Deep neural networks exhibit a simplicity bias, a well-documented tendency to favor simple functions over complex ones. In this work, we cast new light on this phenomenon through the lens of the Minimum Description Length principle, formalizing supervised learning as a problem of optimal two-part lossless compression. Our theory explains how simplicity bias governs feature selection in neural networks through a fundamental trade-off between model complexity (the cost of describing the hypothesis) and predictive power (the cost of describing the data). Our framework predicts that as the amount of available training data increases, learners transition through qualitatively different features -- from simple spurious shortcuts to complex features -- only when the reduction in data encoding cost justifies the increased model complexity. Consequently, we identify distinct data regimes where increasing data promotes robustness by ruling out trivial shortcuts, and conversely, regimes where limiting data can act as a form of complexity-based regularization, preventing the learning of unreliable complex environmental cues. We validate our theory on a semi-synthetic benchmark showing that the feature selection of neural networks follows the same trajectory of solutions as optimal two-part compressors.
Figures & tables
Figure 1: (a) In a supervised setting, some features causally relate to the label (e.g., phenotypical characteristics), while others are spurious (e.g., background of the image). A Bayesian solution leverages all available features. (b) Total expected compression cost as a function of training dataset size N . Each line represents the two-part description length of a candidate model. The MDL-optimal learner selects the model achieving the lowest total cost at each N , inducing transitions between qualitatively different solutions as the amount of data increases.
Figure 2: Comparison of the compression view and the learning view for Scenario A and B. (a) Compression envelopes on a log-log scale: each line shows the total two-part description length of a candidate predictor as a function of dataset size N . (b) Accuracy on held-out datasets as a function of N : solid lines are empirical measurements, dashed lines are reported metrics of the MDL-optimal solution. (c) Permutation feature importance (accuracy drop after shuffling each feature): a higher drop indicates stronger reliance on that feature.
Figure 3: Empirical vs. theoretical transition point N for varying task characteristics. Each point is one experimental configuration; the dashed line is the identity Ntheory=Nempirical . (a) Reducing the predictiveness of a feature increases its compression rate disadvantage, causing the transition to occur earlier. (b) Increasing the complexity of a feature inflates its description cost, causing the transition to occur earlier. In both cases, the proximity to the diagonal indicates that our MDL-based theory accurately predicts the observed feature switch as measured by our proxy.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure C.1 : Example samples from the EMNIST-based benchmark we curated, featuring the three types of features we presented in Section 2.3 : the digit shape (robust feature), the color (simple spurious feature) and the watermark (complex environmental-cue). The label is determined by the digit value, while the color and watermark are correlated with the label in an environment-dependent way.
Parameter
Symbol
Description
Values
Environment & Spurious Correlation
flip_prob
pflip
Label noise probability
{0,0.01,…,0.1}
spur_prob
pe
Relative proportion of environments
{0,0.02,…,0.2} or 0.5
uninformative_majority
—
Randomize color in majority env
True , False
Watermark Configuration
watermark_bank_size
K
Distinct patterns per environment
1 to 100 (log-spaced)
Appendix
Table E.1 : Hyperparameters for the Watermarked CMNIST Experiments
We address the fundamental question of why deep neural networks generalize by establishing a pointwise generalization theory for fully connected networks. This framework resolves long-standing barriers to characterizing the rich nonlinear feature-learning regime and builds a new statistical foundation for representation learning. For each trained model, we characterize the hypothesis via a pointwise Riemannian Dimension, derived from the eigenvalues of the learned feature representations across layers. This establishes a principled framework for deriving hypothesis-dependent, representation-aware generalization bounds. These bounds offer a systematic upgrade over approaches based on model size, products of norms, and infinite-width linearizations, yielding guarantees that are orders of magnitude tighter in both theory and experiment. Analytically, we identify the structural properties and mathematical principles that explain the tractability of deep networks. Empirically, the pointwise Riemannian Dimension exhibits substantial feature compression, decreases with increased over-parameterization, and captures the implicit bias of optimizers. Taken together, our results indicate that deep networks are mathematically tractable in practical regimes and that their generalization is sharply explained by pointwise, feature-spectrum-aware complexity.
We identify a Selection Plateau phenomenon in one-shot neural network pruning: all rank-monotone weight scorers converge to identical accuracy at fixed sparsity, independent of functional form. We propose the Sparsity-Information-Complexity Spectrum (SICS) hypothesis: a sparsity-dependent minimum feature complexity kappa(S) governs plateau escape, with kappa=0 sufficient at low sparsity (S<0.65), kappa=1 dominant at critical sparsity (S~0.7), and kappa=2 necessary at extreme sparsity (S>0.75). On ViT-Small/CIFAR-10, testing nine feature classes across four sparsities, smooth non-monotone features provide +6.6% escape at S=0.7, while only raw features with high-frequency wiggle escape at S=0.8 (+2.6%). A fake non-monotone scorer underperforms the gradient baseline, indicating the requirement is magnitude-independent non-monotonicity. A handcrafted Gaussian bump achieves only +0.006 escape vs. chaos-derived +0.046, indicating rank-alignment is necessary but insufficient. SICS provides a unifying explanation for the performance clustering of diverse pruning methods and suggests that future selection algorithms should adapt feature complexity to target sparsity.
Deep networks often exhibit a preference for "simple" solutions, and such a simplicity bias is widely believed to play a key role in generalization. Yet a broadly applicable, quantitative measure of simplicity remains elusive. We introduce polynomial representations as a distribution-aware, low-dimensional surrogate for neural functions: we approximate a network's predictive behavior along data-dependent interpolation paths using orthogonal polynomial bases, yielding a compact functional representation. We show that the effective degree of this representation serves as a practical simplicity metric that is predictive of generalization across tasks and architectures, and consistently outperforms existing generalization proxies such as sharpness. Finally, polynomial representations naturally yield a differentiable simplicity regularizer, which consistently improves generalization in image and text classification, fine-tuning contrastive vision-language models, and reinforcement learning.
Tianren Zhang, Xiangxin Li, Minghao Xiao +2
Department of Automation, Tsinghua University, Beijing, China.