Estimation and Inference in Distributional Reinforcement Learning

• URL: http://arxiv.org/abs/2309.17262v1
• Date: Fri, 29 Sep 2023 14:14:53 GMT
• Title: Estimation and Inference in Distributional Reinforcement Learning
• Authors: Liangyu Zhang, Yang Peng, Jiadong Liang, Wenhao Yang, Zhihua Zhang
• Abstract summary: We show that a dataset of size $widetilde Oleft(frac|mathcalS||mathcalA|epsilon2 (1-gamma)4right)$ suffices to ensure the Kolmogorov metric and total variation metric between $hatetapi$ and $etapi$ is below $epsilon$ with high probability. Our findings give rise to a unified approach to statistical inference of a wide class of statistical functionals of $etapi$.
• Score: 28.253677740976197
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we study distributional reinforcement learning from the perspective of statistical efficiency. We investigate distributional policy evaluation, aiming to estimate the complete distribution of the random return (denoted $\eta^\pi$) attained by a given policy $\pi$. We use the certainty-equivalence method to construct our estimator $\hat\eta^\pi$, given a generative model is available. We show that in this circumstance we need a dataset of size $\widetilde O\left(\frac{|\mathcal{S}||\mathcal{A}|}{\epsilon^{2p}(1-\gamma)^{2p+2}}\right)$ to guarantee a $p$-Wasserstein metric between $\hat\eta^\pi$ and $\eta^\pi$ is less than $\epsilon$ with high probability. This implies the distributional policy evaluation problem can be solved with sample efficiency. Also, we show that under different mild assumptions a dataset of size $\widetilde O\left(\frac{|\mathcal{S}||\mathcal{A}|}{\epsilon^{2}(1-\gamma)^{4}}\right)$ suffices to ensure the Kolmogorov metric and total variation metric between $\hat\eta^\pi$ and $\eta^\pi$ is below $\epsilon$ with high probability. Furthermore, we investigate the asymptotic behavior of $\hat\eta^\pi$. We demonstrate that the ``empirical process'' $\sqrt{n}(\hat\eta^\pi-\eta^\pi)$ converges weakly to a Gaussian process in the space of bounded functionals on Lipschitz function class $\ell^\infty(\mathcal{F}_{W_1})$, also in the space of bounded functionals on indicator function class $\ell^\infty(\mathcal{F}_{\mathrm{KS}})$ and bounded measurable function class $\ell^\infty(\mathcal{F}_{\mathrm{TV}})$ when some mild conditions hold. Our findings give rise to a unified approach to statistical inference of a wide class of statistical functionals of $\eta^\pi$.

Related papers

• Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples [24.45016514352055]
  We study the sample-complexity of learning $T+1$ functions $f_star(t) circ g_star$ from a function class $mathcal F times mathcal G$. We show that as the number of tasks $T$ increases, both the sample requirement and risk bound converge to that of $r$-dimensional regression.
  arXiv  Detail & Related papers  (2024-10-15T03:20:19Z)
• Multiple-policy Evaluation via Density Estimation [30.914344538340412]
  We propose an algorithm named $mathrmCAESAR$ for this problem. Up to low order and logarithmic terms $mathrmCAESAR$ achieves a sample complexity $tildeOleft(fracH4epsilon2sum_h=1Hmax_kin[K]sum_s,afrac(d_hpik(s,a))2mu*_h(s,a)right)$, where $dpi
  arXiv  Detail & Related papers  (2024-03-29T23:55:25Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Private Convex Optimization via Exponential Mechanism [16.867534746193833]
  We show that modifying the exponential mechanism by adding an $ellcave2$ regularizer to $F(x)$ recovers both the known optimal empirical risk and population loss under $(epsilon,delta)$-DP. We also show how to implement this mechanism using $widetildeO(n min(d, n)) queries to $f_i(x) for the DP-SCO where $n$ is the number of samples/optimal and $d is the ambient dimension.
  arXiv  Detail & Related papers  (2022-03-01T06:51:03Z)
• A Law of Robustness beyond Isoperimetry [84.33752026418045]
  We prove a Lipschitzness lower bound $Omega(sqrtn/p)$ of robustness of interpolating neural network parameters on arbitrary distributions. We then show the potential benefit of overparametrization for smooth data when $n=mathrmpoly(d)$. We disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=exp(omega(d))$.
  arXiv  Detail & Related papers  (2022-02-23T16:10:23Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Non-Parametric Estimation of Manifolds from Noisy Data [1.0152838128195467]
  We consider the problem of estimating a $d$ dimensional sub-manifold of $mathbbRD$ from a finite set of noisy samples. We show that the estimation yields rates of convergence of $n-frack2k + d$ for the point estimation and $n-frack-12k + d$ for the estimation of tangent space.
  arXiv  Detail & Related papers  (2021-05-11T02:29:33Z)
• From Smooth Wasserstein Distance to Dual Sobolev Norm: Empirical Approximation and Statistical Applications [18.618590805279187]
  We show that $mathsfW_p(sigma)$ is controlled by a $pth order smooth dual Sobolev $mathsfd_p(sigma)$. We derive the limit distribution of $sqrtnmathsfd_p(sigma)(hatmu_n,mu)$ in all dimensions $d$, when $mu$ is sub-Gaussian.
  arXiv  Detail & Related papers  (2021-01-11T17:23:24Z)
• Optimal Mean Estimation without a Variance [103.26777953032537]
  We study the problem of heavy-tailed mean estimation in settings where the variance of the data-generating distribution does not exist. We design an estimator which attains the smallest possible confidence interval as a function of $n,d,delta$.
  arXiv  Detail & Related papers  (2020-11-24T22:39:21Z)
• Agnostic Learning of a Single Neuron with Gradient Descent [92.7662890047311]
  We consider the problem of learning the best-fitting single neuron as measured by the expected square loss. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$.
  arXiv  Detail & Related papers  (2020-05-29T07:20:35Z)

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.