🤖 AI Summary
Traditional empirical risk minimization focuses solely on a single optimal solution, failing to capture the multistability and uncertainty inherent in data-driven learning. This work reframes the empirical loss function as an interaction potential and constructs an energy-based model grounded in Gibbs measures on a Cayley tree, thereby establishing—for the first time—a rigorous connection between loss landscapes and probabilistic inference on tree-structured graphs. By leveraging nonlinear integral fixed-point equations, data-dependent kernels inducing compact operators, and phase transition analysis, the study theoretically proves existence and uniqueness of solutions in the one-dimensional setting. Numerical experiments further demonstrate the coexistence of multiple solution branches under non-separable kernels, revealing that data can induce multiple learning states and associated phase transitions.
📝 Abstract
We introduce a data-driven probabilistic framework for learning systems based on Gibbs measures on hierarchical structures. Unlike standard empirical risk minimization, where a dataset is used to identify a single optimal parameter, our approach transforms the empirical loss function into an interaction potential defining an energy-based model. The resulting Gibbs distribution describes a family of equilibrium learning states generated by the data.
We formulate the consistency conditions of the associated finite-volume distributions and derive nonlinear integral fixed-point equations whose solutions characterize the admissible learning states. These equations provide a rigorous connection between empirical loss landscapes and probabilistic inference on trees. For translation-invariant solutions, the problem reduces to the analysis of positive compact operators induced by data-dependent kernels, allowing us to establish existence and uniqueness conditions in the one-dimensional setting.
Furthermore, we show that hierarchical learning systems may exhibit phase-transition phenomena: for certain empirical kernels on Cayley trees, multiple Gibbs measures emerge beyond a critical inverse temperature, corresponding to distinct equilibrium prediction regimes. Numerical experiments with non-separable kernels illustrate the appearance of multiple solution branches and demonstrate the coexistence of several data-induced learning states.
Our results provide a new perspective on energy-based learning, where data do not merely determine an optimal model through minimization but define an entire probabilistic landscape of possible inference states.