Many problems in AI are simplified by clever representations of sensory or symbolic input. How to discover such representations automatically, from large amounts of unlabeled data, remains a fundamental challenge. The goal of statistical methods for dimensionality reduction is to detect and discover low dimensional structure in high dimensional data. In this paper, we review a recently proposed...

This work considers the problem of provably optimal reinforcement learning for (episodic) finite horizon MDPs, i.e. how an agent learns to maximize his/her (long term) reward in an uncertain environment. The main contribution is in providing a novel algorithm — Variance-reduced Upper Confidence Q-learning (vUCQ) — which enjoys a regret bound of Õ( √ HSAT +HSA), where the T is the number of time...

We consider the fundamental problem in non-convex 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 non-convex optimization remain to be full gradient descent that converges in O(1/ε) iterations for smooth objectives, and stochastic gradient descent that converges ...

Stochastic gradient optimization is a class of widely used algorithms for training machine learning models. To optimize an objective, it uses the noisy gradient computed from the random data samples instead of the true gradient computed from the entire dataset. However, when the variance of the noisy gradient is large, the algorithm might spend much time bouncing around, leading to slower conve...

Efficiency-Improvement Techniques Ref: Law and Kelton, Chapter 11 1. Variance Reduction and Efficiency Improvement The techniques that we will look at typically are designed to reduce the variance of the estimator for the performance measure that we are trying to estimate via simulation. (Reduction of the variance leads to narrower confidence intervals, and hence less computational effort is re...

Stochastic gradient-based Monte Carlo methods such as stochastic gradient Langevin dynamics are useful tools for posterior inference on large scale datasets in many machine learning applications. These methods scale to large datasets by using noisy gradients calculated using a mini-batch or subset of the dataset. However, the high variance inherent in these noisy gradients degrades performance ...

We study nonconvex finite-sum problems and analyze stochastic variance reduced gradient (Svrg) methods for them. Svrg and related methods have recently surged into prominence for convex optimization given their edge over stochastic gradient descent (Sgd); but their theoretical analysis almost exclusively assumes convexity. In contrast, we prove non-asymptotic rates of convergence (to stationary...

Policy evaluation is a crucial step in many reinforcement-learning procedures, which estimates a value function that predicts states’ longterm value under a given policy. In this paper, we focus on policy evaluation with linear function approximation over a fixed dataset. We first transform the empirical policy evaluation problem into a (quadratic) convex-concave saddle point problem, and then ...

Based on the dual formulation by Rogers (2002), Monte Carlo algorithms to estimate the high-biased and low-biased estimates for American option prices are proposed. Bounds for pricing errors and the variance of biased estimators are shown to be dependent on hedging martingales. These martingales are applied to (1) simultaneously reduce the error bound and the variance of the high-biased estimat...

Variance reduction (VR) methods boost the performance of stochastic gradient descent (SGD) by enabling the use of larger, constant stepsizes and preserving linear convergence rates. However, current variance reduced SGD methods require either high memory usage or an exact gradient computation (using the entire dataset) at the end of each epoch. This limits the use of VR methods in practical dis...