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.

Replica Symmetry Breaking and Algorithmic Thresholds in Empirical Risk Minimization under Multi-Index Model

Modern machine learning models are trained by optimizing high-dimensional non-convex empirical risk functions. Such cost functions can have a multitude of local optima and yet, gradient-based optimization appears to converge to near-global optima. Within a simple supervised learning setting, we develop a precise picture of which parts of the empirical risk landscape are accessible by polynomial-time algorithms. We are given i.i.d. pairs $\{(\boldsymbol{x}_i,y_i):\; 1 \le i\le n\}$ with $\boldsymbol{x}_i\in \mathbb{R}^d$ standard Gaussian feature vectors, and $y_i\in\mathbb{R}$ response variables that depend on $\boldsymbol{x}_i$ through their projections on an unknown $k$-dimensional subspace. We use empirical risk minimization to learn a model that depends on an $m$-dimensional projection of the data (e.g., an $m$-neurons neural network). We propose an incremental approximate message passing (IAMP) algorithm and precisely characterize the training error it achieves, as well as the relation between test and training error, in the high dimensional asymptotics $n,d\to\infty$, with $n/d\toα\in (0, +\infty)$. Based on earlier work in related models, we expect that the performance achieved by our algorithm is optimal among polynomial-time algorithms.