Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond

• URL: http://arxiv.org/abs/2502.04575v1
• Date: Fri, 07 Feb 2025 00:05:28 GMT
• Title: Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond
• Authors: Wei Guo, Molei Tao, Yongxin Chen,
• Abstract summary: Given an unnormalized probability density $piproptomathrme-V$, estimating its normalizing constant $Z=int_mathbbRdmathrme-V(x)mathrmdx$ or free energy $F=-log Z$ is a crucial problem in Bayesian statistics, statistical mechanics, and machine learning. We propose a new normalizing constant estimation algorithm based on reverse diffusion samplers and establish a framework for analyzing its complexity.
• Score: 28.931489333515618
• License:
• Abstract: Given an unnormalized probability density $\pi\propto\mathrm{e}^{-V}$, estimating its normalizing constant $Z=\int_{\mathbb{R}^d}\mathrm{e}^{-V(x)}\mathrm{d}x$ or free energy $F=-\log Z$ is a crucial problem in Bayesian statistics, statistical mechanics, and machine learning. It is challenging especially in high dimensions or when $\pi$ is multimodal. To mitigate the high variance of conventional importance sampling estimators, annealing-based methods such as Jarzynski equality and annealed importance sampling are commonly adopted, yet their quantitative complexity guarantees remain largely unexplored. We take a first step toward a non-asymptotic analysis of annealed importance sampling. In particular, we derive an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\mathcal{A}}^2}{\varepsilon^4}\right)$ for estimating $Z$ within $\varepsilon$ relative error with high probability, where $\beta$ is the smoothness of $V$ and $\mathcal{A}$ denotes the action of a curve of probability measures interpolating $\pi$ and a tractable reference distribution. Our analysis, leveraging Girsanov theorem and optimal transport, does not explicitly require isoperimetric assumptions on the target distribution. Finally, to tackle the large action of the widely used geometric interpolation of probability distributions, we propose a new normalizing constant estimation algorithm based on reverse diffusion samplers and establish a framework for analyzing its complexity.

Related papers

• Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination [35.67742880001828]
  We give the first computationally efficient algorithm for high-dimensional robust mean estimation with mean-shift contamination. Our algorithm has near-optimal sample complexity, runs in sample-polynomial time, and approximates the target mean to any desired accuracy.
  arXiv  Detail & Related papers  (2025-02-20T17:53:13Z)
• Dimension-free Private Mean Estimation for Anisotropic Distributions [55.86374912608193]
  Previous private estimators on distributions over $mathRd suffer from a curse of dimensionality. We present an algorithm whose sample complexity has improved dependence on dimension.
  arXiv  Detail & Related papers  (2024-11-01T17:59:53Z)
• Sum-of-squares lower bounds for Non-Gaussian Component Analysis [33.80749804695003]
  Non-Gaussian Component Analysis (NGCA) is the statistical task of finding a non-Gaussian direction in a high-dimensional dataset. Here we study the complexity of NGCA in the Sum-of-Squares framework.
  arXiv  Detail & Related papers  (2024-10-28T18:19:13Z)
• Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling [28.931489333515618]
  We establish an oracle complexity of $widetildeOleft(fracdbeta2cal A2varepsilon6right)$ for the simple annealed Langevin Monte Carlo algorithm. We show that $cal A$ represents the action of a curve of probability measures interpolating the target distribution $pi$ and a readily sampleable distribution.
  arXiv  Detail & Related papers  (2024-07-24T02:15:48Z)
• Robust computation of optimal transport by $\beta$-potential regularization [79.24513412588745]
  Optimal transport (OT) has become a widely used tool in the machine learning field to measure the discrepancy between probability distributions. We propose regularizing OT with the beta-potential term associated with the so-called $beta$-divergence. We experimentally demonstrate that the transport matrix computed with our algorithm helps estimate a probability distribution robustly even in the presence of outliers.
  arXiv  Detail & Related papers  (2022-12-26T18:37:28Z)
• 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)
• The Sample Complexity of Robust Covariance Testing [56.98280399449707]
  We are given i.i.d. samples from a distribution of the form $Z = (1-epsilon) X + epsilon B$, where $X$ is a zero-mean and unknown covariance Gaussian $mathcalN(0, Sigma)$. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples. We prove a sample complexity lower bound of $Omega(d2)$ for $epsilon$ an arbitrarily small constant and $gamma
  arXiv  Detail & Related papers  (2020-12-31T18:24:41Z)
• Optimal Sub-Gaussian Mean Estimation in $\mathbb{R}$ [5.457150493905064]
  We present a novel estimator with sub-Gaussian convergence. Our estimator does not require prior knowledge of the variance. Our estimator construction and analysis gives a framework generalizable to other problems.
  arXiv  Detail & Related papers  (2020-11-17T02:47:24Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

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.