The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks

Key points

• Tempering the likelihood, by raising it to the power $1/T$ for a temperature $T < 1$, is equivalent to scaling the KL term by $T$.1
• We ask in this paper how fast $T$ must decrease with $M$ to counteract this degeneracy and strike a good balance between the two terms.1
• For single-hidden-layer linear networks with isotropic Gaussian priors, we derive the limiting predictive distribution under schedules of the form $T = τ/M^{c}$, with constants $τ, c > 0$, as $M \to \infty$ and compare it with the untempered neural network Gaussian process (NNGP) posterior, the infinite-width limit of the exact posterior.1
• With suitable choices of $τ,c$, one can recover either the NNGP posterior expectation or its variance.1

Sources (1)

• [1]The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 6, 09:24 PM

  In wide Bayesian neural networks, Gaussian mean-field variational inference is prone to "prior dominance": the Kullback-Leibler (KL) regularization term of the ELBO outweighs the expected log-likelihood, and the variational predictive distribution collapses to the prior predictive as the width $M$ grows.

  Tempering the likelihood, by raising it to the power $1/T$ for a temperature $T < 1$, is equivalent to scaling the KL term by $T$.

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