[Paper Review] Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring

Sungjin Ahn, Anoop Korattikara+1 more|UvA-DARE (University of Amsterdam)|Jun 27, 2012

Markov Chains and Monte Carlo Methods11 references114 citations

TL;DR

This paper proposes Stochastic Gradient Fisher Scoring (SGFS), a Bayesian posterior sampling method that uses stochastic gradients to efficiently generate samples from a posterior distribution using only mini-batches of data. By leveraging the Bayesian Central Limit Theorem, SGFS adapts its behavior to achieve fast mixing at high rates (sampling from a normal approximation) and improved convergence via preconditioning at low rates, while also functioning as an efficient optimizer during burn-in.

ABSTRACT

In this paper we address the following question: Can we approximately sample from a Bayesian posterior distribution if we are only allowed to touch a small mini-batch of data-items for every sample we generate?. An algorithm based on the Langevin equation with stochastic gradients (SGLD) was previously proposed to solve this, but its mixing rate was slow. By leveraging the Bayesian Central Limit Theorem, we extend the SGLD algorithm so that at high mixing rates it will sample from a normal approximation of the posterior, while for slow mixing rates it will mimic the behavior of SGLD with a pre-conditioner matrix. As a bonus, the proposed algorithm is reminiscent of Fisher scoring (with stochastic gradients) and as such an efficient optimizer during burn-in.

Motivation & Objective

• Address the challenge of sampling from Bayesian posteriors when only mini-batches of data can be accessed per sample.
• Improve upon Stochastic Gradient Langevin Dynamics (SGLD), which suffers from slow mixing rates.
• Leverage the Bayesian Central Limit Theorem to adapt sampling behavior based on mixing speed.
• Design a method that combines efficient posterior sampling with effective optimization during burn-in phases.
• Develop a stochastic gradient variant of Fisher scoring to enhance convergence and sampling efficiency.

Proposed method

• Propose a stochastic gradient-based algorithm that approximates the posterior distribution using only small data batches.
• Integrate the Bayesian Central Limit Theorem to dynamically adjust sampling behavior: normal approximation at high mixing rates, SGLD-like dynamics at low rates.
• Use a preconditioning matrix derived from the Fisher information to improve mixing and convergence speed.
• Formulate the update rule as a stochastic gradient Fisher scoring step, combining gradient ascent with curvature information.
• Maintain a burn-in phase where the algorithm acts as an efficient optimizer by mimicking Fisher scoring.
• Ensure the method remains computationally efficient by avoiding full-data computations and relying on mini-batch gradients.

Experimental results

Research questions

• RQ1 Can we achieve efficient Bayesian posterior sampling using only mini-batch data access per sample?
• RQ2 How can we improve the mixing rate of stochastic gradient-based posterior samplers like SGLD?
• RQ3 Can we leverage the Bayesian Central Limit Theorem to adapt sampling behavior based on mixing speed?
• RQ4 Does a stochastic gradient version of Fisher scoring improve convergence and sampling efficiency?
• RQ5 Can the same algorithm serve both as a sampler and an optimizer during burn-in?

Key findings

• The proposed Stochastic Gradient Fisher Scoring (SGFS) algorithm achieves faster mixing rates compared to standard SGLD by adapting to the posterior curvature.
• At high mixing rates, SGFS samples from a normal approximation of the posterior, enabling rapid exploration.
• At low mixing rates, SGFS emulates SGLD with a preconditioner, improving convergence and reducing random walk behavior.
• The algorithm functions as an efficient optimizer during burn-in due to its Fisher scoring-inspired update rule.
• SGFS maintains computational efficiency by relying solely on mini-batch gradients, avoiding full-data computations.
• Empirical results show improved sampling efficiency and faster convergence in posterior estimation compared to baseline SGLD.