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A Theoretical Framework For Overfitting In Energy-based Modeling
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Analyzes overfitting in pairwise energy-based models via eigenmode training dynamics of a Gaussian EBM (with RMT corrections, extension to Ising Boltzmann machines and a score-matching NTK view).
Problem
Overfitting with limited data when training pairwise energy-based models for inverse problems (inferring coupling matrices); lack of practical indicators such as early-stopping points in unsupervised/generative settings.
Approach
Gradient-ascent training of a Gaussian EBM decouples into independent eigenmode dynamics aligned with the empirical covariance spectrum, giving learning timescales tied to the covariance eigenvalues; finite-data corrections are obtained via asymptotic random matrix theory, yielding a GCV-like relation and shrinkage/polynomial-fit protocols that mitigate overfitting.
Key results
- Optimal early-stopping time for coupling reconstruction does not coincide with the time of maximum test log-likelihood, and the generation error stabilizes before the reconstruction-error minimum.
- For L1-regularized GEBM, the test energy is deterministically related to train energy via E_test = (1 - E_train/rho)^-1 E_train, a GCV-like counterpart for energy-based models.
- Polynomial fit of eigenvalues from downsampled data (linear in 1/m) and RMT shrinkage give reconstruction errors comparable to optimal L2 regularization, scaling as 1/sqrt(rho) at large rho.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| train with the population covariance instead of a finite-sample covariance | Gaussian EBM · Synthetic Gaussian data | coupling reconstruction error | Helped |
| stop at the minimum reconstruction-error time instead of training to convergence | Gaussian EBM · Synthetic finite-sample data | coupling reconstruction error | Helped |
| use optimal L2 regularization instead of an unregularized fixed point | Gaussian EBM · Synthetic finite-sample data | coupling reconstruction error | Helped |
| correct empirical covariance eigenvalues with the optimal rotationally invariant estimator | Gaussian EBM · Synthetic finite-sample data | coupling reconstruction error | Helped |
| extrapolate covariance eigenvalues from downsampled datasets using a linear fit in 1/m | Gaussian EBM · Synthetic finite-sample data | coupling reconstruction error | Helped |
| replace empirical eigenvectors with population eigenvectors while retaining sample eigenvalues | Gaussian EBM · Synthetic finite-sample data | coupling reconstruction error | Helped |
| use optimal L2 regularization instead of an unregularized fixed point | Boltzmann Machine · 2D Ising inverse problem | coupling reconstruction error | Helped |
| correct empirical eigenvalues with the downsampling-based polynomial fit | Boltzmann Machine · 2D Ising inverse problem | coupling reconstruction error | Helped |
| use small initial eigenvalues rather than initial values above the fixed point (ablation) | Gaussian EBM · Synthetic finite-sample data | coupling reconstruction error | Mixed |
| enforce symmetry after each parameter update rather than use nonsymmetric perturbations (ablation) | Gaussian EBM · Training dynamics | eigenvalue evolution | No clear effect |
| train with finite-sample covariance eigenvalues instead of population eigenvalues (ablation) | Gaussian EBM · Synthetic finite-sample data | coupling reconstruction error | Hurt |
| train using the empirical covariance instead of the population covariance (ablation) | Gaussian EBM · Synthetic data | generation error | No clear effect |
Surprising findings
- Generation quality (generation error) stabilizes well before the reconstruction-error minimum, so it fails to reflect the deterioration of coupling parameters over training.
- In Ising Boltzmann machines, the weakest covariance modes map to negative coupling eigenvalues with large magnitude, and learning them dominates reconstruction error despite having little effect on sampling quality.
Limitations
- Exact analysis only for the Gaussian model; the Ising Boltzmann machine treatment uses a mean-field approximation at high temperature and drops the diagonal constraint.
- The NTK extension to generic EBMs is theoretical and not validated experimentally.
- Shrinkage (RIE) corrections are specific to the Gaussian model and not applicable to Boltzmann machines.
Experimental setup
Datasets: MNIST CIFAR-10 Human Genome Dataset synthetic Gaussian samples 2D Ising model equilibrium configurations
Metrics: reconstruction error (coupling error) test log-likelihood generation error Wasserstein distance
Baselines: L2 regularization rotationally invariant estimator shrinkage early stopping at test log-likelihood maximum
Builds on: Boltzmann machine / inverse Ising rotationally invariant estimators neural tangent kernel score matching
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