Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators

Kook, Yunbum; Lee, Yin Tat; Shen, Ruoqi; Vempala, Santosh

doi:10.48550/arxiv.2210.07219

Cited by 2 publications

(2 citation statements)

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“…For example, [GV22] studied the Riemmanian Langevin dynamics assuming access to an oracle to sample from Brownian motion on a manifold, whose complexity heavily depends on the manifold. Further, the convergence rate of Riemannian Hamiltonian Monte Carlo (RHMC) in polytopes was studied in [LV18], and a discretized version was analyzed in [KLSV22]; the results apply to a limited family of distributions, and the convergence rate is fairly large. For RHMC to converge to the correct target distribution, sophisticated discretization methods such as Implicit Midpoint Method are necessary.…”

Section: Prior Workmentioning

confidence: 99%

Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler

Gopi¹,

Lee²,

Liu³

et al. 2023

Preprint

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The development of efficient sampling algorithms catering to non-Euclidean geometries has been a challenging endeavor, as discretization techniques which succeed in the Euclidean setting do not readily carry over to more general settings. We develop a non-Euclidean analog of the recent proximal sampler of [LST21b], which naturally induces regularization by an object known as the log-Laplace transform (LLT) of a density. We prove new mathematical properties (with an algorithmic flavor) of the LLT, such as strong convexity-smoothness duality and an isoperimetric inequality, which are used to prove a mixing time on our proximal sampler matching [LST21b] under a warm start. As our main application, we show our warm-started sampler improves the value oracle complexity of differentially private convex optimization in ℓ p and Schatten-p norms for p ∈ [1, 2] to match the Euclidean setting [GLL22], while retaining state-of-the-art excess risk bounds [GLL + 23]. We find our investigation of the LLT to be a promising proof-of-concept of its utility as a tool for designing samplers, and outline directions for future exploration.

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Section: Prior Workmentioning

confidence: 99%

Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler

Gopi¹,

Lee²,

Liu³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The following theorem (see [15]) illustrates how one-step coupling with the isoperimetry leads to a lower bound on the s-conductance. Its proof is similar to that of Lemma 13 in [21] and can be found in full detail in Appendix E.1.…”

Section: Markov Chainsmentioning

confidence: 99%

The Network Visibility Problem

Khashayar

Gomez-Rodriguez

2021

ACM Trans. Inf. Syst.

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Social media is an attention economy where broadcasters are constantly competing for attention in their followers’ feeds. Broadcasters are likely to elicit greater attention from their followers if their posts remain visible at the top of their followers’ feeds for a longer period of time. However, this depends on the rate at which their followers receive information in their feeds, which in turn depends on the broadcasters they follow. Motivated by this observation and recent calls for fairness of exposure in social networks, in this article, we look at the task of recommending links from the perspective of visibility optimization. Given a set of candidate links provided by a link recommendation algorithm, our goal is to find a subset of those links that would provide the highest visibility to a set of broadcasters. To this end, we first show that this problem reduces to maximizing a nonsubmodular nondecreasing set function under matroid constraints. Then, we show that the set function satisfies a notion of approximate submodularity that allows the standard greedy algorithm to enjoy theoretical guarantees. Experiments on both synthetic and real data gathered from Twitter show that the greedy algorithm is able to consistently outperform several competitive baselines.

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Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators

Cited by 2 publications

References 0 publications

Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler

Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler

The Network Visibility Problem

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