting, where we use quantile regression to provide optimistic estimates of the performance of an unobserved hyperparameter setting, and combine this with a distance metric between unobserved and observed hyperparameter settings to …
National University of Defense Technology（国防科技大学）；Tsinghua University（清华大学）；University of Utah（犹他大学）
Heavy ball momentum is crucial in accelerating (stochastic) gradient-based optimization algorithms for machine learning. Existing heavy ball momentum is usually weighted by a uniform hyperparameter, which relies on excessive tuning. Moreover, the calibrated fixed hyperparameter may not lead to optimal performance. In this paper, to eliminate the effort for tuning the momentum-related hyperparameter, we propose a new adaptive momentum inspired by the optimal choice of the heavy ball momentum for quadratic optimization. Our proposed adaptive heavy ball momentum can improve stochastic gradient de…
One of the most popular training algorithms for deep neural networks is the Adaptive Moment Estimation (Adam) introduced by Kingma and Ba. Despite its success in many applications there is no satisfactory convergence analysis: only local convergence can be shown for batch mode under some restrictions on the hyperparameters, counterexamples exist for incremental mode. Recent results show that for simple quadratic objective functions limit cycles of period 2 exist in batch mode, but only for atypical hyperparameters, and only for the algorithm without bias correction. %More general there are sev…
NYU Abu Dhabi（纽约大学阿布扎比分校）；Queen’s University Belfast（贝尔法斯特女王大学）
Machine Learning (ML) architectures have been applied to several applications that involve sensitive data, where a guarantee of users' data privacy is required. Differentially Private Stochastic Gradient Descent (DPSGD) is the state-of-the-art method to train privacy-preserving models. However, DPSGD comes at a considerable accuracy loss leading to sub-optimal privacy/utility trade-offs. Towards investigating new ground for better privacy-utility trade-off, this work questions; (i) if models' hyperparameters have any inherent impact on ML models' privacy-preserving properties, and (ii) if mode…
Gradient-based hyperparameter optimization has earned a widespread popularity in the context of few-shot meta-learning, but remains broadly impractical for tasks with long horizons (many gradient steps), due to memory scaling and gradient degradation issues. A common workaround is to learn hyperparameters online, but this introduces greediness which comes with a significant performance drop. We propose forward-mode differentiation with sharing (FDS), a simple and efficient algorithm which tackles memory scaling issues with forward-mode differentiation, and gradient degradation issues by sharin…
CEREMADE（CEREMADE（巴黎多芬大学数学与决策研究中心））；CNRS（法国国家科学研究中心）；Université Paris-Dauphine（巴黎多芬大学）；PSL Research University（巴黎文理研究大学）；École Normale Supérieure（巴黎高等师范学院）
Most models in machine learning contain at least one hyperparameter to control for model complexity. Choosing an appropriate set of hyperparameters is both crucial in terms of model accuracy and computationally challenging. In this work we propose an algorithm for the optimization of continuous hyperparameters using inexact gradient information. An advantage of this method is that hyperparameters can be updated before model parameters have fully converged. We also give sufficient conditions for the global convergence of this method, based on regularity conditions of the involved functions and …
Variance Reduction for Stochastic Gradient Generalized Non-reversible Langevin Monte Carlo Algorithms
FinTech Thrust, Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州）金融科技方向）；School of Mathematical Science, Fudan University（复旦大学数学学院）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）
The University of Edinburgh（爱丁堡大学）；Samsung AI Research Centre, Cambridge（三星人工智能研究中心）
Gradient-based meta-learning and hyperparameter optimization have seen significant progress recently, enabling practical end-to-end training of neural networks together with many hyperparameters. Nevertheless, existing approaches are relatively expensive as they need to compute second-order derivatives and store a longer computational graph. This cost prevents scaling them to larger network architectures. We present EvoGrad, a new approach to meta-learning that draws upon evolutionary techniques to more efficiently compute hypergradients. EvoGrad estimates hypergradient with respect to hyperpa…
Department of Electrical Engineering（电气工程系）；Stanford University（斯坦福大学）
Cross-validation is the workhorse of modern applied statistics and machine learning, as it provides a principled framework for selecting the model that maximizes generalization performance. In this paper, we show that the cross-validation risk is differentiable with respect to the hyperparameters and training data for many common machine learning algorithms, including logistic regression, elastic-net regression, and support vector machines. Leveraging this property of differentiability, we propose a cross-validation gradient method (CVGM) for hyperparameter optimization. Our method enables eff…
