A Complete Recipe for Stochastic Gradient MCMC (1506.04696v2)

Abstract: Many recent Markov chain Monte Carlo (MCMC) samplers leverage continuous dynamics to define a transition kernel that efficiently explores a target distribution. In tandem, a focus has been on devising scalable variants that subsample the data and use stochastic gradients in place of full-data gradients in the dynamic simulations. However, such stochastic gradient MCMC samplers have lagged behind their full-data counterparts in terms of the complexity of dynamics considered since proving convergence in the presence of the stochastic gradient noise is non-trivial. Even with simple dynamics, significant physical intuition is often required to modify the dynamical system to account for the stochastic gradient noise. In this paper, we provide a general recipe for constructing MCMC samplers--including stochastic gradient versions--based on continuous Markov processes specified via two matrices. We constructively prove that the framework is complete. That is, any continuous Markov process that provides samples from the target distribution can be written in our framework. We show how previous continuous-dynamic samplers can be trivially "reinvented" in our framework, avoiding the complicated sampler-specific proofs. We likewise use our recipe to straightforwardly propose a new state-adaptive sampler: stochastic gradient Riemann Hamiltonian Monte Carlo (SGRHMC). Our experiments on simulated data and a streaming Wikipedia analysis demonstrate that the proposed SGRHMC sampler inherits the benefits of Riemann HMC, with the scalability of stochastic gradient methods.

• The paper introduces a complete framework that uses a diffusion matrix and a curl matrix to ensure continuous MCMC samplers maintain the desired invariant distribution.
• It unifies various methods, including HMC and SGHMC, by formulating a recipe that overcomes challenges of stochastic gradient noise without requiring Metropolis-Hastings corrections.
• Experimental results validate the approach by demonstrating faster convergence and robust exploration in complex posterior landscapes for large-scale Bayesian inference.

A Complete Recipe for Stochastic Gradient MCMC

The paper "A Complete Recipe for Stochastic Gradient MCMC," authored by Yi-An Ma, Tianqi Chen, and Emily B. Fox, provides a comprehensive framework for constructing Markov chain Monte Carlo (MCMC) samplers using continuous Markov processes. This framework notably encompasses stochastic gradient MCMC variants by specifying two matrices: a positive semidefinite diffusion matrix and a skew-symmetric curl matrix.

The motivation behind this work centers on overcoming the challenges associated with simple dynamics and stochastic gradient noise in MCMC samplers. The authors propose a methodology that constructs MCMC samplers which naturally maintain the target posterior distribution as their invariant distribution.

Key Contributions

1. Unified Framework: The authors introduce a unifying framework that can encompass all continuous-dynamic MCMC methods. By defining the dynamics in terms of a diffusion matrix $D(z)$ and a curl matrix $Q(z)$, the framework ensures that the stationary distribution, which samples the target distribution, is preserved.
2. Completeness: A critical feature of this approach is its completeness. For any continuous Markov process expected to yield a desired stationary distribution, the framework can identify appropriate matrices, $D(z)$ and $Q(z)$, confirming that the proposed method is exhaustive in terms of constructing valid samplers.
3. General Recipe for Dynamics: The framework suggests that any choice of these matrices will maintain the correct invariant distribution if the process remains ergodic. Moreover, the authors constructively demonstrate how many classical and recent MCMC methods fit into this formulation.
4. Stochastic Gradient Extension: The authors effectively extend this methodology to stochastic gradient MCMC samplers. They avoid the computational burdens associated with Metropolis-Hastings corrections by presenting a strictly stochastic sampling procedure.

Practical Implementations

The paper applies its framework to both well-known and newly proposed algorithms:

• Hamiltonian Monte Carlo (HMC) and Variants: The authors reframe traditional HMC along with recent enhancements like Stochastic Gradient HMC (SGHMC) within their model. They manage to highlight potential pitfalls of straightforward stochastic gradient applications, emphasizing the need for corrections related to diffusion and noise.
• Stochastic Gradient Riemann Hamiltonian Monte Carlo (SGRHMC): As a direct application of their approach, the authors propose an innovative sampler, SGRHMC, which promises efficient exploration of parameter spaces by incorporating geometric information through the Fisher Information Metric.

Experimental Validation

Experimental results underscore their theoretical claims, focusing on both synthetic data and practical applications such as online Latent Dirichlet Allocation (LDA). The SGRHMC sampler showcases superior performance in complex posterior landscapes, demonstrating quicker convergence and effectiveness.

Implications and Future Directions

The proposed framework has significant implications:

• Practical Impact: By providing an efficient and scalable approach, this methodology can substantially benefit large-scale Bayesian inference problems.
• Theoretical Advancements: The completeness of the framework suggests avenues for future research in improving the dynamics and exploring uncharted regions within the matrices' space.
• Further Research: The exploration of optimal $D(z)$ and $Q(z)$ tailored for specific problem spaces remains an open area that could lead to more efficient methods.

In summary, the paper delineates a robust structure for constructing stochastic gradient MCMC that is both theoretically sound and practically applicable. By presenting a complete and adaptable framework, the research paves the way for further advancements in probabilistic inference through MCMC methodologies.