A Stein Identity for q-Gaussians with Bounded Support
This work simplifies gradient estimation for non-Gaussian distributions, potentially benefiting Bayesian deep learning and sharpness-aware minimization, but it is incremental as it extends prior results.
The paper tackled the problem of applying Stein's identity to non-Gaussian distributions by deriving a new Stein identity for bounded-support q-Gaussians, resulting in gradient estimators with forms similar to Gaussian ones and reduced variance in experiments.
Stein's identity is a fundamental tool in machine learning with applications in generative models, stochastic optimization, and other problems involving gradients of expectations under Gaussian distributions. Less attention has been paid to problems with non-Gaussian expectations. Here, we consider the class of bounded-support $q$-Gaussians and derive a new Stein identity leading to gradient estimators which have nearly identical forms to the Gaussian ones, and which are similarly easy to implement. We do this by extending the previous results of Landsman, Vanduffel, and Yao (2013) to prove new Bonnet- and Price-type theorems for q-Gaussians. We also simplify their forms by using escort distributions. Our experiments show that bounded-support distributions can reduce the variance of gradient estimators, which can potentially be useful for Bayesian deep learning and sharpness-aware minimization. Overall, our work simplifies the application of Stein's identity for an important class of non-Gaussian distributions.