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Posterior sampling with the spike-and-slab prior [MB88], a popular multimodal distribution used to model uncertainty in variable selection, is considered the theoretical gold standard method for Bayesian sparse linear regression [CPS09, Roc18]. However, designing provable algorithms for performing this sampling task is notoriously challenging. Existing posterior samplers for Bayesian sparse variable selection tasks either require strong assumptions about the signal-to-noise ratio (SNR) [YWJ16], only work when the measurement count grows at least linearly in the dimension [MW24], or rely on heuristic approximations to the posterior. We give the first provable algorithms for spike-and-slab posterior sampling that apply for any SNR, and use a measurement count sublinear in the problem dimension. Concretely, assume we are given a measurement matrix X∈ℝn×d and noisy observations y=Xθ⋆+ξ of a signal θ⋆ drawn from a spike-and-slab prior π with a Gaussian diffuse density and expected sparsity k, where ξ∼(𝟘n,σ2In). We give a polynomial-time high-accuracy sampler for the posterior π(⋅∣X,y), for any SNR σ−1 > 0, as long as n≥k3⋅polylog(d) and X is drawn from a matrix ensemble satisfying the restricted isometry property. We further give a sampler that runs in near-linear time ≈nd in the same setting, as long as n≥k5⋅polylog(d). To demonstrate the flexibility of our framework, we extend our result to spike-and-slab posterior sampling with Laplace diffuse densities, achieving similar guarantees when σ=O(1k) is bounded.
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Scalable Spike-and-Slab
Spike-and-slab priors are commonly used for Bayesian variable selection, due to their interpretability and favorable statistical properties. However, existing samplers for spike-and-slab posteriors incur prohibitive computational costs when the number of variables is large. In this article, we propose Scalable Spike-and-Slab (S^3), a scalable Gibbs sampling implementation for high-dimensional Bayesian regression with the continuous spike-and-slab prior of George & McCulloch (1993). For a dataset with n observations and p covariates, S^3 has order max{n^2 p_t, np} computational cost at iteration t where p_t never exceeds the number of covariates switching spike-and-slab states between iterations t and t-1 of the Markov chain. This improves upon the order n^2 p per-iteration cost of state-of-the-art implementations as, typically, p_t is substantially smaller than p. We apply S^3 on synthetic and real-world datasets, demonstrating orders of magnitude speed-ups over existing exact samplers and significant gains in inferential quality over approximate samplers with comparable cost.
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- Award ID(s):
- 1844695
- PAR ID:
- 10469611
- Editor(s):
- Chaudhuri, Kamalika and
- Publisher / Repository:
- PMLR
- Date Published:
- Edition / Version:
- 39
- Volume:
- 162
- Page Range / eLocation ID:
- 2021-2040
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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