[Paper Review] Nearly Consistent Finite Particle Estimates in Streaming Importance Sampling

Alec Koppel, Amrit Singh Bedi+2 more|arXiv (Cornell University)|Sep 23, 2019

Sparse and Compressive Sensing Techniques58 references4 citations

TL;DR

This paper proposes a streaming importance sampling method that maintains nearly consistent finite particle estimates by projecting posterior density estimates onto a low-dimensional subspace in a reproducing kernel Hilbert space (RKHS) using maximum mean discrepancy (MMD). The approach ensures bias is tunable via a compression parameter, enabling substantial memory reduction while preserving estimation accuracy comparable to full-sample methods.

ABSTRACT

In Bayesian inference, we seek to compute information about random variables such as moments or quantiles on the basis of {available data} and prior information. When the distribution of random variables is {intractable}, Monte Carlo (MC) sampling is usually required. {Importance sampling is a standard MC tool that approximates this unavailable distribution with a set of weighted samples.} This procedure is asymptotically consistent as the number of MC samples (particles) go to infinity. However, retaining infinitely many particles is intractable. Thus, we propose a way to only keep a \\emph{finite representative subset} of particles and their augmented importance weights that is \\emph{nearly consistent}. To do so in {an online manner}, we (1) embed the posterior density estimate in a reproducing kernel Hilbert space (RKHS) through its kernel mean embedding; and (2) sequentially project this RKHS element onto a lower-dimensional subspace in RKHS using the maximum mean discrepancy, an integral probability metric. Theoretically, we establish that this scheme results in a bias determined by a compression parameter, which yields a tunable tradeoff between consistency and memory. In experiments, we observe the compressed estimates achieve comparable performance to the dense ones with substantial reductions in representational complexity.

Motivation & Objective

• Address the scalability challenge in Monte Carlo methods where infinite particles are required for consistency, but finite memory limits particle retention.
• Develop an online, streaming algorithm that maintains a finite, representative subset of particles with augmented weights to approximate the posterior distribution.
• Ensure theoretical consistency with a bias that is controllable through a compression parameter, enabling a tunable tradeoff between memory usage and estimation accuracy.
• Overcome the intractability of direct measure-space projections by embedding the posterior in an RKHS via kernel mean embedding.
• Achieve finite model order growth in online settings by using an MMD-based orthogonal matching pursuit (OMP) stopping criterion.

Proposed method

• Embed the posterior density estimate into a reproducing kernel Hilbert space (RKHS) using kernel mean embedding to enable linear algebraic operations on probability measures.
• Sequentially project the RKHS representation of the posterior onto a lower-dimensional subspace using the maximum mean discrepancy (MMD), an integral probability metric.
• Apply an MMD-orthogonal matching pursuit (OMP) algorithm to iteratively select particles that are sufficiently far from the current subspace, ensuring diversity and representativeness.
• Use a stopping criterion based on MMD distance: if the distance from a new particle’s feature map to the current subspace is below a threshold ε/g(x(n)), the particle is not added.
• Formulate the projection error as a function of the importance weight and the MMD distance, enabling theoretical control over the bias introduced by compression.
• Establish that the number of particles in the dictionary remains finite and bounded by $ Oig((W/ u)^{2p}ig) $, where $ W $ is a bound on the unnormalized importance weight and $ u $ is the compression threshold.

Experimental results

Research questions

• RQ1 Can we maintain asymptotic consistency in importance sampling while using only a finite number of particles?
• RQ2 How can we ensure that the compressed particle set remains representative of the true posterior distribution in an online, streaming setting?
• RQ3 What is the theoretical bias introduced by finite particle compression, and can it be quantified and controlled?
• RQ4 Can the MMD-based projection criterion guarantee finite model order growth in the dictionary size under bounded importance weights?
• RQ5 To what extent can memory complexity be reduced without sacrificing estimation accuracy in streaming Bayesian inference?

Key findings

• The proposed method achieves nearly consistent estimates with a bias that is explicitly bounded by the compression parameter ε, enabling a tunable tradeoff between memory and accuracy.
• The number of particles retained in the dictionary is finite and upper-bounded by $ Cig((W/ u)ig)^{2p} $, where $ W $ is a bound on the unnormalized importance weight and $ u $ is the MMD threshold.
• The MMD-OMP stopping criterion ensures that new particles are only added when they contribute a significant distance from the current subspace, preventing redundant or redundant particle accumulation.
• Theoretical analysis shows that model order growth (i.e., dictionary size increase) occurs only when the MMD distance exceeds $ rac{ u}{g(oldsymbol{x}(n))} $, which is guaranteed to be bounded away from zero under bounded importance weights.
• Empirical results demonstrate that the compressed estimates achieve performance comparable to full-sample importance sampling, with significant reductions in representational complexity.
• The method ensures that the projection error is bounded by $ u $, and the resulting estimate converges to the true posterior as $ u o 0 $, establishing consistency in the limit of vanishing compression.