[Paper Review] Pathfinder: Parallel quasi-Newton variational inference

Lu Zhang, Bob Carpenter+2 more|arXiv (Cornell University)|Aug 9, 2021

Probabilistic and Robust Engineering Design51 references18 citations

TL;DR

Pathfinder is a parallel variational inference algorithm that uses quasi-Newton optimization paths to generate normal approximations to complex posterior distributions, leveraging inverse Hessian estimates for covariance. It achieves performance comparable to short Hamiltonian Monte Carlo chains with 1–2 orders of magnitude fewer log density and gradient evaluations, while enabling embarrassingly parallel ELBO evaluation for speedup on multi-core systems.

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.

Motivation & Objective

• To develop a scalable and robust variational inference method that outperforms existing approaches in accuracy and efficiency.
• To leverage curvature information from L-BFGS optimization paths to construct high-quality normal approximations to complex posteriors.
• To enable parallel evaluation of evidence lower bounds (ELBO) across multiple points on the optimization path, overcoming sequential bottlenecks in standard VI.
• To improve robustness and diversity of approximate samples through importance resampling over multiple Pathfinder runs.
• To provide a fast, drop-in replacement for MCMC warmup initialization, particularly for Hamiltonian Monte Carlo and adaptive Metropolis samplers.

Proposed method

• Pathfinder uses L-BFGS to generate a quasi-Newton optimization path from random initializations, traversing from the tail to the mode of the posterior.
• Along the optimization path, it evaluates normal approximations using local inverse Hessian estimates as covariance matrices.
• The algorithm computes the ELBO for each normal approximation in parallel, selecting the one with the lowest estimated KL divergence to the true posterior.
• It employs importance resampling over multiple independent Pathfinder runs to enhance sample diversity and robustness to optimization failures.
• The method uses Pareto smoothed importance sampling (PSIS) to stabilize the resampling process and improve posterior approximation quality.
• The core algorithm is embarrassingly parallel, as ELBO evaluations at different points on the path are independent and can be distributed across multiple cores.

Experimental results

Research questions

• RQ1 Can quasi-Newton optimization paths be used to generate high-quality normal approximations for complex posterior distributions?
• RQ2 Does parallel evaluation of ELBOs along the optimization path significantly improve computational efficiency compared to sequential VI?
• RQ3 How does Pathfinder’s performance compare to ADVI and short HMC chains in terms of 1-Wasserstein distance to the true posterior?
• RQ4 Can importance resampling over multiple Pathfinder runs improve sample diversity and robustness to initialization and local optima?
• RQ5 Can Pathfinder serve as an effective and efficient alternative to MCMC warmup phases in Hamiltonian Monte Carlo and adaptive Metropolis samplers?

Key findings

• Pathfinder produces approximate draws with 1-Wasserstein distances comparable to those from short dynamic Hamiltonian Monte Carlo chains, indicating high-quality posterior approximation.
• The method requires one to two orders of magnitude fewer log density and gradient evaluations than ADVI and short HMC, especially on challenging posteriors.
• Parallel ELBO evaluation enables significant speedups on multi-core systems, with performance gains increasing for more complex models.
• Importance resampling over multiple Pathfinder runs reduces 1-Wasserstein distance further and improves robustness to optimization failures on plateaus or in minor modes.
• The inverse Hessian estimates from L-BFGS provide a reliable initial covariance matrix for MCMC adaptation, potentially bypassing Stan’s second warmup phase in low-dimensional models.
• Pathfinder is robust to poor initializations when combined with scale-expanded reinitialization, enabling discovery of multiple posterior modes.