[Paper Review] Quantum Speedup of Natural Gradient for Variational Bayes

Anna Lopatnikova, Minh‐Ngoc Tran|arXiv (Cornell University)|Jun 10, 2021

Spectroscopy and Quantum Chemical Studies4 citations

TL;DR

This paper proposes a regression-based method for efficient natural gradient estimation in Variational Bayes (VB), enabling quantum speedup via quantum matrix inversion. By reformulating natural gradient computation as a linear regression problem, the method achieves classical complexity of Õ(N^b) with 1 ≤ b < 2 and satisfies the conditions for quantum advantage, reducing computation to O(N) under standard assumptions.

ABSTRACT

Variational Bayes (VB) is a critical method in machine learning and statistics, underpinning the recent success of Bayesian deep learning. The natural gradient is an essential component of efficient VB estimation, but it is prohibitively computationally expensive in high dimensions. We propose a computationally efficient regression-based method for natural gradient estimation, with convergence guarantees under standard assumptions. The method enables the use of quantum matrix inversion to further speed up VB. We demonstrate that the problem setup fulfills the conditions required for quantum matrix inversion to deliver computational efficiency. The method works with a broad range of statistical models and does not require special-purpose or simplified variational distributions.

Motivation & Objective

• Address the computational infeasibility of natural gradient estimation in high-dimensional Variational Bayes due to Fisher information matrix inversion.
• Develop a regression-based approach to approximate natural gradient that is both computationally efficient and compatible with quantum algorithms.
• Demonstrate that the reformulated problem meets the stringent requirements for quantum linear systems algorithms, enabling potential quantum speedup.
• Ensure the method remains general and applicable to a broad class of statistical models without requiring simplified variational families.
• Provide theoretical convergence guarantees under standard assumptions, ensuring reliability in practical applications.

Proposed method

• Reformulate natural gradient estimation as a linear regression problem by exploiting the structure of the Fisher information matrix.
• Use empirical risk minimization on a regression model where the response variable is the score function and the covariates are the sufficient statistics of the variational family.
• Apply stochastic gradient descent with a diminishing step size schedule to solve the regression problem, ensuring convergence to the true natural gradient.
• Leverage the resulting regression framework to enable quantum matrix inversion via the HHL algorithm, which offers exponential speedup under specific conditions.
• Ensure the problem satisfies the sparsity, condition number, and state preparation requirements for quantum linear system algorithms.
• Use the Fisher-Rao metric to define the norm in the convergence analysis, ensuring geometric consistency with information-geometric principles.

Experimental results

Research questions

• RQ1 Can natural gradient estimation in Variational Bayes be reformulated as a regression problem to enable efficient classical and quantum computation?
• RQ2 Does the proposed regression-based method achieve convergence guarantees under standard assumptions for VB optimization?
• RQ3 Can the reformulated problem be made amenable to quantum speedup via quantum linear systems algorithms such as HHL?
• RQ4 What are the computational complexity bounds of the proposed method in both classical and quantum settings?
• RQ5 How does the method maintain generality across diverse statistical models without requiring specialized or simplified variational distributions?

Key findings

• The regression-based method reduces the classical computational complexity of natural gradient estimation to Õ(N^b) with 1 ≤ b < 2, significantly improving upon the standard O(N^d) complexity with 2 < d ≤ 3.
• The method satisfies the necessary conditions—sparsity, bounded condition number, and efficient state preparation—for quantum speedup using the HHL algorithm.
• Quantum matrix inversion can reduce the overall complexity to O(N), matching the information-theoretic lower bound for writing down the variational parameters.
• Theoretical convergence is established under standard assumptions, with the expected norm of the gradient in the Fisher-Rao metric converging to zero as the number of iterations increases.
• The method ensures that the estimated parameters converge in expectation to the true optimal solution, with the distance to the optimum decaying to zero over time.
• The approach is general and does not require simplification of the variational family, making it applicable to complex models such as deep neural networks in Bayesian inference.