Conventional wisdom attributes the mysterious generalization abilities of overparameterized neural networks to gradient descent (and its variants). The recent volume hypothesis challenges this view: it posits that these generalization abilities persist even when gradient descent is replaced by Guess & Check (G&C), i.e., by randomly drawing weight settings until one that fits the training data is found. The validity of the volume hypothesis for wide and deep neural networks remains an open question. In this paper, we theoretically investigate this question for matrix factorization (with linear and non-linear activation): a canonical testbed in neural network theory. We first prove that generalization under G&C deteriorates with increasing width, establishing what is, to our knowledge, the first canonical case where G&C is provably inferior to gradient descent. Conversely, we prove that generalization under G&C improves with increasing depth, revealing a stark contrast between wide and deep networks, which we further validate empirically. These findings suggest that even in simple settings, there may not be a simple answer to the question of whether neural networks need gradient descent to generalize well.

A.1 Proof of Theorem 2 Consider an instance of semi-supervised active regression with the labeled dataset Xlab being empty, and with the unlabeled dataset composed of k copies of the standard basis vector ei for each i = 1,,d.

Building upon thedataset distillation framework,wemakeakeyobservationthata sharedcommonrepresentation allows for more efficient and effective distillation.

In statistical learning, models are classified as regular or singular depending on whether the mapping from parameters to probability distributions is injective. Most models with hierarchical structures or latent variables are singular, for which conventional criteria such as the Akaike Information Criterion and the Bayesian Information Criterion are inapplicable due to the breakdown of normal approximations for the likelihood and posterior. To address this, the Widely Applicable Information Criterion (WAIC) and the Widely Applicable Bayesian Information Criterion (WBIC) have been proposed. Since WAIC and WBIC are computed using posterior distributions at different temperature settings, separate posterior sampling is generally required. In this paper, we theoretically derive an asymptotic equation that links WAIC and WBIC, despite their dependence on different posteriors. This equation yields an asymptotically unbiased expression of WAIC in terms of the posterior distribution used for WBIC. The result clarifies the structural relationship between these criteria within the framework of singular learning theory, and deepens understanding of their asymptotic behavior. This theoretical contribution provides a foundation for future developments in the computational efficiency of model selection in singular models.

We propose a method to optimize language model pre-training data mixtures through efficient approximation of the cross-entropy loss corresponding to each candidate mixture via a Mixture of Data Experts (MDE). We use this approximation as a source of additional features in a regression model, trained from observations of model loss for a small number of mixtures. Experiments with Transformer decoder-only language models in the range of 70M to 1B parameters on the SlimPajama dataset show that our method achieves significantly better performance than approaches that train regression models using only the mixture rates as input features. Combining this improved optimization method with an objective that takes into account cross-entropy on end task data leads to superior performance on few-shot downstream evaluations. We also provide theoretical insights on why aggregation of data expert predictions can provide good approximations to model losses for data mixtures.

Monotone learning refers to learning processes in which expected performance consistently improves as more training data is introduced. Non-monotone behavior of machine learning has been the topic of a series of recent works, with various proposals that ensure monotonicity by applying transformations or wrappers on learning algorithms. In this work, from a different perspective, we tackle the topic of monotone learning within the framework of Probably Approximately Correct (PAC) learning theory. Following the mechanism that estimates sample complexity of a PAC-learnable problem, we derive a performance lower bound for that problem, and prove the monotonicity of that bound as the sample sizes increase. By calculating the lower bound distribution, we are able to prove that given a PAC-learnable problem with a hypothesis space that is either of finite size or of finite VC dimension, any learning algorithm based on Empirical Risk Minimization (ERM) is monotone if training samples are independent and identically distributed (i.i.d.). We further carry out an experiment on two concrete machine learning problems, one of which has a finite hypothesis set, and the other of finite VC dimension, and compared the experimental data for the empirical risk distributions with the estimated theoretical bound. The results of the comparison have confirmed the monotonicity of learning for the two PAC-learnable problems.

We study the generalization properties of binary logistic classification in a simplified setting, for which a "memorizing" and "generalizing" solution can always be strictly defined, and elucidate empirically and analytically the mechanism underlying Grokking in its dynamics. We analyze the asymptotic long-time dynamics of logistic classification on a random feature model with a constant label and show that it exhibits Grokking, in the sense of delayed generalization and non-monotonic test loss. We find that Grokking is amplified when classification is applied to training sets which are on the verge of linear separability. Even though a perfect generalizing solution always exists, we prove the implicit bias of the logisitc loss will cause the model to overfit if the training data is linearly separable from the origin. For training sets that are not separable from the origin, the model will always generalize perfectly asymptotically, but overfitting may occur at early stages of training. Importantly, in the vicinity of the transition, that is, for training sets that are almost separable from the origin, the model may overfit for arbitrarily long times before generalizing. We gain more insights by examining a tractable one-dimensional toy model that quantitatively captures the key features of the full model. Finally, we highlight intriguing common properties of our findings with recent literature, suggesting that grokking generally occurs in proximity to the interpolation threshold, reminiscent of critical phenomena often observed in physical systems.

Understanding the underlying correlations in complex datasets is the main challenge of statistical learning. Assuming that training and generalization data are drawn from a similar distribution, the discrepancy between training and generalization metrics quantifies how well a model extracts meaningful features from the training data, and what portion of its reasoning is based on idiosyncrasies in the training data. Traditionally, one would expect that once a neural network (NN) training converges to a low loss value, the generalization error should either plateau, for good models, or deteriorate for models that overfit. Surprisingly, [18] found that a shallow transformer trained on algorithmic datasets features drastically different dynamics. The network first overfits the training data, achieving low and stable training loss with high generalization error for an extended period, then suddenly and rapidly transitions to a perfect generalization phase. This counter-intuitive phenomenon, dubbed grokking, has recently garnered much attention and many underlying mechanisms have been proposed as possible explanations. These include the difficulty of representation learning [10], the scale of parameters at initialization [11], spikes in loss ("slingshots") [21], random walks among optimal solutions [15], and the simplicity of the generalising solution [16, Appendix E]. In this paper we take a different approach, leveraging the simplest possible models which still display grokking - linear estimators.