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Generalization in Learning: Eyring Formula and Predator-prey Model
S. V. Kozyrev, I. A. Lopatin, Alexander Pechen
TL;DR
The paper asks how physical and biological models can explain generalization, overfitting control, and delayed generalization. It applies Langevin and predator–prey interpretations, and reports better approximation than gradient descent for one noisy-data function-recovery problem.
Problem
The paper investigates how physical and biological approaches can explain generalization and control overfitting, including the delayed-generalization phenomenon of grokking.
Source: §1 · pp. 1–2Method
It interprets stochastic gradient descent through Langevin dynamics and GAN learning through a predator–prey model, and analyzes grokking using thermodynamic arguments.
Source: §1 · p. 1, §4 · p. 5, §6 · pp. 10–11Results
For noisy-data recovery of an unknown function by a fixed-degree polynomial, the predator–prey model provides better approximation than gradient descent.
Source: §1 · p. 2Takeaways & Limitations
The paper presents wide-minimum selection and predator-driven displacement from narrow minima as explanations for improved generalization within its models and numerical example.
Source: §6 · pp. 10–11The grokking account assumes that zero-risk-region entropy decreases linearly with sample size and that the zero-risk manifold has sufficiently high entropy to contain the desired algorithmic solution.
Source: §3 · pp. 4–5
Abstract
from arXiv · showhide
In this work, generalization in machine learning is investigated using a predator-prey model. We overview biological and chemical interpretation of the stochastic gradient Langevin dynamics and generative adversarial network. For a particular example of recovering an unknown function by a polynomial of fixed degree from a set of noisy data, the predator-prey model provides better approximation compared to the gradient descent method. Further, thermodynamical arguments, in particular Eyring formula, are used to explain grokking (delayed generalization) phenomenon in machine learning.
1 Introduction
The paper uses physical and biological interpretations of stochastic gradient learning and GANs to study overfitting control, and applies thermodynamic arguments to delayed generalization. For noisy-data function recovery, its predator–prey model gives better approximation than gradient descent.
1 Introduction
- Stochastic gradient descent is treated as stochastic gradient Langevin dynamics in a risk potential, allowing thermodynamic arguments to explain selection of wide minima and improved generalization.The Eyring formula is used to explain why stochastic gradient optimization favors wider minima.Source: §1 · pp. 1–2
- In the GAN interpretation, the generator and discriminator act as prey and predator, with the predator pushing the prey out of narrow minima.Source: §1 · p. 2
- The paper explains grokking, or delayed generalization, using statistical-physics ideas: fast memorization is modeled as Brownian motion in a potential, followed by motion with zero potential.Source: §1 · p. 2
- For fixed-degree polynomial recovery of an unknown function from noisy data, the predator–prey model provides better approximation than gradient descent.Source: §1 · p. 2
2 Stochastic gradient descent and generalization
The section formulates learning as empirical-risk minimization and describes stochastic gradient descent through Langevin dynamics. It then uses the Eyring formula to connect noise-driven transitions among minima with the selection of wider, more stable solutions.
2.1 Stochastic gradient Langevin dynamics
- Learning is framed as minimizing empirical risk over a sample, with a nonnegative loss and a hypothesis parameter in R^m.Source: §2.1 · p. 2
- Gradient descent follows the negative risk gradient, while stochastic gradient Langevin dynamics adds small Gaussian noise and has a continuous-time diffusion description.The dynamics use Brownian motion with temperature θ; the sample variable may be omitted when it is not important.Source: §2.1 · p. 2
- The diffusion equation has a stationary Gibbs distribution proportional to e^-βf, where β is inverse temperature.Source: §2.1 · p. 3
2.2 Generalization and Eyring formula
- Overfitting is associated with low training risk but high validation risk, and the section relates generalization to wide, flat minima and algorithmic stability.Source: §2.2 · p. 3
- The Eyring formula describes noise-driven transition rates between potential wells in terms of the free-energy difference between the transition state and initial state.Free energy is defined as F = E − θS, with E the energy and S the entropy.Source: §2.2 · p. 3
- Higher-entropy, wider wells have lower transition rates and free energy, so stochastic gradient Langevin dynamics more effectively captures them, favoring algorithmically stable solutions.Source: §2.2 · p. 3
3 Grokking, thermodynamics and Eyring formula
The section models grokking as a two-phase process: training first memorizes the sample, then moves along a zero-risk manifold toward a high-entropy region. An Eyring-formula argument approximates how grokking time depends on sample size.
