Statistical Physics of Learning

Many real-world datasets exhibit unusually large values far more frequently than predicted by Gaussian models. Heavy-tailed distributions capture this behavior, yet evaluating learning performance under them remains challenging because rare, large feature entries retain non-vanishing effects even in high dimensions. Even in the canonical setting of empirical risk minimization for linear regression with entry-wise i.i.d. symmetric $α$-stable data, a precise asymptotic characterization of prediction has been lacking. In this work, we introduce a functional order parameter that describes the random effective problem associated with each coefficient. Using the replica method, we fully characterize the generalization error in the proportional high-dimensional limit where the sample size and feature dimension diverge at a fixed ratio. Additionally, this analysis establishes a heavy-tail universality law, scaling laws relating typical errors to prediction reliability, and the Bayes-optimal prediction error. In addition to characterizing the effects of extreme entries on the learning process, our method applies broadly to other systems with persistent local heterogeneity.

Auxiliary learning is an optimization paradigm in which a neural network's performance on a target task is improved by jointly training it on additional tasks. However, the mechanisms behind this improvement remain poorly understood. We study this problem using a teacher-student framework and derive a closed system of differential equations describing the dynamics of online stochastic gradient descent in the large-input limit. For linear networks, we obtain a closed-form expression for the generalization error to leading order in the learning rate, quantifying how task correlations and label noise determine the benefit of auxiliary learning. For non-linear activation functions, we develop a fluctuation-dissipation analytical theory that establishes a general relation linking the main and auxiliary errors to the corresponding single-task error. Numerical experiments support the theoretical predictions and show how auxiliary tasks improve generalization by balancing the forcing dynamics towards the optimal solution with gradient noise.

We trained 64 independently seeded networks in four configurations, continuing each to sustained convergence or a 40,000-epoch ceiling. We then asked whether an RSB-inspired distribution of pairwise weight overlaps changes across the grokking transition. It is the alignment step, not the overlap statistic, that determines what this registered probe can report. The registered implementation permutes hidden units without the corresponding bias and head-internal permutations and therefore does not preserve the network function. Every q_wt value computed through this alignment inherits the defect; q_fn does not, because it is computed from predictions of the unpermuted models. The numerical-precision requirement also failed, and an audit found protocol deviations. Consequently, the pre-registered rule gives no verdict: registered outcome UNDETERMINED (reason code C0_INSTRUMENT_INVALID). These data provide neither a confirmatory null nor a validated reading of the Parisi order parameter. Only frac40 cleared the 12/16 checkpoint-completeness requirement. For this configuration, a post-hoc criterion applied to the same data gave a Hartigan-dip interval containing zero (95% CI for Delta dip = [-0.017, 0.034]), whereas the overlap standard deviation increased by a factor of about 5.6. A post-hoc calibration assigns the dip test zero power at the simulated separations; the interval is therefore uninformative, not evidence of no change. The standard-deviation ratio is the only statistic here with power at the observed effect. Ensemble loss was near-flat only under the pre-specified 1% threshold. Finally, grokking rates of 0/16, 11/16 and 16/16 remain descriptive because train fraction is confounded with split identity.

The test error of a model plotted against its number of parameters $d$ falls, peaks when the model can just fit the training data, and falls again, exhibiting the double descent phenomenon. We explain the phenomenon with statistical mechanics. The training trajectory of a stochastic gradient-based method is a particle wandering over the energy landscape of the training loss at an induced temperature $T$, and a run that has equilibrated visits every parameter vector of a given training loss equally often, the fundamental postulate of statistical mechanics, with probability given by the Boltzmann distribution. Because training starts at an initial point and has only finite time to diffuse, it carries an effective weight decay, which makes every parameter a quadratic degree of freedom. The equipartition theorem then distributes the energy among the $d$ degrees of freedom in shares of $T/2$, so at a fixed training loss adding parameters lowers the temperature and drives the Boltzmann distribution toward the stationary path. Finally, adding parameters can only lower the $L^2$ norm of the stationary path, so a solution sampled at fixed loss is less likely to be large with increasing $d$, effectively increasing weight regularization.

