We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include as special cases 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 $\widetilde{O}(\lvert V\rvert)$\footnote{Throughout, $\widetilde{O}(\cdot)$ hides polylogarithmic factors in $n$.} time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. This is the first nearlylinear runtime in the output size, which is clearly optimal. For a determinantal point process on $k$sized subsets of a ground set of $n$ elements, defined via an $n\times n$ kernel matrix, we show how to approximately sample in $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega1})$ time preprocessing, where $\omega<2.372864$ is the matrix multiplication exponent. The time to compute just the weight of the output set is simply $\simeq k^\omega$, a natural barrier that suggests our runtime might be optimal for determinantal point processes as well. As a corollary, we even improve the state of the art for obtaining a single sample from a determinantal point process, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega1})$.
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, the domain size can be reduced to nearly linear in the output size $t=\widetilde{O}(k)$, improving the state of the art from $t= \widetilde{O}(k^2)$ for general strongly Rayleigh distributions and the more specialized $t=\widetilde{O}(k^{1.5})$ for spanning tree distributions. Our reduction involves sampling from $\widetilde{O}(1)$ domainsparsified distributions, all of which can be produced efficiently assuming approximate overestimates for marginals of $\mu$ are known and stored in a convenient data structure. Having access to marginals is the discrete analog of having access to the mean and covariance of a continuous distribution, or equivalently knowing ``isotropy'' for the distribution, the key behind optimal samplers in the continuous setting based on the famous KannanLov\'aszSimonovits (KLS) conjecture. We view our result as analogous in spirit to the KLS conjecture and its consequences for sampling, but rather for discrete strongly Rayleigh measures.
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From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization
We establish a connection between sampling and optimization on discrete domains. For a family of distributions $\mu$ defined on size $k$ subsets of a ground set of elements, that is closed under external fields, we show that rapid mixing of natural local random walks implies the existence of simple approximation algorithms to find $\max \mu(\cdot)$. More precisely, we show that if $t$step downup random walks have spectral gap at least inverse polynomially large, then $t$step local search finds $\max \mu(\cdot)$ within a factor of $k^{O(k)}$. As the main application of our result, we show that $2$step local search achieves a nearlyoptimal $k^{O(k)}$factor approximation for MAP inference on nonsymmetric $k$DPPs. This is the first nontrivial multiplicative approximation algorithm for this problem. In our main technical result, we show that an exchange inequality, a concept rooted in discrete convex analysis, can be derived from fast mixing of local random walks. We further advance the state of the art on the mixing of random walks for nonsymmetric DPPs and more generally sectorstable distributions, by obtaining the tightest possible bound on the step size needed for polynomialtime mixing of random walks. We bring the step size down by a factor of $2$ compared to prior works, and consequently get a quadratic improvement on the runtime of local search steps; this improvement is potentially of independent interest in sampling applications.
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 Award ID(s):
 2045354
 NSFPAR ID:
 10393963
 Editor(s):
 Loh, PoLing; Raginsky, Maxim
 Date Published:
 Journal Name:
 Proceedings of Thirty Fifth Conference on Learning Theory
 Volume:
 178
 Page Range / eLocation ID:
 55965618
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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