Neural Information Processing Systems http://nips.cc/

We study the free energy for pure and mixed spherical $p$-spin models with i.i.d.\ disorder. In the mixed case, each $p$-interaction layer is assumed either to have regularly varying tails with exponent $α_p$ or to satisfy a finite $2p$-th moment condition. For the pure spherical $p$-spin model with regularly varying disorder of tail index $α$, we introduce a tail-adapted normalization that interpolates between the classical Gaussian scaling and the extreme-value scale, and we prove a sharp universality dichotomy for the quenched free energy. In the subcritical regime $α<2p$, the thermodynamics is driven by finitely many extremal couplings and the free energy converges to a non-degenerate random limit described by the NIM (non-intersecting monomial) model, depending only on extreme-order statistics. At the critical exponent $α=2p$, we obtain a random one-dimensional TAP-type variational formula capturing the coexistence of an extremal spike and a universal Gaussian bulk on spherical slices. In the supercritical regime $α>2p$ (more generally, under a finite $2p$-th moment assumption), the free energy is universal and agrees with the deterministic Crisanti--Sommers/Parisi value of the corresponding Gaussian model, as established in [Sawhney-Sellke'24]. We then extend the subcritical and critical results to mixed spherical models in which each $p$-layer is either heavy-tailed with $α_p\le 2p$ or has finite $2p$-th moment. In particular, we derive a TAP-type variational representation for the mixed model, yielding a unified universality classification of the quenched free energy across tail exponents and mixtures.

Federated Learning (FL) is a distributed machine learning setting that requires multiple clients to collaborate on training a model while maintaining data privacy. The unaddressed inherent sparsity in data and models often results in overly dense models and poor generalizability under data and client participation heterogeneity. We propose FL with an L0 constraint on the density of non-zero parameters, achieved through a reparameterization using probabilistic gates and their continuous relaxation: originally proposed for sparsity in centralized machine learning. We show that the objective for L0 constrained stochastic minimization naturally arises from an entropy maximization problem of the stochastic gates and propose an algorithm based on federated stochastic gradient descent for distributed learning. We demonstrate that the target density (rho) of parameters can be achieved in FL, under data and client participation heterogeneity, with minimal loss in statistical performance for linear and non-linear models: Linear regression (LR), Logistic regression (LG), Softmax multi-class classification (MC), Multi-label classification with logistic units (MLC), Convolution Neural Network (CNN) for multi-class classification (MC). We compare the results with a magnitude pruning-based thresholding algorithm for sparsity in FL. Experiments on synthetic data with target density down to rho = 0.05 and publicly available RCV1, MNIST, and EMNIST datasets with target density down to rho = 0.005 demonstrate that our approach is communication-efficient and consistently better in statistical performance.

Singular learning theory (SLT) \citep{watanabe2009algebraic,watanabe2018mathematical} provides a rigorous asymptotic framework for Bayesian models with non-identifiable parameterizations, yet the statistical meaning of its second-order invariant, the \emph{singular fluctuation}, has remained unclear. In this work, we show that singular fluctuation admits a precise and natural interpretation as a \emph{specific heat}: the second derivative of the Bayesian free energy with respect to temperature. Equivalently, it measures the posterior variance of the log-likelihood observable under the tempered Gibbs posterior. We further introduce a collection of related thermodynamic quantities, including entropy flow, prior susceptibility, and cross-susceptibility, that together provide a detailed geometric diagnosis of singular posterior structure. Through extensive numerical experiments spanning discrete symmetries, boundary singularities, continuous gauge freedoms, and piecewise (ReLU) models, we demonstrate that these thermodynamic signatures cleanly distinguish singularity types, exhibit stable finite-sample behavior, and reveal phase-transition--like phenomena as temperature varies. We also show empirically that the widely used WAIC estimator \citep{watanabe2010asymptotic, watanabe2013widely} is exactly twice the thermodynamic specific heat at unit temperature, clarifying its robustness in singular models.Our results establish a concrete bridge between singular learning theory and statistical mechanics, providing both theoretical insight and practical diagnostics for modern Bayesian models.