In the kernel regime, neural-network learning inherits its preferences from a frozen spectrum. During feature learning, this spectrum evolves, yet networks retain systematic biases toward simple, smooth directions. We develop a function-space statistical framework explaining the origin of these preferences, treating functions and their learning operators as macroscopic variables, with parameterization entering through the multiplicity of parameter configurations realizing each function. For mean-squared loss, error relaxes exactly under the evolving learning operator $M=JJ^\ast$. Training stochasticity induces a Gaussian weight over function-space states, while parameter multiplicity contributes an entropic operator $B$, defined by the curvature of its log multiplicity. A local Laplace expansion yields the fluctuation free energy $Φ_{\mathrm{fluc}}(M;B)=\frac{σ_ξ^2}{2}\log\det(M^{-1}+B)+\mathrm{const}$, analogous to an Occam factor. Under mild statistical conditions, this free energy is rotationally stationary exactly when $[M,B]=0$, is minimized by pairing large eigenvalues of $M$ with small eigenvalues of $B$, and generates a local restoring force against mismatch. Learning is therefore biased toward faster relaxation along entropically cheaper directions. This preference strengthens with training noise and vanishes in the deterministic limit, beyond gradient-flow accounts of operator alignment. For ReLU networks, we relate entropic curvature to the minimal rearrangement of activation boundaries required for a functional change and bound this structural cost by directional smoothness. Consequently, smooth directions are preferentially learned faster, in a data-adaptive manner, even as the learning operator evolves.

This chapter reconstructs the Hopfield network as a physical theory of memory rather than merely an early neural-network algorithm. It begins with the problem as it stood before 1982-threshold logic, Hebbian association, correlation memories, and recurrent binary networks-and isolates what Hopfield's synthesis added: a dynamical definition of content-addressable memory, a symmetric recurrent architecture with a Lyapunov function, a Hebbian embedding of patterns in its couplings, and a physical account of basins, robustness, and graceful degradation. The binary and graded-response energy functions are derived in full, together with the signal-crosstalk decomposition governing pattern stability, the mean-field theory of retrieval at extensive load, and the zero-temperature retrieval spinodal at (alpha 0.138) established by Amit, Gutfreund, and Sompolinsky. The energy-based program is then followed through analog optimization networks, polynomial dense associative memories, exponential interactions, and modern continuous Hopfield updates, including the precise conditions under which the update becomes scaled dot-product attention. Throughout, capacity claims are tied to their disorder ensemble, scaling limit, and success criterion, showing why numerically different storage limits need not conflict. A closing assessment distinguishes established results from surviving principles, assumption-bound limitations, and open problems, treating the Hopfield network as an effective theory whose symmetry, locality, and point-neuron assumptions delimit its biological reach. Fixed-seed numerical experiments expose the mechanisms discussed but do not substitute for analytical results.

What a finite learning device has recorded and what will hold value for it on future tasks are not the same quantity. We develop a typed accounting for finite-state learning devices that separates four components: a training-side fit functional $Φ_{\mathrm{fit}}$, the record-correlation stock $J_{D}=I(M;D)$, an update-side search ledger $σ_{M}$, and an operational capital value $V(M;T,b)$. This value is the work gap between an informed protocol class and a blind class obtained by deleting the memory-read port and re-optimizing from scratch. (I) Separation: for every $n$, there is a device family on which record correlation and world correlation grow by $n\ln 2$ while the capital gain is exactly zero. In the $\mathrm{flat}^{*}$ regime, data-free updates never increase $V$. (II) Capitalization ledger: an exact $\mathrm{flat}^{*}$ extraction identity and a universal ledger identity give, for (F5$'$)-stable $M$-local updates under a no-discarded-record-correlation condition (f), the bound $η_{\mathrm{cap}}\le 1$ for the capitalization efficiency $η_{\mathrm{cap}}=ΔV/(k T\,σ_{M})$, together with necessary and sufficient conditions for equality. (III) Value retention: for the retention gap $L_{\mathrm{gen}}$ and retention ratio $ρ_{\mathrm{gen}}$ (the former carries no sign constraint; the latter is defined for positive training-side value and is not confined to $[0,1]$) we give a two-layer alignment domain: an exact exchange rate between value and the side-information-adjusted record fit $I(M';D\mid Y)$ without any record-side-information independence assumption, and a raw record-stock exchange rate under a joint side-information neutrality condition $(M,D)\perp Y$, whose boundary is marked by an explicit one-time-pad witness. These are statements about finite-device value retention under task-distribution shift, not a theory of statistical generalization.