3 Grokking, thermodynamics and Eyring formula
- In the cited grokking experiments, networks first memorized training data while overfitting, then generalized after continued training; grokking time increased as sample size decreased.The experiments included modular arithmetic and other structured tasks, with overparameterized networks.Source: §3 · p. 4
- The proposed account separates fast memorization, driven toward the zero-risk manifold, from slower Brownian motion along that manifold toward a high-entropy region containing the algorithmic solution.The zero-risk manifold shrinks as additional sample constraints are imposed.Source: §3 · pp. 4–5
- The model assumes that zero-risk-region entropy decreases linearly as sample size grows, corresponding to an exponential decrease in region volume under successive constraints.A multidimensional-cube example illustrates how each added constraint can remove a similar fraction of the region's volume.Source: §3 · p. 5
- Eyring’s formula approximates reciprocal grokking time as a function of sample size, with the initial-region free energy growing linearly and transition-state free energy changing weakly as the sample grows.The argument focuses on entropic contributions to free energy and relates increasing initial-state entropy to exponentially longer transition time.Source: §3 · p. 5
4 The GAN Model and generalization
The GAN is formulated as a minimax problem, with stochastic gradient Langevin dynamics describing parameter updates. The generator–discriminator interaction is interpreted as a predator–prey model that may reduce overfitting by pushing the discriminator away from narrow maxima.
4 The GAN Model and generalization
- The GAN objective is minimax, with y minimized and x maximized; the discriminator and generator are parameterized probability distributions over Z.x and y are the respective hypotheses for the discriminator and generator.Source: §4 · p. 5
- Stochastic gradient Langevin dynamics reduces the GAN optimization problem to SDEs for random walks in discriminator and generator parameter spaces.The associated Wiener processes act on those parameter spaces, and the SDEs use derivatives of V with respect to its two arguments.Source: §4 · p. 6
- For one-dimensional x and y with V = ωxy, the zero-noise system’s solution oscillates around the minimax.The passage relates these oscillations to prior discussions of gradient-descent GAN optimization [8] [9] [10].Source: §4 · p. 6
- The generator may suppress overfitting by pushing the discriminator out of narrow maxima, an interaction interpreted as predator–prey dynamics.The discriminator tends toward local maxima of V1(x), while the generator follows it.Source: §4 · p. 6
5 Reduction of overfitting for function recovery with a predator-prey model
In noisy function recovery with fixed-degree polynomials, the predator–prey procedure lowers test loss and hidden-function deviation relative to gradient descent. The section interprets its behavior through oscillations among nearby minima and reports that improvement requires sufficiently many iterations.
5 Reduction of overfitting for function recovery with a predator-prey model
- For polynomial degrees d = 1, 3, 5, and 7, d = 1 underfits, d = 3 learns normally, and d = 5 and 7 overfit the noisy data.The experiment approximates a hidden function from noisy data using fixed-degree polynomials.Source: §5 · pp. 6–7
- MSE on the test sample fell from 0.161 to 0.106, while deviation from the hidden function fell from 0.0594 to 0.0334 with the predator–prey method.The comparison selects the best model by minimum test loss over the last N iterations for ordinary gradient descent and predator–prey training.Source: Figure 3 · p. 9, §5 · p. 9, Figure 4 · p. 10
- The simulations show oscillations between nearby minima, whose test losses can differ by 1.5–2 times, allowing the method to search among extrema and select better ones.The authors describe these regions as wide lacunas containing several closely located minima.Source: §5 · pp. 9–10
- The improvement is significant only after sufficiently many iterations, and the reported overfitting reduction can reach 1.5–2 times.The authors suggest that oversized models may first fit general patterns and later fit sample-specific noise.Source: §5 · p. 10
6 Conclusion
The conclusion presents wide-minimum selection as an explanation for improved generalization in stochastic gradient descent. It also connects thermodynamic arguments to grokking and describes a predator–prey model whose predator pushes prey out of narrow maxima.
6 Conclusion
- Stochastic gradient descent, viewed as Brownian motion in a potential, explains selection of wide minima and improved generalization.Source: §6 · p. 10
- Thermodynamic methods are used to explain grokking, or delayed generalization.Source: §6 · pp. 10–11
- The predator–prey model improves generalization by pushing prey out of narrow maxima, a claim supported by a numerical simulation for a simple learning problem.The conclusion presents this simulation as justification for the claim.Source: §6 · p. 11