KSD Aggregated Goodness-of-fit Test

TL;DR

This paper introduces KSDAgg, a new goodness-of-fit test based on Kernel Stein Discrepancy that aggregates multiple kernels, avoids data splitting, and achieves near-optimal power with practical computation methods.

Contribution

The paper proposes KSDAgg, a novel aggregation-based goodness-of-fit test that enhances power and computational efficiency over existing KSD methods.

Findings

KSDAgg outperforms existing KSD-based tests on synthetic data.

It achieves near-minimax optimal rates for certain density classes.

The method effectively selects kernel bandwidths without data splitting.

Abstract

We investigate properties of goodness-of-fit tests based on the Kernel Stein Discrepancy (KSD). We introduce a strategy to construct a test, called KSDAgg, which aggregates multiple tests with different kernels. KSDAgg avoids splitting the data to perform kernel selection (which leads to a loss in test power), and rather maximises the test power over a collection of kernels. We provide non-asymptotic guarantees on the power of KSDAgg: we show it achieves the smallest uniform separation rate of the collection, up to a logarithmic term. For compactly supported densities with bounded model score function, we derive the rate for KSDAgg over restricted Sobolev balls; this rate corresponds to the minimax optimal rate over unrestricted Sobolev balls, up to an iterated logarithmic term. KSDAgg can be computed exactly in practice as it relies either on a parametric bootstrap or on a wild bootstrap to estimate the quantiles and the level corrections. In particular, for the crucial choice of bandwidth of a fixed kernel, it avoids resorting to arbitrary heuristics (such as median or standard deviation) or to data splitting. We find on both synthetic and real-world data that KSDAgg outperforms other state-of-the-art quadratic-time adaptive KSD-based goodness-of-fit testing procedures.