A scientific theory of deep learning, comprising learning dynamics and statistical properties of learned models, is rapidly gaining attention. One of the corner stones of this development are analytically solvable toy models, allowing for the fully tractable analysis of the learning dynamics. Here we analytically investigate such a toy model using the regularization strength as a tunable external parameter - akin to external fields in statistical physics. In previous studies, (i) an onset of learning transition was predicted analytically and (ii) it was phenomenologically/numerically established that tuning the regularization strength can result in a cascade of phase transitions. The number of those transitions was linked to the geometry of the loss landscape determined by the model complexity. Setting up a rigorous framework underpinning the previous numerical observations, our investigation reveals a precise connection between those cascades of phase transitions, learnable features and the underlying geometry. We provide analytic predictions of these phase transitions as well as tractable order parameters related to learned features. At the level of the minimal model, we connect this macroscopic perspective (that can be condensed into an effective description) to the microscopic perspective in terms of the geometry of the loss landscape characterized by the Hessian spectrum. Thus, the presented model provides a platform to explore and sharpen advances made in the scientific theory of deep learning rooted in statistical physics concepts.

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

Normally the statistical mechanics of learning treats constraints on weight distributions as restrictions that shrink the space of possible solutions. Therefore, it reduces model capacity. In this paper we would like to take a contrary approach, which, however, is based on the earlier work on distribution-constrained perceptrons. Rather than treating a prescribed weight distribution as a mere restriction, we propose that it defines the intrinsic geometry upon which learning naturally unfolds. We formulate both deep neural networks and variational quantum circuits as gradient flows on a product of Wasserstein manifolds -- one classical Wasserstein space for each layer and one quantum Wasserstein space for the circuit parameters. Within this geometry, the capacity reduction, which was previously associated with distributional constraints, appears as the metric structure of the constraint manifold itself. We develop a hierarchical mean-field description for deep networks, extend the framework to the quantum setting using the quantum Wasserstein distance of order 1, and introduce two such practical algorithms, Hierarchical DisCo-SGD and Quantum DisCo, that follow approximate geodesics on the manifold of the product itself. Experiments on teacher-student problems, standard image classification tasks, and small variational quantum classifiers show that respecting these distributional geometries improves generalization, stabilizes training, and reduces the severity of barren plateaus compared with unconstrained and purely norm-based baselines. This approach firstly reframes structural constraints as geometric priors and suggests a route for incorporating biological, spectral, or hardware-derived distributional information into both learning systems, viz., classical and quantum learning.

The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. This phenomenon is all the more surprising given that generalization error diverges when the number of model parameters approaches a critical value from below. Here we use dynamical mean field theory to show that this so-called "double descent" behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. The corresponding response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately.

Artificial neural networks (NNs) and machine learning (ML) algorithms are poorly understood from a theoretical perspective, which makes it difficult to fully realize their potential and overcome their weaknesses. For instance, ML algorithms train NN weights by moving them along a low-dimensional subspace of their allowed values, but this implicitly low-dimensional learning structure is not properly exploited to improve training because its nature is not well understood. Moreover, trained NNs are easily confused by pervasive adversarial attacks whose theoretical underpinnings are still unclear. This thesis aims to improve our theoretical understanding of NNs and ML, with a particular focus on adversarial attacks and implicitly low-dimensional learning. For this purpose, we use mathematical tools from statistical mechanics to study different types of NNs and ways in which they can fit the data. In particular, we study two classes of models that fit the data with various degrees of learning and memorization: dense associative memory (DAM) and restricted Boltzmann machines (RBM). In the process, we investigate connections between different versions of these models that are useful to make analytical investigations more efficient.

