Learning to Accelerate by the Methods of Step-size Planning
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Gradient descent is slow to converge for ill-conditioned problems and non-convex problems. An important technique for acceleration is step-size adaptation. The first part of this paper contains a detailed review of step-size adaptation methods, including Polyak step-size, L4, LossGrad, Adam and IDBD. In the second part of this paper, we propose a new class of methods of accelerating gradient descent that are quite different from existing techniques. The new methods, which we call step-size planning, use the update experience to learn an improved way of updating the parameters. The methods organize the experience into K steps away from each other to facilitate planning. From the past experience, our planning algorithm, Csawg, learns a step-size model which is a form of multi-step machine that predicts future updates. We extends Csawg to applying step-size planning multiple steps, which leads to further speedup. We discuss and highlight the projection power of the diagonal-matrix step-size for future large scale applications. We show for a convex problem, our methods can surpass the convergence rate of Nesterov's accelerated gradient, 1 - √(μ/L), where μ, L are the strongly convex factor of the loss function F and the Lipschitz constant of F'. On the classical non-convex Rosenbrock function, our planning methods achieve zero error below 500 gradient evaluations, while gradient descent takes about 10000 gradient evaluations to reach a 10^-3 accuracy. We discuss the connection of step-size planing to planning in reinforcement learning, in particular, Dyna architectures. We leave convergence and convergence rate proofs and applications of the methods to high-dimensional problems for future work.
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