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Register a new userExplicit Gradient Learning
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Added by
Mor Sinay
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Black-Box Optimization (BBO) methods can find optimal policies for systems that interact with complex environments with no analytical representation. As such, they are of interest in many Artificial Intelligence (AI) domains. Yet classical BBO methods fall short in high-dimensional non-convex problems. They are thus often overlooked in real-world AI tasks. Here we present a BBO method, termed Explicit Gradient Learning (EGL), that is designed to optimize high-dimensional ill-behaved functions. We derive EGL by finding weak-spots in methods that fit the objective function with a parametric Neural Network (NN) model and obtain the gradient signal by calculating the parametric gradient. Instead of fitting the function, EGL trains a NN to estimate the objective gradient directly. We prove the convergence of EGL in convex optimization and its robustness in the optimization of integrable functions. We evaluate EGL and achieve state-of-the-art results in two challenging problems: (1) the COCO test suite against an assortment of standard BBO methods; and (2) in a high-dimensional non-convex image generation task.
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Stochastic gradient descent (SGD) is a popular and efficient method with wide applications in training deep neural nets and other nonconvex models. While the behavior of SGD is well understood in the convex learning setting, the existing theoretical results for SGD applied to nonconvex objective functions are far from mature. For example, existing results require to impose a nontrivial assumption on the uniform boundedness of gradients for all iterates encountered in the learning process, which is hard to verify in practical implementations. In this paper, we establish a rigorous theoretical foundation for SGD in nonconvex learning by showing that this boundedness assumption can be removed without affecting convergence rates. In particular, we establish sufficient conditions for almost sure convergence as well as optimal convergence rates for SGD applied to both general nonconvex objective functions and gradient-dominated objective functions. A linear convergence is further derived in the case with zero variances.
We study the training of regularized neural networks where the regularizer can be non-smooth and non-convex. We propose a unified framework for stochastic proximal gradient descent, which we term ProxGen, that allows for arbitrary positive preconditioners and lower semi-continuous regularizers. Our framework encompasses standard stochastic proximal gradient methods without preconditioners as special cases, which have been extensively studied in various settings. Not only that, we present two important update rules beyond the well-known standard methods as a byproduct of our approach: (i) the first closed-form proximal mappings of $ell_q$ regularization ($0 leq q leq 1$) for adaptive stochastic gradient methods, and (ii) a revised version of ProxQuant that fixes a caveat of the original approach for quantization-specific regularizers. We analyze the convergence of ProxGen and show that the whole family of ProxGen enjoys the same convergence rate as stochastic proximal gradient descent without preconditioners. We also empirically show the superiority of proximal methods compared to subgradient-based approaches via extensive experiments. Interestingly, our results indicate that proximal methods with non-convex regularizers are more effective than those with convex regularizers.
Stochastic gradient descent (SGD), which dates back to the 1950s, is one of the most popular and effective approaches for performing stochastic optimization. Research on SGD resurged recently in machine learning for optimizing convex loss functions and training nonconvex deep neural networks. The theory assumes that one can easily compute an unbiased gradient estimator, which is usually the case due to the sample average nature of empirical risk minimization. There exist, however, many scenarios (e.g., graphs) where an unbiased estimator may be as expensive to compute as the full gradient because training examples are interconnected. Recently, Chen et al. (2018) proposed using a consistent gradient estimator as an economic alternative. Encouraged by empirical success, we show, in a general setting, that consistent estimators result in the same convergence behavior as do unbiased ones. Our analysis covers strongly convex, convex, and nonconvex objectives. We verify the results with illustrative experiments on synthetic and real-world data. This work opens several new research directions, including the development of more efficient SGD updates with consistent estimators and the design of efficient training algorithms for large-scale graphs.
We study a new aggregation operator for gradients coming from a mini-batch for stochastic gradient (SG) methods that allows a significant speed-up in the case of sparse optimization problems. We call this method AdaBatch and it only requires a few lines of code change compared to regular mini-batch SGD algorithms. We provide a theoretical insight to understand how this new class of algorithms is performing and show that it is equivalent to an implicit per-coordinate rescaling of the gradients, similarly to what Adagrad methods can do. In theory and in practice, this new aggregation allows to keep the same sample efficiency of SG methods while increasing the batch size. Experimentally, we also show that in the case of smooth convex optimization, our procedure can even obtain a better loss when increasing the batch size for a fixed number of samples. We then apply this new algorithm to obtain a parallelizable stochastic gradient method that is synchronous but allows speed-up on par with Hogwild! methods as convergence does not deteriorate with the increase of the batch size. The same approach can be used to make mini-batch provably efficient for variance-reduced SG methods such as SVRG.
Deterministic Policy Gradient (DPG) removes a level of randomness from standard randomized-action Policy Gradient (PG), and demonstrates substantial empirical success for tackling complex dynamic problems involving Markov decision processes. At the same time, though, DPG loses its ability to learn in a model-free (i.e., actor-only) fashion, frequently necessitating the use of critics in order to obtain consistent estimates of the associated policy-reward gradient. In this work, we introduce Zeroth-order Deterministic Policy Gradient (ZDPG), which approximates policy-reward gradients via two-point stochastic evaluations of the $Q$-function, constructed by properly designed low-dimensional action-space perturbations. Exploiting the idea of random horizon rollouts for obtaining unbiased estimates of the $Q$-function, ZDPG lifts the dependence on critics and restores true model-free policy learning, while enjoying built-in and provable algorithmic stability. Additionally, we present new finite sample complexity bounds for ZDPG, which improve upon existing results by up to two orders of magnitude. Our findings are supported by several numerical experiments, which showcase the effectiveness of ZDPG in a practical setting, and its advantages over both PG and Baseline PG.
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