A Spectral Approach to Gradient Estimation for Implicit Distributions

Abstract

Recently there have been increasing interests in learning and inference with implicit distributions (i.e., distributions without tractable densities). To this end, we develop a gradient estimator for implicit distributions based on Stein's identity and a spectral decomposition of kernel operators, where the eigenfunctions are approximated by the Nyström method. Unlike the previous works that only provide estimates at the sample points, our approach directly estimates the gradient function, thus allows for a simple and principled out-of-sample extension. We provide theoretical results on the error bound of the estimator and discuss the bias-variance tradeoff in practice. The effectiveness of our method is demonstrated by applications to gradient-free Hamiltonian Monte Carlo and variational inference with implicit distributions. Finally, we discuss the intuition behind the estimator by drawing connections between the Nyström method and kernel PCA, which indicates that the estimator can automatically adapt to the geometry of the underlying distribution.

arXiv commentTo appear at ICML 2018

References in corpus (16)

• Adam: A Method for Stochastic Optimization
• Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks
• GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium
• Weight Uncertainty in Neural Networks
• Adversarial Feature Learning
• Adversarially Learned Inference
• Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm
• Learning in Implicit Generative Models
• Measuring Sample Quality with Stein's Method
• Hierarchical Implicit Models and Likelihood-Free Variational Inference
• Variational Inference using Implicit Distributions
• Operator Variational Inference
• Gradient Estimators for Implicit Models
• Provable Bayesian Inference via Particle Mirror Descent
• ZhuSuan: A Library for Bayesian Deep Learning
• High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation

Cited by in corpus (20)

• The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
• Monte Carlo Gradient Estimation in Machine Learning
• Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models
• Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation
• Adversarial Distributional Training for Robust Deep Learning
• Repulsive Deep Ensembles are Bayesian
• Latent Variable Modelling with Hyperbolic Normalizing Flows
• Variational Implicit Processes
• Mutual Information Gradient Estimation for Representation Learning
• LiBRe: A Practical Bayesian Approach to Adversarial Detection
• Bidirectional Generative Modeling Using Adversarial Gradient Estimation
• Blindness of score-based methods to isolated components and mixing proportions
• Posterior Meta-Replay for Continual Learning
• Efficient Learning of Generative Models via Finite-Difference Score Matching
• Autoregressive Score Matching
• Sampling with Mirrored Stein Operators
• Variationally Inferred Sampling Through a Refined Bound for Probabilistic Programs
• Active Slices for Sliced Stein Discrepancy
• Accurate and Reliable Forecasting using Stochastic Differential Equations
• KernelNet: A Data-Dependent Kernel Parameterization for Deep Generative Modeling