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1. Stein Variational Gradient Descent With Matrix-Valued Kernels

   Wang, Dilin; Tang, Ziyang; Bajaj, Chandrajit; Liu, Qiang (January 2019, Conference on Neural Information Processing Systems)

   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. 

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2. Sampling with Trusthworthy Constraints: A Variational Gradient Framework

   Xingchao Liu, Xin Tong (January 2021, Advances in neural information processing systems)

   Sampling-based inference and learning techniques, especially Bayesian inference, provide an essential approach to handling uncertainty in machine learning (ML). As these techniques are increasingly used in daily life, it becomes essential to safeguard the ML systems with various trustworthy-related constraints, such as fairness, safety, interpretability. Mathematically, enforcing these constraints in probabilistic inference can be cast into sampling from intractable distributions subject to general nonlinear constraints, for which practical efficient algorithms are still largely missing. In this work, we propose a family of constrained sampling algorithms which generalize Langevin Dynamics (LD) and Stein Variational Gradient Descent (SVGD) to incorporate a moment constraint specified by a general nonlinear function. By exploiting the gradient flow structure of LD and SVGD, we derive two types of algorithms for handling constraints, including a primal-dual gradient approach and the constraint controlled gradient descent approach. We investigate the continuous-time mean-field limit of these algorithms and show that they have O(1/t) convergence under mild conditions. Moreover, the LD variant converges linearly assuming that a log Sobolev like inequality holds. Various numerical experiments are conducted to demonstrate the efficiency of our algorithms in trustworthy settings. 

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3. Nonlinear Stein Variational Gradient Descent for Learning Diversified Mixture Models

   Wang, Dilin; Liu, Qiang (July 2019, International conference of machine learning)

   Diversification has been shown to be a powerful mechanism for learning robust models in non- convex settings. A notable example is learning mixture models, in which enforcing diversity between the different mixture components allows us to prevent the model collapsing phenomenon and capture more patterns from the observed data. In this work, we present a variational approach for diversity-promoting learning, which leverages the entropy functional as a natural mechanism for enforcing diversity. We develop a simple and efficient functional gradient-based algorithm for optimizing the variational objective function, which provides a significant generalization of Stein variational gradient descent (SVGD). We test our method on various challenging real world problems, including deep embedded clustering and deep anomaly detection. Empirical results show that our method provides an effective mechanism for diversity-promoting learning, achieving substantial improvement over existing methods. 

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4. Profiling Pareto Front With Multi-Objective Stein Variational Gradient Descent

   Xingchao Liu; Xin Tong; Qiang Liu (January 2021, Advances in neural information processing systems)

   Finding diverse and representative Pareto solutions from the Pareto front is a key challenge in multi-objective optimization (MOO). In this work, we propose a novel gradient-based algorithm for profiling Pareto front by using Stein variational gradient descent (SVGD). We also provide a counterpart of our method based on Langevin dynamics. Our methods iteratively update a set of points in a parallel fashion to push them towards the Pareto front using multiple gradient descent, while encouraging the diversity between the particles by using the repulsive force mechanism in SVGD, or diffusion noise in Langevin dynamics. Compared with existing gradient-based methods that require predefined preference functions, our method can work efficiently in high dimensional problems, and can obtain more diverse solutions evenly distributed in the Pareto front. Moreover, our methods are theoretically guaranteed to converge to the Pareto front. We demonstrate the effectiveness of our method, especially the SVGD algorithm, through extensive experiments, showing its superiority over existing gradient-based algorithms. 

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5. Path-Guided Particle-based Sampling

   Fan, Mingzhou; Zhou, Ruida; Tian, Chao; Qian, Xiaoning (July 2024, International Conference on Machine Learning (ICML), PMLR 235, 2024)

   Particle-based Bayesian inference methods by sampling from a partition-free target (posterior) distribution, e.g., Stein variational gradient descent (SVGD), have attracted significant attention. We propose a path-guided particle-based sampling (PGPS) method based on a novel Logweighted Shrinkage (LwS) density path linking an initial distribution to the target distribution. We propose to utilize a Neural network to learn a vector field motivated by the Fokker-Planck equation of the designed density path. Particles, initiated from the initial distribution, evolve according to the ordinary differential equation defined by the vector field. The distribution of these particles is guided along a density path from the initial distribution to the target distribution. The proposed LwS density path allows for an efficient search of modes of the target distribution while canonical methods fail. We theoretically analyze the Wasserstein distance of the distribution of the PGPS-generated samples and the target distribution due to approximation and discretization errors. Practically, the proposed PGPS-LwS method demonstrates higher Bayesian inference accuracy and better calibration ability in experiments conducted on both synthetic and real-world Bayesian learning tasks, compared to baselines, such as SVGD and Langevin dynamics, etc. 

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