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Register a new userLearning Stochastic Optimal Policies via Gradient Descent
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Added by
Stefano Massaroli
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We systematically develop a learning-based treatment of stochastic optimal control (SOC), relying on direct optimization of parametric control policies. We propose a derivation of adjoint sensitivity results for stochastic differential equations through direct application of variational calculus. Then, given an objective function for a predetermined task specifying the desiderata for the controller, we optimize their parameters via iterative gradient descent methods. In doing so, we extend the range of applicability of classical SOC techniques, often requiring strict assumptions on the functional form of system and control. We verify the performance of the proposed approach on a continuous-time, finite horizon portfolio optimization with proportional transaction costs.
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Representations are fundamental to artificial intelligence. The performance of a learning system depends on the type of representation used for representing the data. Typically, these representations are hand-engineered using domain knowledge. More recently, the trend is to learn these representations through stochastic gradient descent in multi-layer neural networks, which is called backprop. Learning the representations directly from the incoming data stream reduces the human labour involved in designing a learning system. More importantly, this allows in scaling of a learning system for difficult tasks. In this paper, we introduce a new incremental learning algorithm called crossprop, which learns incoming weights of hidden units based on the meta-gradient descent approach, that was previously introduced by Sutton (1992) and Schraudolph (1999) for learning step-sizes. The final update equation introduces an additional memory parameter for each of these weights and generalizes the backprop update equation. From our experiments, we show that crossprop learns and reuses its feature representation while tackling new and unseen tasks whereas backprop relearns a new feature representation.
We propose to optimize neural networks with a uniformly-distributed random learning rate. The associated stochastic gradient descent algorithm can be approximated by continuous stochastic equations and analyzed within the Fokker-Planck formalism. In the small learning rate regime, the training process is characterized by an effective temperature which depends on the average learning rate, the mini-batch size and the momentum of the optimization algorithm. By comparing the random learning rate protocol with cyclic and constant protocols, we suggest that the random choice is generically the best strategy in the small learning rate regime, yielding better regularization without extra computational cost. We provide supporting evidence through experiments on both shallow, fully-connected and deep, convolutional neural networks for image classification on the MNIST and CIFAR10 datasets.
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 propose a new algorithm called Parle for parallel training of deep networks that converges 2-4x faster than a data-parallel implementation of SGD, while achieving significantly improved error rates that are nearly state-of-the-art on several benchmarks including CIFAR-10 and CIFAR-100, without introducing any additional hyper-parameters. We exploit the phenomenon of flat minima that has been shown to lead to improved generalization error for deep networks. Parle requires very infrequent communication with the parameter server and instead performs more computation on each client, which makes it well-suited to both single-machine, multi-GPU settings and distributed implementations.
Non-convex optimization problems are challenging to solve; the success and computational expense of a gradient descent algorithm or variant depend heavily on the initialization strategy. Often, either random initialization is used or initialization rules are carefully designed by exploiting the nature of the problem class. As a simple alternative to hand-crafted initialization rules, we propose an approach for learning good initialization rules from previous solutions. We provide theoretical guarantees that establish conditions that are sufficient in all cases and also necessary in some under which our approach performs better than random initialization. We apply our methodology to various non-convex problems such as generating adversarial examples, generating post hoc explanations for black-box machine learning models, and allocating communication spectrum, and show consistent gains over other initialization techniques.
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