William L. Tong

Jacob A. Zavatone-Veth, William L. Tong, Cengiz Pehlevan · harvard

Understanding how feature learning affects generalization is among the foremost goals of modern deep learning theory. Here, we study how the ability to learn representations affects the generalization performance of a simple class of models: deep Bayesian linear neural networks trained on unstructured Gaussian data. By comparing deep random feature models to deep networks in which all layers are trained, we provide a detailed characterization of the interplay between width, depth, data density, and prior mismatch. We show that both models display sample-wise double-descent behavior in the presence of label noise. Random feature models can also display model-wise double-descent if there are narrow bottleneck layers, while deep networks do not show these divergences. Random feature models can have particular widths that are optimal for generalization at a given data density, while making neural networks as wide or as narrow as possible is always optimal. Moreover, we show that the leading-order correction to the kernel-limit learning curve cannot distinguish between random feature models and deep networks in which all layers are trained. Taken together, our findings begin to elucidate how architectural details affect generalization performance in this simple class of deep regression models.

William L. Tong, Ege Cakar, Cengiz Pehlevan

Recent years have witnessed meteoric progress in reasoning models: neural networks that generate intermediate reasoning traces (RTs) before producing a final output. Despite the rapid advancement, our understanding of how RTs support reasoning, and the limits of this paradigm, remain incomplete. To promote greater clarity, we introduce PITA: a novel large-scale dataset of over 23 million statements in propositional logic and their corresponding proofs. As a benchmark for robust reasoning, we focus on length generalization: if a model is trained to determine truth or falsity on statements with proofs up to fixed length, how well does it generalize to statements requiring longer proofs? We propose notions of (1) task depth and (2) task breadth, which measure respectively (1) the number of steps required to solve an example from a task and (2) the number of unique examples across a task. We vary these quantities across subsets of PITA, and find that RT models generalize well on broad and shallow subsets, while deteriorating on narrow and deep subsets relative to non-RT baselines. To determine whether our results are idiosyncratic to PITA or indicative of general phenomena, we compare our results to a simple synthetic task based on syllogisms. Our resulting theory suggests fundamental scalings that limit how well RT models perform on deep tasks, and highlights their generalization strengths on broad tasks. Our findings overall identify fundamental benefits and limitations inherent in using reasoning traces.

William L. Tong, Cengiz Pehlevan

In-context learning (ICL), the remarkable ability to solve a task from only input exemplars, is often assumed to be a unique hallmark of Transformer models. By examining commonly employed synthetic ICL tasks, we demonstrate that multi-layer perceptrons (MLPs) can also learn in-context. Moreover, MLPs, and the closely related MLP-Mixer models, learn in-context comparably with Transformers under the same compute budget in this setting. We further show that MLPs outperform Transformers on a series of classical tasks from psychology designed to test relational reasoning, which are closely related to in-context classification. These results underscore a need for studying in-context learning beyond attention-based architectures, while also challenging prior arguments against MLPs' ability to solve relational tasks. Altogether, our results highlight the unexpected competence of MLPs in a synthetic setting, and support the growing interest in all-MLP alternatives to Transformer architectures. It remains unclear how MLPs perform against Transformers at scale on real-world tasks, and where a performance gap may originate. We encourage further exploration of these architectures in more complex settings to better understand the potential comparative advantage of attention-based schemes.

Benjamin S. Ruben, William L. Tong, Hamza Tahir Chaudhry et al.

Given a fixed budget for total model size, one must choose between training a single large model or combining the predictions of multiple smaller models. We investigate this trade-off for ensembles of random-feature ridge regression models in both the overparameterized and underparameterized regimes. Using deterministic equivalent risk estimates, we prove that when a fixed number of parameters is distributed among $K$ independently trained models, the ridge-optimized test risk increases with $K$. Consequently, a single large model achieves optimal performance. We then ask when ensembles can achieve \textit{near}-optimal performance. In the overparameterized regime, we show that, to leading order, the test error depends on ensemble size and model size only through the total feature count, so that overparameterized ensembles consistently achieve near-optimal performance. To understand underparameterized ensembles, we derive scaling laws for the test risk as a function of total parameter count when the ensemble size and parameters per ensemble member are jointly scaled according to a ``growth exponent'' $\ell$. While the optimal error scaling is always achieved by increasing model size with a fixed ensemble size, our analysis identifies conditions on the kernel and task eigenstructure under which near-optimal scaling laws can be obtained by joint scaling of ensemble size and model size.

William L. Tong, Cengiz Pehlevan

Equality reasoning is ubiquitous and purely abstract: sameness or difference may be evaluated no matter the nature of the underlying objects. As a result, same-different (SD) tasks have been extensively studied as a starting point for understanding abstract reasoning in humans and across animal species. With the rise of neural networks that exhibit striking apparent proficiency for abstractions, equality reasoning in these models has also gained interest. Yet despite extensive study, conclusions about equality reasoning vary widely and with little consensus. To clarify the underlying principles in learning SD tasks, we develop a theory of equality reasoning in multi-layer perceptrons (MLP). Following observations in comparative psychology, we propose a spectrum of behavior that ranges from conceptual to perceptual outcomes. Conceptual behavior is characterized by task-specific representations, efficient learning, and insensitivity to spurious perceptual details. Perceptual behavior is characterized by strong sensitivity to spurious perceptual details, accompanied by the need for exhaustive training to learn the task. We develop a mathematical theory to show that an MLP's behavior is driven by learning richness. Rich-regime MLPs exhibit conceptual behavior, whereas lazy-regime MLPs exhibit perceptual behavior. We validate our theoretical findings in vision SD experiments, showing that rich feature learning promotes success by encouraging hallmarks of conceptual behavior. Overall, our work identifies feature learning richness as a key parameter modulating equality reasoning, and suggests that equality reasoning in humans and animals may similarly depend on learning richness in neural circuits.