Pathfinder: Parallel quasi-Newton variational inference (2108.03782v4)

Abstract: We propose Pathfinder, a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the inverse Hessian estimates produced by the optimizer. Pathfinder returns draws from the approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior. We evaluate Pathfinder on a wide range of posterior distributions, demonstrating that its approximate draws are better than those from automatic differentiation variational inference (ADVI) and comparable to those produced by short chains of dynamic Hamiltonian Monte Carlo (HMC), as measured by 1-Wasserstein distance. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors. Importance resampling over multiple runs of Pathfinder improves the diversity of approximate draws, reducing 1-Wasserstein distance further and providing a measure of robustness to optimization failures on plateaus, saddle points, or in minor modes. The Monte Carlo KL divergence estimates are embarrassingly parallelizable in the core Pathfinder algorithm, as are multiple runs in the resampling version, further increasing Pathfinder's speed advantage with multiple cores.

Overview of "Pathfinder: Parallel Quasi-Newton Variational Inference"

The paper "Pathfinder: Parallel quasi-Newton variational inference," introduces a novel algorithm named Pathfinder, designed for approximating sampling from differentiable log densities, which is a core task in Bayesian computation. This paper proposes a distinct approach to variational inference (VI), leveraging quasi-Newton optimization techniques to enhance efficiency, scalability, and robustness in the inference process.

Pathfinder operates by following an optimization trajectory starting from a random initialization point, aiming to establish normal approximations to the target density while capitalizing on local covariance information gleaned from quasi-Newton inverse Hessian estimates. The algorithm identifies the normal approximation that minimizes the Kullback-Leibler (KL) divergence relative to the true posterior, effectively striking a balance between computational feasibility and inferential accuracy.

Algorithmic Approach

Pathfinder differentiates itself from traditional automatic differentiation variational inference (ADVI) by employing a quasi-Newton method (specifically L-BFGS) to trace the optimization path. The trajectory traverses from the posterior's tail through the body to a mode or pole, during which local Gaussian approximations are evaluated. The algorithm exploits the inverse Hessian estimates from these paths to propose candidate distributions.

Significantly, Pathfinder requires one to two orders of magnitude fewer evaluations of log density and gradient computations relative to ADVI and short chains of dynamic Hamiltonian Monte Carlo (HMC), thus offering substantial computational advantages, especially in challenging posterior landscapes. Additionally, importance resampling aids in mitigating the risk of optimization failures associated with plateaus, saddle points, or minor modes, ensuring robustness and diversity in the approximated draws.

A critical innovation is Pathfinder's capacity for embarrassingly parallel evaluation of Monte Carlo KL divergence via multiple cores, significantly accelerating its processing compared to traditional sequential VI algorithms. This parallelizable aspect of Pathfinder could be especially advantageous in contexts where computational resources can be leveraged for concurrent evaluations.

Experimental Evaluation

The authors conducted extensive experiments using a diverse set of 20 Bayesian models, assessing Pathfinder's performance against both ADVI (mean-field and full-rank versions) and initial phases of dynamic HMC. The results, evaluated via 1-Wasserstein distance, demonstrated that Pathfinder's approximations ranged from comparable to notably superior, especially in computational efficiency. Cases featuring posteriors with complex geometry, such as Neal's Funnel or models with multimodal distributions, further highlighted Pathfinder's advantage in navigating the high probability regions effectively.

Practical and Theoretical Implications

The introduction of Pathfinder has several implications:

1. Enhanced Initialization for MCMC: Pathfinder could serve as a potent initializer for MCMC algorithms, potentially reducing the need for exhaustive warm-up phases, thereby increasing overall efficiency in sampling from posterior distributions.
2. Broader Applicability in Variational Inference: The integration of quasi-Newton optimization into the VI framework offers a robust alternative for cases where traditional stochastic gradient descent approaches might falter due to high variance or need for excessively small step sizes.
3. Advancements in Parallel Computing for Bayesian Inference: By facilitating parallel computation of KL divergence estimates, Pathfinder opens new avenues for optimizing resource allocation and reducing computation times in Bayesian workflow-settings.
4. Potential Adaptation to High-Dimensional Inference Problems: The scalability of Pathfinder in terms of computational cost provides a promising outlook for tackling high-dimensional models prevalent in contemporary data-intensive fields.

Future Directions

Given its robust framework, Pathfinder could be extended to address several current challenges and opportunities in Bayesian computation:

• Integration with Discrete Parameter Models: Extending Pathfinder's methodology to incorporate discrete parameter spaces might widen its applicability across different models and contexts.
• Multi-path Extensions for Enhanced Robustness: The paper's exploration of a multi-path variant suggests further investigations into multi-path optimization could enhance robustness against local optima in complex posteriors.
• Adaptive Schemes for Initial Distribution Choice: Developing adaptive strategies for selecting initialization distributions to maximize the exploration of posterior spaces can augment Pathfinder's effectiveness, especially in heavily multimodal contexts.

In summary, Pathfinder represents a significant step forward in variational inference methods, offering a practical, efficient, and parallelizable solution to posterior approximation. Its marriage of quasi-Newton optimization with VI stands poised to influence ongoing advancements in the field of statistical computation, particularly in effectively managing the computational demands of modern Bayesian inference.