Optimal Sublinear Sampling of Spanning Trees and Determinantal Point
Processes via Average-Case Entropic Independence
- URL: http://arxiv.org/abs/2204.02570v1
- Date: Wed, 6 Apr 2022 04:11:26 GMT
- Title: Optimal Sublinear Sampling of Spanning Trees and Determinantal Point
Processes via Average-Case Entropic Independence
- Authors: Nima Anari, Yang P. Liu, Thuy-Duong Vuong
- Abstract summary: We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions.
For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $widetildeO(lvert Vrvert)$ time per sample.
For a determinantal point process on subsets of size $k$ of a ground set of $n$ elements, we show how to approximately sample in $widetildeO(komega)$ time after an initial $widetildeO(nk
- Score: 3.9586758145580014
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We design fast algorithms for repeatedly sampling from strongly Rayleigh
distributions, which include random spanning tree distributions and
determinantal point processes. For a graph $G=(V, E)$, we show how to
approximately sample uniformly random spanning trees from $G$ in
$\widetilde{O}(\lvert V\rvert)$ time per sample after an initial
$\widetilde{O}(\lvert E\rvert)$ time preprocessing. For a determinantal point
process on subsets of size $k$ of a ground set of $n$ elements, we show how to
approximately sample in $\widetilde{O}(k^\omega)$ time after an initial
$\widetilde{O}(nk^{\omega-1})$ time preprocessing, where $\omega<2.372864$ is
the matrix multiplication exponent. We even improve the state of the art for
obtaining a single sample from determinantal point processes, from the prior
runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to
$\widetilde{O}(nk^{\omega-1})$.
In our main technical result, we achieve the optimal limit on domain
sparsification for strongly Rayleigh distributions. In domain sparsification,
sampling from a distribution $\mu$ on $\binom{[n]}{k}$ is reduced to sampling
from related distributions on $\binom{[t]}{k}$ for $t\ll n$. We show that for
strongly Rayleigh distributions, we can can achieve the optimal
$t=\widetilde{O}(k)$. Our reduction involves sampling from $\widetilde{O}(1)$
domain-sparsified distributions, all of which can be produced efficiently
assuming convenient access to approximate overestimates for marginals of $\mu$.
Having access to marginals is analogous to having access to the mean and
covariance of a continuous distribution, or knowing "isotropy" for the
distribution, the key assumption behind the Kannan-Lov\'asz-Simonovits (KLS)
conjecture and optimal samplers based on it. We view our result as a moral
analog of the KLS conjecture and its consequences for sampling, for discrete
strongly Rayleigh measures.
Related papers
- On the query complexity of sampling from non-log-concave distributions [2.4253233571593547]
We study the problem of sampling from a $d$-dimensional distribution with density $p(x)propto e-f(x)$, which does not necessarily satisfy good isoperimetric conditions.
We show that for a wide range of parameters, sampling is strictly easier than optimization by a super-exponential factor in the dimension $d$.
arXiv Detail & Related papers (2025-02-10T06:54:16Z) - On the sampling complexity of coherent superpositions [0.4972323953932129]
We consider the problem of sampling from the distribution of measurement outcomes when applying a POVM to a superposition.
We give an algorithm which $-$ given $O(chi |c|2 log1/delta)$ such samples and calls to oracles to evaluate the probability density functions.
arXiv Detail & Related papers (2025-01-28T16:56:49Z) - Polynomial time sampling from log-smooth distributions in fixed dimension under semi-log-concavity of the forward diffusion with application to strongly dissipative distributions [9.48556659249574]
We provide a sampling algorithm with complexity in fixed dimension.
We prove that our algorithm achieves an expected $epsilon$ error in $KL$ divergence.
As an application, we derive an exponential complexity improvement for the problem of sampling from an $L$-log-smooth distribution.
arXiv Detail & Related papers (2024-12-31T17:51:39Z) - 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) - Statistical-Computational Trade-offs for Density Estimation [60.81548752871115]
We show that for a broad class of data structures their bounds cannot be significantly improved.
This is a novel emphstatistical-computational trade-off for density estimation.
arXiv Detail & Related papers (2024-10-30T15:03:33Z) - Simple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via Gauss-Jordan [49.1574468325115]
We revisit the sample and computational complexity of completing a rank-1 tensor in $otimes_i=1N mathbbRd$.
We present a characterization of the problem which admits an algorithm amounting to Gauss-Jordan on a pair of random linear systems.
arXiv Detail & Related papers (2024-08-10T04:26:19Z) - Robust Mean Estimation Without Moments for Symmetric Distributions [7.105512316884493]
We show that for a large class of symmetric distributions, the same error as in the Gaussian setting can be achieved efficiently.
We propose a sequence of efficient algorithms that approaches this optimal error.
Our algorithms are based on a generalization of the well-known filtering technique.
arXiv Detail & Related papers (2023-02-21T17:52:23Z) - 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) - Perfect Sampling from Pairwise Comparisons [26.396901523831534]
We study how to efficiently obtain perfect samples from a discrete distribution $mathcalD$ given access only to pairwise comparisons of elements of its support.
We design a Markov chain whose stationary distribution coincides with $mathcalD$ and give an algorithm to obtain exact samples using the technique of Coupling from the Past.
arXiv Detail & Related papers (2022-11-23T11:20:30Z) - Sparse sketches with small inversion bias [79.77110958547695]
Inversion bias arises when averaging estimates of quantities that depend on the inverse covariance.
We develop a framework for analyzing inversion bias, based on our proposed concept of an $(epsilon,delta)$-unbiased estimator for random matrices.
We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, the estimator $(epsilon,delta)$-unbiased for $(Atop A)-1$ with a sketch of size $m=O(d+sqrt d/
arXiv Detail & Related papers (2020-11-21T01:33:15Z) - 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.