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 @math 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.
Variance Reduction: The variance of stochastic gradients is detrimental to SGD, motivating variance reduction techniques @cite_10 @cite_33 @cite_29 @cite_15 @cite_34 @cite_1 @cite_48 that aim to reduce the variance incurred due to their stochastic process of estimation, and improve the convergence rate mainly for convex optimization while some are extended to non-convex problems @cite_42 @cite_30 @cite_32 . One of the most practical algorithms for better convergence rates includes momentum @cite_37 , modified momentum for accelerated gradient @cite_3 , and stochastic estimation of accelerated gradient (Accelerated-SGD) @cite_38 . These algorithms are more focused on the efficiency in convergence than the generalization of model for accuracy.
@cite_30 @cite_38 @cite_37 @cite_33 @cite_48 @cite_29 @cite_42 @cite_1 @cite_32 @cite_3 @cite_15 @cite_34 @cite_10
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Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov's accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). In general, “fast gradient” methods have provable improvements over gradient descent only for the deterministic case, where the gradients are exact. In the stochastic case, the popular explanations for their wide applicability is that when these fast gradient methods are applied in the stochastic case, they partially mimic their exact gradient counterparts, resulting in some practical gain. This work provides a counterpoint to this belief by proving that there are simple problem instances where these methods cannot outperform SGD despite the best setting of its parameters. These negative problem instances are, in an informal sense, generic; they do not look like carefully constructed pathological instances. These results suggest (along with empirical evidence) that HB or NAG's practical performance gains are a by-product of minibatching. Furthermore, this work provides a viable (and provable) alternative, which, on the same set of problem instances, significantly improves over HB, NAG, and SGD's performance. This algorithm, denoted as ASGD, is a simple to implement stochastic algorithm, based on a relatively less popular version of Nesterov's AGD. Extensive empirical results in this paper show that ASGD has performance gains over HB, NAG, and SGD. Stochastic gradient descent is popular for large scale optimization but has slow convergence asymptotically due to the inherent variance. To remedy this problem, we introduce an explicit variance reduction method for stochastic gradient descent which we call stochastic variance reduced gradient (SVRG). For smooth and strongly convex functions, we prove that this method enjoys the same fast convergence rate as those of stochastic dual coordinate ascent (SDCA) and Stochastic Average Gradient (SAG). However, our analysis is significantly simpler and more intuitive. Moreover, unlike SDCA or SAG, our method does not require the storage of gradients, and thus is more easily applicable to complex problems such as some structured prediction problems and neural network learning. We consider the fundamental problem in nonconvex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order nonconvex optimization remain to be full gradient descent that converges in O(1 e) iterations for smooth objectives, and stochastic gradient descent that converges in O(1 e2) iterations for objectives that are sum of smooth functions. We provide the first improvement in this line of research. Our result is based on the variance reduction trick recently introduced to convex optimization, as well as a brand new analysis of variance reduction that is suitable for non-convex optimization. For objectives that are sum of smooth functions, our first-order minibatch stochastic method converges with an O(1 e) rate, and is faster than full gradient descent by Ω(n1 3). We demonstrate the effectiveness of our methods on empirical risk minimizations with nonconvex loss functions and training neural nets. As application demands for zeroth-order (gradient-free) optimization accelerate, the need for variance reduced and faster converging approaches is also intensifying. This paper addresses these challenges by presenting: a) a comprehensive theoretical analysis of variance reduced zeroth-order (ZO) optimization, b) a novel variance reduced ZO algorithm, called ZO-SVRG, and c) an experimental evaluation of our approach in the context of two compelling applications, black-box chemical material classification and generation of adversarial examples from black-box deep neural network models. Our theoretical analysis uncovers an essential difficulty in the analysis of ZO-SVRG: the unbiased assumption on gradient estimates no longer holds. We prove that compared to its first-order counterpart, ZO-SVRG with a two-point random gradient estimator could suffer an additional error of order @math , where @math is the mini-batch size. To mitigate this error, we propose two accelerated versions of ZO-SVRG utilizing variance reduced gradient estimators, which achieve the best rate known for ZO stochastic optimization (in terms of iterations). Our extensive experimental results show that our approaches outperform other state-of-the-art ZO algorithms, and strike a balance between the convergence rate and the function query complexity. We propose a new stochastic alternating direction method of multipliers (ADMM) algorithm, which incrementally approximates the full gradient in the linearized ADMM formulation. Besides having a low per-iteration complexity as existing stochastic ADMM algorithms, it improves the convergence rate on convex problems from O(1 √T) to O(1 T ), where T is the number of iterations. This matches the convergence rate of the batch ADMM algorithm, but without the need to visit all the samples in each iteration. Experiments on the graph-guided fused lasso demonstrate that the new algorithm is significantly faster than state-of-the-art stochastic and batch ADMM algorithms. Variance Reducing (VR) stochastic methods are fast-converging alternatives to the classical Stochastic Gradient Descent (SGD) for solving large-scale regularized finite sum problems, especially when a highly accurate solution is required. One critical step in VR is the function sampling. State-of-the-art VR algorithms such as SVRG and SAGA, employ either Uniform Probability (UP) or Importance Probability (IP), which is deficient in reducing the variance and hence leads to suboptimal convergence rate. In this paper, we propose a novel sampling scheme that explicitly computes some Adaptive Probability (AP) at each iteration. Analysis shows that, equipped with AP, both SVRG and SAGA yield provably better convergence rate than the ones with UP or IP, which is confirmed in experiments. Additionally, to cut down the per iteration computation load, an efficient variant is proposed by utilizing AP periodically, whose performance is empirically validated. We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard algorithms, both in terms of optimizing the training error and reducing the test error quickly.