基于测度分解的稀疏线性回归可证明高效后验采样
Provably Efficient Posterior Sampling for Sparse Linear Regression via Measure Decomposition
AI总结：
针对多峰后验采样难题，本文提出基于测度分解的方法将其转化为对数凹分布采样，并在样本数与参数比超阈值时证明了算法的多项式时间高效性。
摘要：
我们考虑在给定线性观测 $\boldsymbol{y} = \boldsymbol{X}\boldsymbolθ+\boldsymbol{\varepsilon}$ 的情况下，从 $d$ 维系数向量 $\boldsymbolθ$ 的后验分布中进行采样的问题。通常，此类后验分布是多峰的，因此采样具有挑战性。这一观察促使了对旨在近似后验分布的各种启发式方法的探索。在本文中，我们研究了一种不同的方法，该方法基于将后验分解为简单乘积测度的对数凹混合。这种分解使我们能够将多峰分布的采样转化为对数凹分布的采样，后者易于处理且已被详细研究。我们证明，在对先验的温和条件下，对于随机设计，当每个参数的样本数 $n/d$ 超过常数阈值时，这种测度分解通常是可行的。因此，我们在先前未知的情形下获得了一种可证明的高效（多项式时间）采样算法。数值模拟证实该算法是实用的，并表明与最先进的方法相比，它具有吸引人的统计特性。
英文摘要：
We consider the problem of sampling from the posterior distribution of a $d$-dimensional coefficient vector $\boldsymbolθ$, given linear observations $\boldsymbol{y} = \boldsymbol{X}\boldsymbolθ+\boldsymbol{\varepsilon}$. In general, such posteriors are multimodal, and therefore challenging to sample from. This observation has prompted the exploration of various heuristics that aim at approximating the posterior distribution. In this paper, we study a different approach based on decomposing the posterior distribution into a log-concave mixture of simple product measures. This decomposition allows us to reduce sampling from a multimodal distribution of interest to sampling from a log-concave one, which is tractable and has been investigated in detail. We prove that, under mild conditions on the prior, for random designs, such measure decomposition is generally feasible when the number of samples per parameter $n/d$ exceeds a constant threshold. We thus obtain a provably efficient (polynomial time) sampling algorithm in a regime where this was previously not known. Numerical simulations confirm that the algorithm is practical, and reveal that it has attractive statistical properties compared to state-of-the-art methods.
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