Why overparameterised deep networks generalise so remarkably well remains one of the most stubborn open questions in machine learning theory. Classical frameworks like VC dimension and Rademacher complexity predict catastrophic overfitting in modern models, leaving a massive theoretical gap between theory and reality. In this paper, we bridge this divide by introducing a unified framework that links information theory, topology, and statistical mechanics to map the hard limits of deep learning. Central to our approach is the Entropic Learnability Horizon (ELH): a fundamental law stating that a network can only truly learn a target function if the Shannon entropy of the data manifold outpaces the topological entropy of the function's decision boundary, balanced by the von Neumann entropy of the network's weight space. We establish the Shannon-Topological Bottleneck Theorem, proving that when a target boundary's geometric complexity exceeds this informational horizon, the system undergoes a sudden entropic phase transition. It falls into a state of Informational Frustration - a glassy, rigid memorization phase where generalization becomes thermodynamically impossible. Using this lens, we show that the enigmatic phenomenon of "grokking" is actually an Entropic Release, where weights abruptly reorganise to unlock the bottleneck. Finally, we translate this theory into practice with Entropic Gradient Descent (EGD), an optimization algorithm that dynamically manages weight entropy to keep learning on track. Ultimately, this work repositions entropy not just as a tool for tracking uncertainty but as the fundamental physical currency that dictates whether a machine can learn.

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.

Deep neural networks (DNNs) exhibit first order phase transitions under variations of the L2 regularization strength, with each transition marking the onset of a new learnable feature. Below a critical regularization strength, all features are in principle learnable, but coexisting metastable states, separated by energy barriers, can trap the network and impede convergence. A strength of DNNs is their ability to generalize. But many open questions remain, among them the origin of so called grokking: the abrupt, delayed onset of generalization after prolonged apparent overfitting. We show for linear DNNs that grokking is consistent with hysteresis in first-order L2 phase transitions: using L2 regularization to engineer deliberate trapping, we demonstrate that a model in a low-accuracy metastable state escapes only when SGD noise drives it across an energy barrier, with escape times following Arrhenius scaling. We reproduce grokking-like delayed convergence across two orders of magnitude in escape time by deliberately trapping models in metastable phases. Using sparse sub-sampling we also reproduce the canonical grokking curve where test error eventually approaches the final training error. Our work suggests that the number of metastable states equals the number of learnable features -- one per singular value of the data covariance -- the potential for hysteresis grows naturally with task complexity. We provide evidence that the same mechanism likely operates in general nonlinear DNNs. Our results provide routes toward more efficient learning schemes.

We introduce a theoretical framework for analyzing federated learning in a generative setting through a teacher-multiple interacting students scenario, in which each student receives a distinct realization of the data, either through a different noise corruption or by accessing a different subset, possibly of varying size. Using theoretical tools in equilibrium disordered system, we analytically show that interactions among students systematically enhance learning performance: highly noisy students require fewer samples to recover the underlying pattern, while low-noise students achieve a larger overlap with the ground-truth signal. We derive the optimal Bayesian conditions for teacher recovery as functions of the sample complexity, noise level, and interaction strength, and validate these predictions through numerical simulations. The resulting dynamics can be mapped onto equilibrium sampling in a Restricted Boltzmann Machine with a structured hidden layer, providing a principled theoretical understanding of how interactions improve distributed generative modeling.

