Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used method for learning exponential families with continuous variables due to its computational ease when compared against maximum likelihood estimation. However, theoretical understanding of the statistical properties of score matching is still lacking. In this work, we provide a non-asymptotic sample complexity analysis for learning the structure of exponential families of polynomials with score matching. The derived sample bounds show a polynomial dependence on the model dimension. These bounds are the first of its kind, as all prior work has shown only asymptotic bounds on the sample complexity.

In reinforcement learning (RL), one of the key components is policy evaluation, which aims to estimate the value function (i.e., expected long-term accumulated reward) of a policy. With a good policy evaluation method, the RL algorithms will estimate the value function more accurately and find a better policy. When the state space is large or continuous \emph{Gradient-based Temporal Difference(GTD)} policy evaluation algorithms with linear function approximation are widely used. Considering that the collection of the evaluation data is both time and reward consuming, a clear understanding of the finite sample performance of the policy evaluation algorithms is very important to reinforcement learning. Under the assumption that data are i.i.d.

Neural Information Processing Systems http://nips.cc/

To all reviewers, thank you very much for your thoughtful comments and suggestions. R#1: "...importance of similarity among the selected tasks... " R#1: "...domain randomization, when enough samples are used, is a better alternative to meta-learning... " R#2: "...Theorems 1 and 2 are asymptotic... " Hence, the theorems are NOT asymptotic. We will remove the asymptotic parts for clarity. R#2: 'Assumption 2 ... the per-task optimal models are centered around the corresponding optimal solutions. This assumption can easily be dropped with the cost of including the distance as a term.

Within a statistical learning setting, we propose and study an iterative regularization algorithm for least squares defined by an incremental gradient method. In particular, we show that, if all other parameters are fixed a priori, the number of passes over the data (epochs) acts as a regularization parameter, and prove strong universal consistency, i.e. almost sure convergence of the risk, as well as sharp finite sample bounds for the iterates. Our results are a step towards understanding the effect of multiple epochs in stochastic gradient techniques in machine learning and rely on integrating statistical and optimization results.

This paper investigates what can be inferred about an arbitrary continuous probability distribution from a finite sample of $N$ observations drawn from it. The central finding is that the $N$ sorted sample points partition the real line into $N+1$ segments, each carrying an expected probability mass of exactly $1/(N+1)$. This non-parametric result, which follows from fundamental properties of order statistics, holds regardless of the underlying distribution's shape. This equal-probability partition yields a discrete entropy of $\log_2(N+1)$ bits, which quantifies the information gained from the sample and contrasts with Shannon's results for continuous variables. I compare this partition-based framework to the conventional ECDF and discuss its implications for robust non-parametric inference, particularly in density and tail estimation.