Gaussian Cooling and Dikin Walks: The Interior-Point Method for Logconcave Sampling
arXiv (Cornell University)(2023)
摘要
The connections between (convex) optimization and (logconcave) sampling have
been considerably enriched in the past decade with many conceptual and
mathematical analogies. For instance, the Langevin algorithm can be viewed as a
sampling analogue of gradient descent and has condition-number-dependent
guarantees on its performance. In the early 1990s, Nesterov and Nemirovski
developed the Interior-Point Method (IPM) for convex optimization based on
self-concordant barriers, providing efficient algorithms for structured convex
optimization, often faster than the general method. This raises the following
question: can we develop an analogous IPM for structured sampling problems?
In 2012, Kannan and Narayanan proposed the Dikin walk for uniformly sampling
polytopes, and an improved analysis was given in 2020 by Laddha-Lee-Vempala.
The Dikin walk uses a local metric defined by a self-concordant barrier for
linear constraints. Here we generalize this approach by developing and adapting
IPM machinery together with the Dikin walk for poly-time sampling algorithms.
Our IPM-based sampling framework provides an efficient warm start and goes
beyond uniform distributions and linear constraints. We illustrate the approach
on important special cases, in particular giving the fastest algorithms to
sample uniform, exponential, or Gaussian distributions on a truncated PSD cone.
The framework is general and can be applied to other sampling algorithms.
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关键词
psd cone,metric dikin walk
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