Hard-label classification is usually trained with smooth surrogate losses, most prominently softmax cross-entropy. We isolate an asymptotic mechanism by which this mismatch between smooth surrogate and discrete labels produces power-law learning curves in an online teacher-student model. After subtracting the mean logit, the thermodynamic-limit dynamics close in centered variables: a growing centered student-teacher alignment $D$ and the residual student variance $Δ$. At late times, examples away from teacher decision boundaries are already classified confidently and contribute exponentially little. Only boundary layers of width $O(D^{-1})$ remain active, while the noise of fixed-learning-rate online gradient descent maintains a nonzero $Δ$. As a function of the training time $α$ the late-time solution yields a $α^{-1/3}$ power law not only for the test loss but also for the generalization error $ε_g$, i.e., one minus test accuracy. This is much slower than the $α^{-1}$ Bayes-optimal reference for the same model. We further show that learning-rate schedules can improve the generalization error towards a $ε_g \sim α^{-1/2}$ power law. Simulations support the predicted order parameter dynamics and learning curves. Controlled experiments with correlated Gaussian inputs and whitened pretrained features show that data structure can dominate transients. Therefore, our result is an asymptotic, complementary mechanism rather than an alternative to spectral explanations of neural scaling laws.

Energy-based models (EBMs) are flexible generative architectures inspired by statistical physics, but their learning and generative properties remain poorly understood. Here, we analyze a solvable EBM in the high-dimensional limit: the spherical Boltzmann machine (SBM). Combining tools from random matrix theory and dynamical mean-field theory, we: solve exact equations describing the training dynamics of the SBM; compute the Bayesian evidence, which acts as a partition function in parameter space and encodes global properties of the trained model; and uncover cascades of phase transitions that occur both during training and as a function of hyperparameters, related to successive alignment and condensation of the top modes of the coupling matrix to the data. We connect these transitions to sampling-time generative phenomena in a teacher-student scenario, including: sampling temperature tuning, double descent as a function of regularization strength, tempered posterior effects, and out-of-equilibrium effects during training that induce biases in the trained model. We provide numerical evidence demonstrating that all these phenomena appear in standard generative architectures, beyond the SBM.

These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable $φ$ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of $φ$ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of $nβ$) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape.

Understanding the training dynamics of deep neural networks remains a major open problem, with physics-inspired approaches offering promising insights. Building on this perspective, we develop a thermodynamic framework to describe the stationary distributions of stochastic gradient descent (SGD) with weight decay for scale-invariant neural networks, a setting that both reflects practical architectures with normalization layers and permits theoretical analysis. We establish analogies between training hyperparameters (e.g., learning rate, weight decay) and thermodynamic variables such as temperature, pressure, and volume. Starting with a simplified isotropic noise model, we uncover a close correspondence between SGD dynamics and ideal gas behavior, validated through theory and simulation. Extending to training of neural networks, we show that key predictions of the framework, including the behavior of stationary entropy, align closely with experimental observations. This framework provides a principled foundation for interpreting training dynamics and may guide future work on hyperparameter tuning and the design of learning rate schedulers.

Learning is a complex dynamical process shaped by a range of interconnected decisions. Careful design of hyperparameter schedules for artificial neural networks or efficient allocation of cognitive resources by biological learners can dramatically affect performance. Yet, theoretical understanding of optimal learning strategies remains sparse, especially due to the intricate interplay between evolving metaparameters and nonlinear learning dynamics. The search for optimal protocols is further hindered by the high dimensionality of the learning space, often resulting in predominantly heuristic, difficult to interpret, and computationally demanding solutions. Here, we combine statistical physics with control theory in a unified theoretical framework to identify optimal learning protocols in prototypical neural network models. In the high-dimensional limit, we derive closed-form ordinary differential equations that track online stochastic gradient descent through low-dimensional order parameters. We formulate the design of learning protocols as an optimal control problem directly on the dynamics of the order parameters with the goal of minimizing the generalization error. This formulation encompasses a variety of learning scenarios, optimization constraints, and control budgets. We apply it to representative cases, including optimal curricula, adaptive dropout regularization and noise schedules in denoising autoencoders. We find nontrivial yet interpretable strategies highlighting how optimal protocols mediate learning trade-offs. Our results establish a principled foundation for understanding and designing optimal protocols and suggest a path toward a theory of meta-learning grounded in statistical physics.