Near Minimax-Optimal Distributional Temporal Difference Algorithms and The Freedman Inequality in Hilbert Spaces
- URL: http://arxiv.org/abs/2403.05811v2
- Date: Thu, 14 Mar 2024 09:24:51 GMT
- Title: Near Minimax-Optimal Distributional Temporal Difference Algorithms and The Freedman Inequality in Hilbert Spaces
- Authors: Yang Peng, Liangyu Zhang, Zhihua Zhang,
- Abstract summary: We propose a non-parametric distributional TD algorithm (NTD) for a $gamma$-discounted infinite-horizon Markov decision process.
We establish a novel Freedman's inequality in Hilbert spaces, which would be of independent interest.
- Score: 24.03281329962804
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Distributional reinforcement learning (DRL) has achieved empirical success in various domains. One of the core tasks in the field of DRL is distributional policy evaluation, which involves estimating the return distribution $\eta^\pi$ for a given policy $\pi$. The distributional temporal difference (TD) algorithm has been accordingly proposed, which is an extension of the temporal difference algorithm in the classic RL literature. In the tabular case, \citet{rowland2018analysis} and \citet{rowland2023analysis} proved the asymptotic convergence of two instances of distributional TD, namely categorical temporal difference algorithm (CTD) and quantile temporal difference algorithm (QTD), respectively. In this paper, we go a step further and analyze the finite-sample performance of distributional TD. To facilitate theoretical analysis, we propose a non-parametric distributional TD algorithm (NTD). For a $\gamma$-discounted infinite-horizon tabular Markov decision process, we show that for NTD we need $\tilde{O}\left(\frac{1}{\varepsilon^{2p}(1-\gamma)^{2p+1}}\right)$ iterations to achieve an $\varepsilon$-optimal estimator with high probability, when the estimation error is measured by the $p$-Wasserstein distance. This sample complexity bound is minimax optimal (up to logarithmic factors) in the case of the $1$-Wasserstein distance. To achieve this, we establish a novel Freedman's inequality in Hilbert spaces, which would be of independent interest. In addition, we revisit CTD, showing that the same non-asymptotic convergence bounds hold for CTD in the case of the $p$-Wasserstein distance.
Related papers
- Finite Time Analysis of Temporal Difference Learning for Mean-Variance in a Discounted MDP [1.0923877073891446]
We consider the problem of policy evaluation for variance in a discounted reward Markov decision process.
For this problem, a temporal difference (TD) type learning algorithm with linear function approximation (LFA) exists in the literature.
We derive finite sample bounds that hold (i) in the mean-squared sense; and (ii) with high probability, when tail iterate averaging is employed.
arXiv Detail & Related papers (2024-06-12T05:49:53Z) - Semi-Discrete Optimal Transport: Nearly Minimax Estimation With Stochastic Gradient Descent and Adaptive Entropic Regularization [38.67914746910537]
We prove an $mathcalO(t-1)$ lower bound rate for the OT map, using the similarity between Laguerre cells estimation and density support estimation.
To nearly achieve the desired fast rate, we design an entropic regularization scheme decreasing with the number of samples.
arXiv Detail & Related papers (2024-05-23T11:46:03Z) - Robust Estimation under the Wasserstein Distance [28.792608997509376]
We introduce a new outlier-robust Wasserstein distance $mathsfW_pvarepsilon$ which allows for $varepsilon$ outlier mass to be removed from its input distributions.
We show that minimum distance estimation under $mathsfW_pvarepsilon$ achieves minimax optimal robust estimation risk.
arXiv Detail & Related papers (2023-02-02T17:20:25Z) - A gradient estimator via L1-randomization for online zero-order
optimization with two point feedback [93.57603470949266]
We present a novel gradient estimator based on two function evaluation and randomization.
We consider two types of assumptions on the noise of the zero-order oracle: canceling noise and adversarial noise.
We provide an anytime and completely data-driven algorithm, which is adaptive to all parameters of the problem.
arXiv Detail & Related papers (2022-05-27T11:23:57Z) - Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process.
We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator.
We prove minimax lower bounds over sub-classes of MRPs.
arXiv Detail & Related papers (2021-09-24T14:48:20Z) - Wasserstein distance estimates for the distributions of numerical
approximations to ergodic stochastic differential equations [0.3553493344868413]
We study the Wasserstein distance between the in distribution of an ergodic differential equation and the distribution in the strongly log-concave case.
This allows us to study in a unified way a number of different approximations proposed in the literature for the overdamped and underdamped Langevin dynamics.
arXiv Detail & Related papers (2021-04-26T07:50:04Z) - Variance-Reduced Off-Policy TDC Learning: Non-Asymptotic Convergence
Analysis [27.679514676804057]
We develop a variance reduction scheme for the two time-scale TDC algorithm in the off-policy setting.
Experiments demonstrate that the proposed variance-reduced TDC achieves a smaller convergence error than both the conventional TDC and the variance-reduced TD.
arXiv Detail & Related papers (2020-10-26T01:33:05Z) - Faster Convergence of Stochastic Gradient Langevin Dynamics for
Non-Log-Concave Sampling [110.88857917726276]
We provide a new convergence analysis of gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave.
At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain.
arXiv Detail & Related papers (2020-10-19T15:23:18Z) - On the Almost Sure Convergence of Stochastic Gradient Descent in
Non-Convex Problems [75.58134963501094]
This paper analyzes the trajectories of gradient descent (SGD)
We show that SGD avoids saddle points/manifolds with $1$ for strict step-size policies.
arXiv Detail & Related papers (2020-06-19T14:11:26Z) - Non-asymptotic Convergence of Adam-type Reinforcement Learning
Algorithms under Markovian Sampling [56.394284787780364]
This paper provides the first theoretical convergence analysis for two fundamental RL algorithms of policy gradient (PG) and temporal difference (TD) learning.
Under general nonlinear function approximation, PG-AMSGrad with a constant stepsize converges to a neighborhood of a stationary point at the rate of $mathcalO(log T/sqrtT)$.
Under linear function approximation, TD-AMSGrad with a constant stepsize converges to a neighborhood of the global optimum at the rate of $mathcalO(log T/sqrtT
arXiv Detail & Related papers (2020-02-15T00:26:49Z) - Differentially Quantized Gradient Methods [53.3186247068836]
We show that Differentially Quantized Gradient Descent (DQ-GD) attains a linear contraction factor of $maxsigma_mathrmGD, rhon 2-R$.
No algorithm within a certain class can converge faster than $maxsigma_mathrmGD, 2-R$.
arXiv Detail & Related papers (2020-02-06T20:40:53Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.