Information Geometry of Message Passing

We show that the natural-gradient stationary condition of variational inference has an edge-local form on a Forney-style factor graph. We start from the Bethe free energy and constrain a selected edge marginal to an exponential family. At a stationary point, the natural parameter of that edge equals the sum of two projected messages, one from each incident factor. Each projected message is the natural-gradient projection of the exact belief-propagation log-message at the current receiving marginal, or equivalently, the gradient of its expectation in the so-called mean coordinates. We call the resulting scheme natural-gradient message passing (NGMP). The rule is local; each edge may carry its own exponential family, and the message a factor sends depends on the marginal that receives it. Compared with variational message passing, NGMP keeps the part of the exact message that the receiving family can represent instead of averaging the factor under the neighboring beliefs. The two coincide when the uncertainty on the edges entering a non-conjugate factor vanishes, and NGMP is more accurate when that uncertainty persists, for example, along a partially observed latent chain or when parameters are filtered through successive data batches. Experiments on Poisson smoothing, heteroskedastic regression, and hourly ETTh forecasting confirm this and show that the gain appears mainly in uncertainty calibration.

On the robustness of noisy solutions in non-convex neural networks

Optimization in non-convex neural network models is strongly influenced by the geometry of the solution space: sparse, isolated, point-like clusters are typically algorithmically inaccessible, whereas wide and flat regions can be found efficiently despite being relatively rare. At zero temperature this picture has been formalized in binary perceptrons through the overlap gap property (OGP), which limits algorithmic access to configurations with zero training error above a critical constraint density $α_{\rm OGP}$. Here we extend this description to finite temperature, where a positive training error is allowed and statistically penalized. We first show that the frozen one-step replica-symmetry-breaking solution, dominating the zero temperature equilibrium measure, survives at any finite temperature. We furthermore derive a general criterion, based on the smoothness of the single-pattern Gibbs weight near the decision boundary, that determines when a finite-temperature relaxation of the loss removes freezing. We then extend the OGP construction to finite temperature and show that dense, algorithmically accessible regions of finite-energy configurations persist beyond $α_{\rm OGP}$, up to a threshold $α_{\rm OGP}(ε)$ that grows with the allowed training error $ε$. Finally, in the teacher-student setting, we show that these wide, finite-energy regions still retain good generalization. Using a finite energy message-passing algorithm, we demonstrate numerically that thermal noise enables effective generalization in the regime of constraint densities where both recovering the teacher and finding a zero temperature solution are computationally hard.