Faster Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walk

• URL: http://arxiv.org/abs/2206.09384v1
• Date: Sun, 19 Jun 2022 11:33:07 GMT
• Title: Faster Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walk
• Authors: Oren Mangoubi, Nisheeth K. Vishnoi
• Abstract summary: We consider the problem of sampling from a $d$-dimensional log-concave distribution $pi(theta) propto e-f(theta)$ constrained to a polytope $K$ defined by $m$ inequalities. Our main result is a "soft-warm'' variant of the Dikin walk Markov chain that requires at most $O((md + d L2 R2) times MDomega-1) log(fracwdelta)$ arithmetic operations to sample from $pi$
• Score: 28.431572772564518
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of sampling from a $d$-dimensional log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a polytope $K$ defined by $m$ inequalities. Our main result is a "soft-threshold'' variant of the Dikin walk Markov chain that requires at most $O((md + d L^2 R^2) \times md^{\omega-1}) \log(\frac{w}{\delta}))$ arithmetic operations to sample from $\pi$ within error $\delta>0$ in the total variation distance from a $w$-warm start, where $L$ is the Lipschitz-constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega$ is the matrix-multiplication constant. When a warm start is not available, it implies an improvement of $\tilde{O}(d^{3.5-\omega})$ arithmetic operations on the previous best bound for sampling from $\pi$ within total variation error $\delta$, which was obtained with the hit-and-run algorithm, in the setting where $K$ is a polytope given by $m=O(d)$ inequalities and $LR = O(\sqrt{d})$. When a warm start is available, our algorithm improves by a factor of $d^2$ arithmetic operations on the best previous bound in this setting, which was obtained for a different version of the Dikin walk algorithm. Plugging our Dikin walk Markov chain into the post-processing algorithm of Mangoubi and Vishnoi (2021), we achieve further improvements in the dependence of the running time for the problem of generating samples from $\pi$ with infinity distance bounds in the special case when $K$ is a polytope.

Related papers

• Outlier Robust Multivariate Polynomial Regression [27.03423421704806]
  We are given a set of random samples $(mathbfx_i,y_i) in [-1,1]n times mathbbR$ that are noisy versions of $(mathbfx_i,p(mathbfx_i)$. The goal is to output a $hatp$, within an $ell_in$-distance of at most $O(sigma)$ from $p$.
  arXiv  Detail & Related papers  (2024-03-14T15:04:45Z)
• Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space [10.292118864147097]
  We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which seeks to approximate a distribution $pi$ over $mathbbRd$ by a product measure $pistar$.
  arXiv  Detail & Related papers  (2023-12-05T16:02:04Z)
• $\ell_p$-Regression in the Arbitrary Partition Model of Communication [59.89387020011663]
  We consider the randomized communication complexity of the distributed $ell_p$-regression problem in the coordinator model. For $p = 2$, i.e., least squares regression, we give the first optimal bound of $tildeTheta(sd2 + sd/epsilon)$ bits. For $p in (1,2)$,we obtain an $tildeO(sd2/epsilon + sd/mathrmpoly(epsilon)$ upper bound.
  arXiv  Detail & Related papers  (2023-07-11T08:51:53Z)
• Near Sample-Optimal Reduction-based Policy Learning for Average Reward MDP [58.13930707612128]
  This work considers the sample complexity of obtaining an $varepsilon$-optimal policy in an average reward Markov Decision Process (AMDP) We prove an upper bound of $widetilde O(H varepsilon-3 ln frac1delta)$ samples per state-action pair, where $H := sp(h*)$ is the span of bias of any optimal policy, $varepsilon$ is the accuracy and $delta$ is the failure probability.
  arXiv  Detail & Related papers  (2022-12-01T15:57:58Z)
• Reward-Mixing MDPs with a Few Latent Contexts are Learnable [75.17357040707347]
  We consider episodic reinforcement learning in reward-mixing Markov decision processes (RMMDPs) Our goal is to learn a near-optimal policy that nearly maximizes the $H$ time-step cumulative rewards in such a model.
  arXiv  Detail & Related papers  (2022-10-05T22:52:00Z)
• Active Sampling for Linear Regression Beyond the $\ell_2$ Norm [70.49273459706546]
  We study active sampling algorithms for linear regression, which aim to query only a small number of entries of a target vector. We show that this dependence on $d$ is optimal, up to logarithmic factors. We also provide the first total sensitivity upper bound $O(dmax1,p/2log2 n)$ for loss functions with at most degree $p$ growth.
  arXiv  Detail & Related papers  (2021-11-09T00:20:01Z)
• Sampling from Log-Concave Distributions with Infinity-Distance Guarantees and Applications to Differentially Private Optimization [33.38289436686841]
  We present an algorithm that outputs a point from a distributionO(varepsilon)$close to $$ in infinity-distance. We also present a "soft-pi" version of the Dikin walk which may be independent interest.
  arXiv  Detail & Related papers  (2021-11-07T13:44:50Z)
• Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry [69.24618367447101]
  Up to logarithmic factors the optimal excess population loss of any $(varepsilon,delta)$-differently private is $sqrtlog(d)/n + sqrtd/varepsilon n.$ We show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)
• Learning Mixtures of Spherical Gaussians via Fourier Analysis [0.5381004207943596]
  We find that a bound on the sample and computational complexity was previously unknown when $omega(1) leq d leq O(log k)$. These authors also show that the sample of complexity of a random mixture of gaussians in a ball of radius $d$ in $d$ dimensions, when $d$ is $Theta(sqrtd)$ in $d$ dimensions, when $d$ is at least $poly(k, frac1delta)$.
  arXiv  Detail & Related papers  (2020-04-13T08:06:29Z)

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.