In stochastic optimization, a common tool to deal sequentially with large sample is to consider the well-known stochastic gradient algorithm. Nevertheless, since the stepsequence is the same for each direction, this can lead to bad results in practice in case of ill-conditionned problem. To overcome this, adaptive gradient algorithms such that Adagrad or Stochastic Newton algorithms should be prefered. This paper is devoted to the non asymptotic analyis of these adaptive gradient algorithms for strongly convex objective. All the theoretical results will be adapted to linear regression and regularized generalized linear model for both Adagrad and Stochastic Newton algorithms.
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Evolution strategy (ES) is one of promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally $L$-strongly convex functions with $U$-Lipschitz continuous gradient are derived as $\exp\left(-\Omega_{d\to\infty}\left(\frac{L}{d\cdot U}\right)\right)$ and $\exp\left(-\frac1d\right)$, respectively. Notably, any prior knowledge on the mathematical properties of the objective function such as Lipschitz constant is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require them.
We introduce a streaming framework for analyzing stochastic approximation/optimization problems. This streaming framework is analogous to solving optimization problems using time-varying mini-batches that arrive sequentially. We provide non-asymptotic convergence rates of various gradient-based algorithms; this includes the famous Stochastic Gradient (SG) descent (a.k.a. Robbins-Monro algorithm), mini-batch SG and time-varying mini-batch SG algorithms, as well as their iterated averages (a.k.a. Polyak-Ruppert averaging). We show i) how to accelerate convergence by choosing the learning rate according to the time-varying mini-batches, ii) that Polyak-Ruppert averaging achieves optimal convergence in terms of attaining the Cramer-Rao lower bound, and iii) how time-varying mini-batches together with Polyak-Ruppert averaging can provide variance reduction and accelerate convergence simultaneously, which is advantageous for many learning problems, such as online, sequential, and large-scale learning. We further demonstrate these favorable effects for various time-varying mini-batches.
In this paper, we study a class of bilevel optimization problems, also known as simple bilevel optimization, where we minimize a smooth objective function over the optimal solution set of another convex constrained optimization problem. Several iterative methods have been developed for tackling this class of problems. Alas, their convergence guarantees are either asymptotic for the upper-level objective, or the convergence rates are slow and sub-optimal. To address this issue, in this paper, we introduce a novel bilevel optimization method that locally approximates the solution set of the lower-level problem via a cutting plane, and then runs a conditional gradient update to decrease the upper-level objective. When the upper-level objective is convex, we show that our method requires ${\mathcal{O}}(\max\{1/\epsilon_f,1/\epsilon_g\})$ iterations to find a solution that is $\epsilon_f$-optimal for the upper-level objective and $\epsilon_g$-optimal for the lower-level objective. Moreover, when the upper-level objective is non-convex, our method requires ${\mathcal{O}}(\max\{1/\epsilon_f^2,1/(\epsilon_f\epsilon_g)\})$ iterations to find an $(\epsilon_f,\epsilon_g)$-optimal solution. We also prove stronger convergence guarantees under the H\"olderian error bound assumption on the lower-level problem. To the best of our knowledge, our method achieves the best-known iteration complexity for the considered class of bilevel problems.
Decentralized minimax optimization has been actively studied in the past few years due to its application in a wide range of machine learning models. However, the current theoretical understanding of its convergence rate is far from satisfactory since existing works only focus on the nonconvex-strongly-concave problem. This motivates us to study decentralized minimax optimization algorithms for the nonconvex-nonconcave problem. To this end, we develop two novel decentralized stochastic variance-reduced gradient descent ascent algorithms for the finite-sum nonconvex-nonconcave problem that satisfies the Polyak-{\L}ojasiewicz (PL) condition. In particular, our theoretical analyses demonstrate how to conduct local updates and perform communication to achieve the linear convergence rate. To the best of our knowledge, this is the first work achieving linear convergence rates for decentralized nonconvex-nonconcave problems. Finally, we verify the performance of our algorithms on both synthetic and real-world datasets. The experimental results confirm the efficacy of our algorithms.
In this paper, we study the stochastic linear bandit problem under the additional requirements of differential privacy, robustness and batched observations. In particular, we assume an adversary randomly chooses a constant fraction of the observed rewards in each batch, replacing them with arbitrary numbers. We present differentially private and robust variants of the arm elimination algorithm using logarithmic batch queries under two privacy models and provide regret bounds in both settings. In the first model, every reward in each round is reported by a potentially different client, which reduces to standard local differential privacy (LDP). In the second model, every action is "owned" by a different client, who may aggregate the rewards over multiple queries and privatize the aggregate response instead. To the best of our knowledge, our algorithms are the first simultaneously providing differential privacy and adversarial robustness in the stochastic linear bandits problem.
Alternating Direction Method of Multipliers (ADMM) is a popular method for solving large-scale Machine Learning problems. Stochastic ADMM was proposed to reduce the per iteration computational complexity, which is more suitable for big data problems. Recently, variance reduction techniques have been integrated with stochastic ADMM in order to get a faster convergence rate, such as SAG-ADMM and SVRG-ADMM. However, their convergence rate is still suboptimal w.r.t the smoothness constant. In this paper, we propose an accelerated stochastic ADMM algorithm with variance reduction, which enjoys a faster convergence than all the existing stochastic ADMM algorithms. We theoretically analyse its convergence rate and show its dependence on the smoothness constant is optimal. We also empirically validate its effectiveness and show its priority over other stochastic ADMM algorithms.
In this paper, we study the conditional stochastic optimization (CSO) problem which covers a variety of applications including portfolio selection, reinforcement learning, robust learning, causal inference, etc. The sample-averaged gradient of the CSO objective is biased due to its nested structure and therefore requires a high sample complexity to reach convergence. We introduce a general stochastic extrapolation technique that effectively reduces the bias. We show that for nonconvex smooth objectives, combining this extrapolation with variance reduction techniques can achieve a significantly better sample complexity than existing bounds. We also develop new algorithms for the finite-sum variant of CSO that also significantly improve upon existing results. Finally, we believe that our debiasing technique could be an interesting tool applicable to other stochastic optimization problems too.
Deep neural networks (DNNs) have achieved unprecedented success in the field of artificial intelligence (AI), including computer vision, natural language processing and speech recognition. However, their superior performance comes at the considerable cost of computational complexity, which greatly hinders their applications in many resource-constrained devices, such as mobile phones and Internet of Things (IoT) devices. Therefore, methods and techniques that are able to lift the efficiency bottleneck while preserving the high accuracy of DNNs are in great demand in order to enable numerous edge AI applications. This paper provides an overview of efficient deep learning methods, systems and applications. We start from introducing popular model compression methods, including pruning, factorization, quantization as well as compact model design. To reduce the large design cost of these manual solutions, we discuss the AutoML framework for each of them, such as neural architecture search (NAS) and automated pruning and quantization. We then cover efficient on-device training to enable user customization based on the local data on mobile devices. Apart from general acceleration techniques, we also showcase several task-specific accelerations for point cloud, video and natural language processing by exploiting their spatial sparsity and temporal/token redundancy. Finally, to support all these algorithmic advancements, we introduce the efficient deep learning system design from both software and hardware perspectives.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
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