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Variational inference matches exact posterior rates for Besov priors

Engineering research
Photo by Mikhail Nilov on Pexels
Research area:Applied mathematicsNumerical methods in inverse problemsInverse problem

What the study found

Variational posteriors for inverse problems with parameters in Besov spaces can achieve convergence rates that match exact Bayesian posteriors under the conditions studied. The authors also report minimax-optimal rates over Besov function classes and better rates than Gaussian priors by a polynomial factor.

Why the authors say this matters

The study suggests that variational inference can provide efficient approximations without losing the convergence-rate guarantees of exact Bayesian inference. The authors conclude that this is relevant for PDE-constrained regression problems and for comparing Besov priors with Gaussian priors.

What the researchers tested

The researchers studied inverse problems for partial differential equations (PDEs) with parameters in Besov spaces, using Besov priors built from random wavelet expansions with p-exponentially distributed coefficients. They analyzed variational inference through a refined prior mass and testing framework and considered common variational families, including Besov-type measures and mean-field families.

What worked and what didn't

Under general conditions on PDE operators, the variational posteriors achieved convergence rates matching those of the exact posterior. The results were also minimax-optimal over B^{alpha}_{pp} classes, and the abstract says these rates outperform Gaussian-prior rates by a polynomial factor. The paper also reports minimax-optimal prediction-loss rates for the two example problems considered.

What to keep in mind

The abstract does not give detailed assumptions, proof limits, or numerical performance details. The examples discussed are Darcy flow and an inverse potential problem for a subdiffusion equation, so the scope described in the abstract is limited to these types of nonlinear inverse problems.

Key points

  • Variational posteriors matched the convergence rates of exact posteriors under the studied conditions.
  • The abstract reports minimax-optimal rates over Besov function classes B^alpha_pp.
  • The paper says these rates improve on Gaussian-prior rates by a polynomial factor.
  • Two example problems were analyzed: Darcy flow and an inverse potential problem for a subdiffusion equation.
  • The abstract reports minimax-optimal prediction-loss rates for the PDE-constrained regression setting.

Disclosure

Research title:
Variational inference matches exact posterior rates for Besov priors
Image credit:
Photo by Mikhail Nilov on Pexels
AI provenance: AI provenance information is not available for this post.

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