Stein Variational Gradient Descent With Matrix-Valued Kernels

Stein variational gradient descent (SVGD) is a particle-based inference algorithm that leverages gradient information for efficient approximate inference. In this work, we enhance SVGD by leveraging preconditioning matrices, such as the Hessian and Fisher information matrix, to incorporate geometric information into SVGD updates. We achieve this by presenting a generalization of SVGD that replaces the scalar-valued kernels in vanilla SVGD with more general matrix-valued kernels. This yields a significant extension of SVGD, and more importantly, allows us to flexibly incorporate various preconditioning matrices to accelerate the exploration in the probability landscape. Empirical results show that our method outperforms vanilla SVGD and a variety of baseline approaches over a range of real-world Bayesian inference tasks.

Splitting Steepest Descent for Growing Neural Architectures

We develop a progressive training approach for neural networks which adaptively grows the network structure by splitting existing neurons to multiple off-springs. By leveraging a functional steepest descent idea, we derive a simple criterion for deciding the best subset of neurons to split and a splitting gradient for optimally updating the off-springs. Theoretically, our splitting strategy is a second-order functional steepest descent for escaping saddle points in an $\infty$-Wasserstein metric space, on which the standard parametric gradient descent is a first-order steepest descent. Our method provides a new computationally efficient approach for optimizing neural network structures, especially for learning lightweight neural architectures in resource-constrained settings.

Energy-Aware Neural Architecture Optimization with Fast Splitting Steepest Descent

Designing energy-efficient networks is of critical importance for enabling state-of-the-art deep learning in mobile and edge settings where the computation and energy budgets are highly limited. Recently, Wu et al. (2019) framed the search of efficient neural architectures into a continuous splitting process: it iteratively splits existing neurons into multiple off-springs to achieve progressive loss minimization, thus finding novel architectures by gradually growing the neural network. However, this method was not specifically tailored for designing energy-efficient networks, and is computationally expensive on large-scale benchmarks. In this work, we substantially improve Wu et al. (2019) in two significant ways: 1) we incorporate the energy cost of splitting different neurons to better guide the splitting process, thereby discovering more energy-efficient network architectures; 2) we substantially speed up the splitting process of Wu et al. (2019), which requires expensive eigen-decomposition, by proposing a highly scalable Rayleigh-quotient stochastic gradient algorithm. Our fast algorithm allows us to reduce the computational cost of splitting to the same level of typical back-propagation updates and enables efficient implementation on GPU. Extensive empirical results show that our method can train highly accurate and energy-efficient networks on challenging datasets such as ImageNet, improving a variety of baselines, including the pruning-based methods and expert-designed architectures.

Improving Neural Language Modeling via Adversarial Training

Recently, substantial progress has been made in language modeling by using deep neural networks. However, in practice, large scale neural language models have been shown to be prone to overfitting. In this paper, we present a simple yet highly effective adversarial training mechanism for regularizing neural language models. The idea is to introduce adversarial noise to the output embedding layer while training the models. We show that the optimal adversarial noise yields a simple closed-form solution, thus allowing us to develop a simple and time efficient algorithm. Theoretically, we show that our adversarial mechanism effectively encourages the diversity of the embedding vectors, helping to increase the robustness of models. Empirically, we show that our method improves on the single model state-of-the-art results for language modeling on Penn Treebank (PTB) and Wikitext-2, achieving test perplexity scores of 46.01 and 38.07, respectively. When applied to machine translation, our method improves over various transformer-based translation baselines in BLEU scores on the WMT14 English-German and IWSLT14 German-English tasks.

Variational Inference with Tail-adaptive f-Divergence

Variational inference with {\alpha}-divergences has been widely used in modern probabilistic machine learning. Compared to Kullback-Leibler (KL) divergence, a major advantage of using {\alpha}-divergences (with positive {\alpha} values) is their mass-covering property. However, estimating and optimizing {\alpha}-divergences require to use importance sampling, which could have extremely large or infinite variances due to heavy tails of importance weights. In this paper, we propose a new class of tail-adaptive f-divergences that adaptively change the convex function f with the tail of the importance weights, in a way that theoretically guarantees finite moments, while simultaneously achieving mass-covering properties. We test our methods on Bayesian neural networks, as well as deep reinforcement learning in which our method is applied to improve a recent soft actor-critic (SAC) algorithm. Our results show that our approach yields significant advantages compared with existing methods based on classical KL and {\alpha}-divergences.

Stein Variational Gradient Descent as Moment Matching

Stein variational gradient descent (SVGD) is a non-parametric inference algorithm that evolves a set of particles to fit a given distribution of interest. We analyze the non-asymptotic properties of SVGD, showing that there exists a set of functions, which we call the Stein matching set, whose expectations are exactly estimated by any set of particles that satisfies the fixed point equation of SVGD. This set is the image of Stein operator applied on the feature maps of the positive definite kernel used in SVGD. Our results provide a theoretical framework for analyzing the properties of SVGD with different kernels, shedding insight into optimal kernel choice. In particular, we show that SVGD with linear kernels yields exact estimation of means and variances on Gaussian distributions, while random Fourier features enable probabilistic bounds for distributional approximation. Our results offer a refreshing view of the classical inference problem as fitting Stein's identity or solving the Stein equation, which may motivate more efficient algorithms.

Stein Variational Message Passing for Continuous Graphical Models

We propose a novel distributed inference algorithm for continuous graphical models, by extending Stein variational gradient descent (SVGD) to leverage the Markov dependency structure of the distribution of interest. Our approach combines SVGD with a set of structured local kernel functions defined on the Markov blanket of each node, which alleviates the curse of high dimensionality and simultaneously yields a distributed algorithm for decentralized inference tasks. We justify our method with theoretical analysis and show that the use of local kernels can be viewed as a new type of localized approximation that matches the target distribution on the conditional distributions of each node over its Markov blanket. Our empirical results show that our method outperforms a variety of baselines including standard MCMC and particle message passing methods.