Stochastic Gradient Descent on a Tree: an Adaptive and Robust Approach to Stochastic Convex Optimization
Online minimization of an unknown convex function over the interval $[0,1]$ is considered under first-order stochastic bandit feedback, which returns a random realization of the gradient of the function at each query point. Without knowing the distribution of the random gradients, a learning algorithm sequentially chooses query points with the objective of minimizing regret defined as the expected cumulative loss of the function values at the query points in excess to the minimum value of the function. An approach based on devising a biased random walk on an infinite-depth binary tree constructed through successive partitioning of the domain of the function is developed. Each move of the random walk is guided by a sequential test based on confidence bounds on the empirical mean constructed using the law of the iterated logarithm. With no tuning parameters, this learning algorithm is robust to heavy-tailed noise with infinite variance and adaptive to unknown function characteristics (specifically, convex, strongly convex, and nonsmooth). It achieves the corresponding optimal regret orders (up to a $\sqrt{\log T}$ or a $\log\log T$ factor) in each class of functions and offers better or matching regret orders than the classical stochastic gradient descent approach which requires the knowledge of the function characteristics for tuning the sequence of step-sizes.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Stochastic Adaptive Gradient Descent Without Descent
We introduce a new adaptive step-size strategy for convex optimization with stochastic gradient that exploits the local geometry of the objective function only by means of a first-order stochastic oracle and without any …
Optimal Adaptive and Accelerated Stochastic Gradient Descent
Stochastic gradient descent (\textsc{Sgd}) methods are the most powerful optimization tools in training machine learning and deep learning models. Moreover, acceleration (a.k.a. momentum) methods and diagonal scaling (a.…
BIG-bench Machine LearningStochastic OptimizationAdaptive Stochastic Gradient Descents on Manifolds with an Application on Weighted Low-Rank Approximation
We prove a convergence theorem for stochastic gradient descents on manifolds with adaptive learning rate and apply it to the weighted low-rank approximation problem.
Kalman Gradient Descent: Adaptive Variance Reduction in Stochastic Optimization
We introduce Kalman Gradient Descent, a stochastic optimization algorithm that uses Kalman filtering to adaptively reduce gradient variance in stochastic gradient descent by filtering the gradient estimates. We present b…
BIG-bench Machine LearningStochastic OptimizationVariational InferenceLocal Quadratic Convergence of Stochastic Gradient Descent with Adaptive Step Size
Establishing a fast rate of convergence for optimization methods is crucial to their applicability in practice. With the increasing popularity of deep learning over the past decade, stochastic gradient descent and its ad…