Measuring Sample Quality with Kernels

By Jackson Gorham, Lester W. Mackey. 2017

263 citations recorded by the source corpus.

Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy measures that provably determine the convergence of a sample to its target distribution. This approach was recently combined with the theory of reproducing kernels to define a closed-form kernel Stein discrepancy (KSD) computable by summing kernel evaluations across pairs of sample points. We develop a theory of weak convergence for KSDs based on Stein's method, demonstrate that commonly used KSDs fail to detect non-convergence even for Gaussian targets, and show that kernels with slowly decaying tails provably determine convergence for a large class of target distributions. The resulting convergence-determining KSDs are suitable for comparing biased, exact, and deterministic sample sequences and simpler to compute and parallelize than alternative Stein discrepancies. We use our tools to compare biased samplers, select sampler hyperparameters, and improve upon existing KSD approaches to one-sample hypothesis testing and sample quality improvement.

Subfields: Artificial Intelligence, Control and Systems Engineering, Statistics and Probability, Evaluation & Benchmarks

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Citations in the corpus

A-NICE-MC: Adversarial Training for MCMC (2017, methodology): “Even evaluating an existing MCMC technique is often challenging, with natural performance metrics intractable to compute [8–10].”
Sliced Score Matching: A Scalable Approach to Density and Score Estimation (2019, methodology): “Two prior methods for score estimation are based on a generalized form of Stein’s identity (Stein, 1981; Gorham & Mackey, 2017): where q ( x ) is a continuously differentiable density and h ( x ) is a function satisfying some regularity conditions.”
Variational (Gradient) Estimate of the Score Function in Energy-based Latent Variable Models (2020, unknown): “Such a form can also be understood as the Stein discrepancy (Gorham & Mackey, 2017) with L 2 constraint on the function space F and has been applied to evaluate GRBMs (Grathwohl et al., 2020b).”
Stein Variational Gradient Descent as Moment Matching (2018, background): “It has been shown that this can be achieved if k ( x , x (cid:48) ) is strictly positive deﬁnite or universal, in a proper technical sense (Liu et al., 2016; Chwialkowski et al., 2016; Gorham & Mackey, 2017; Oates et al., 2017).”
Stein Variational Gradient Descent as Gradient Flow (2017, background): “More recently, Gorham and Mackey [10] (Theorem 8) established a stronger result that Stein discrepancy implies weak convergence on X = R d : let { µ (cid:96) } ∞ =1 be a sequence of probability measures, then for ν p that are distantly dissipative (Deﬁnition 4 of Gorham and Mackey [10]) and a class ”
Robust generalised Bayesian inference for intractable likelihoods (2021, unknown): “Sufficient conditions for convergence control have been established for the Langevin Stein operator, under which the convergence of KSD ( P 𝜃 || P n ) implies the weak convergence of P n to P 𝜃 (Gorham & Mackey, 2017, Theorem 8).”
Stein Neural Sampler (2018, unknown): “The minimum variance unbiased estimator can be written as (2.4) Despite the ease of computation, RKHS is relatively small and may fail to detect non-convergence in higher dimensions (Gorham and Mackey, 2017).”
Stochastic Gradient Markov Chain Monte Carlo (2019, methodology): “The design of appropriate diagnostic tests for stochastic gradient MCMC is a relatively new area of research, and currently methods based on Stein’s discrepancy (Gorham and Mackey, 2015; Gorham et al., 2016; Gorham and Mackey, 2017) are the most popular approach.”
Measuring Sample Quality with Diffusions (2016, methodology): “While we have focused on the graph and classical Stein sets of [35], our diffusion Stein operators can also be paired with the reproducing kernel Hilbert space unit balls advocated in [69, 68, 15, 58, 36] to form tractable kernel diffusion Stein discrepancies.”
Learning the Stein Discrepancy for Training and Evaluating Energy-Based Models without Sampling (2020, methodology): “Taking f to be the supremum over a class of functions F , we can create a quantitative measure: which is known as the Stein Discrepancy (SD) (Gorham & Mackey, 2017).”
Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices (2019, background): “There are recent works that analyze convergence of sampling using various techniques such as reflection coupling [28], kernel methods [29], and higher-order integrators [40], albeit still under some strong conditions such as distant dissipativity, which is similar to strong logconcavity outside a bo”