This technical note considers the sampling of outcomes that provide the greatest amount of information about the structure of underlying world models. This generalisation furnishes a principled approach to structure learning under a plausible set of generative models or hypotheses. In active inference, policies - i.e., combinations of actions - are selected based on their expected free energy, which comprises expected information gain and value. Information gain corresponds to the KL divergence between predictive posteriors with, and without, the consequences of action. Posteriors over models can be evaluated quickly and efficiently using Bayesian Model Reduction, based upon accumulated posterior beliefs about model parameters. The ensuing information gain can then be used to select actions that disambiguate among alternative models, in the spirit of optimal experimental design. We illustrate this kind of active selection or reasoning using partially observed discrete models; namely, a 'three-ball' paradigm used previously to describe artificial insight and 'aha moments' via (synthetic) introspection or sleep. We focus on the sample efficiency afforded by seeking outcomes that resolve the greatest uncertainty about the world model, under which outcomes are generated.

Traditional neural networks, while powerful, rely on biologically implausible learning mechanisms such as global backpropagation. This paper introduces the Structurally Adaptive Predictive Inference Network (SAPIN), a novel computational model inspired by the principles of active inference and the morphological plasticity observed in biological neural cultures. SAPIN operates on a 2D grid where processing units, or cells, learn by minimizing local prediction errors. The model features two primary, concurrent learning mechanisms: a local, Hebbian-like synaptic plasticity rule based on the temporal difference between a cell's actual activation and its learned expectation, and a structural plasticity mechanism where cells physically migrate across the grid to optimize their information-receptive fields. This dual approach allows the network to learn both how to process information (synaptic weights) and also where to position its computational resources (network topology). We validated the SAPIN model on the classic Cart Pole reinforcement learning benchmark. Our results demonstrate that the architecture can successfully solve the CartPole task, achieving robust performance. The network's intrinsic drive to minimize prediction error and maintain homeostasis was sufficient to discover a stable balancing policy. We also found that while continual learning led to instability, locking the network's parameters after achieving success resulted in a stable policy. When evaluated for 100 episodes post-locking (repeated over 100 successful agents), the locked networks maintained an average 82% success rate.

This paper proposes an Interactive Inference Behavior Tree (IIBT) framework that integrates behavior trees (BTs) with active inference under the free energy principle for distributed multi-robot decision-making. The proposed IIBT node extends conventional BTs with probabilistic reasoning, enabling online joint planning and execution across multiple robots. It remains fully compatible with standard BT architectures, allowing seamless integration into existing multi-robot control systems. Within this framework, multi-robot cooperation is formulated as a free-energy minimization process, where each robot dynamically updates its preference matrix based on perceptual inputs and peer intentions, thereby achieving adaptive coordination in partially observable and dynamic environments. The proposed approach is validated through both simulation and real-world experiments, including a multi-robot maze navigation and a collaborative manipulation task, compared against traditional BTs(https://youtu.be/KX_oT3IDTf4). Experimental results demonstrate that the IIBT framework reduces BT node complexity by over 70%, while maintaining robust, interpretable, and adaptive cooperative behavior under environmental uncertainty.

Active inference proposes expected free energy as an objective for planning and decision-making to adequately balance exploitative and explorative drives in learning agents. The exploitative drive, or what an agent wants to achieve, is formalised as the Kullback-Leibler divergence between a variational probability distribution, updated at each inference step, and a preference probability distribution that indicates what states or observations are more likely for the agent, hence determining the agent's goal in a certain environment. In the literature, the questions of how the preference distribution should be specified and of how a certain specification impacts inference and learning in an active inference agent have been given hardly any attention. In this work, we consider four possible ways of defining the preference distribution, either providing the agents with hard or soft goals and either involving or not goal shaping (i.e., intermediate goals). We compare the performances of four agents, each given one of the possible preference distributions, in a grid world navigation task. Our results show that goal shaping enables the best performance overall (i.e., it promotes exploitation) while sacrificing learning about the environment's transition dynamics (i.e., it hampers exploration).

We investigate the phase diagram and memory retrieval capabilities of bipartite energy-based neural networks, namely Restricted Boltzmann Machines (RBMs), as a function of the prior distribution imposed on their hidden units - including binary, multi-state, and ReLU-like activations. Drawing connections to the Hopfield model and employing analytical tools from statistical physics of disordered systems, we explore how the architectural choices and activation functions shape the thermodynamic properties of these models. Our analysis reveals that standard RBMs with binary hidden nodes and extensive connectivity suffer from reduced critical capacity, limiting their effectiveness as associative memories. To address this, we examine several modifications, such as introducing local biases and adopting richer hidden unit priors. These adjustments restore ordered retrieval phases and markedly improve recall performance, even at finite temperatures. Our theoretical findings, supported by finite-size Monte Carlo simulations, highlight the importance of hidden unit design in enhancing the expressive power of RBMs.