How diffusion models generalize beyond their training set is not known, and is somewhat mysterious given two facts: the optimum of the denoising score matching (DSM) objective usually used to train diffusion models is the score function of the training distribution; and the networks usually used to learn the score function are expressive enough to learn this score to high accuracy. We claim that a certain feature of the DSM objective—the fact that its target is not the training distribution's score, but a noisy quantity only equal to it in expectation—strongly impacts whether and to what extent diffusion models generalize. In this paper, we develop a mathematical theory that partly explains this 'generalization through variance' phenomenon. Our theoretical analysis exploits a physics-inspired path integral approach to compute the distributions typically learned by a few paradigmatic under- and overparameterized diffusion models. We find that the distributions diffusion models effectively learn to sample from resemble their training distributions, but with `gaps' filled in, and that this inductive bias is due to the covariance structure of the noisy target used during training. We also characterize how this inductive bias interacts with feature-related inductive biases.

Understanding the behavior of deep reinforcement learning (DRL) agents—particularly as task and agent sophistication increase—requires more than simple comparison of reward curves, yet standard methods for behavioral analysis remain underdeveloped in DRL. We apply tools from neuroscience and ethology to study DRL agents in a novel, complex, partially observable environment, ForageWorld, designed to capture key aspects of real-world animal foraging—including sparse, depleting resource patches, predator threats, and spatially extended arenas. We use this environment as a platform for applying joint behavioral and neural analysis to agents, revealing detailed, quantitatively grounded insights into agent strategies, memory, and planning. Contrary to common assumptions, we find that model-free RNN-based DRL agents can exhibit structured, planning-like behavior purely through emergent dynamics—without requiring explicit memory modules or world models. Our results show that studying DRL agents like animals—analyzing them with neuroethology-inspired tools that reveal structure in both behavior and neural dynamics—uncovers rich structure in their learning dynamics that would otherwise remain invisible. We distill these tools into a general analysis framework linking core behavioral and representational features to diagnostic methods, which can be reused for a wide range of tasks and agents. As agents grow more complex and autonomous, bridging neuroscience, cognitive science, and AI will be essential—not just for understanding their behavior, but for ensuring safe alignment and maximizing desirable behaviors that are hard to measure via reward. We show how this can be done by drawing on lessons from how biological intelligence is studied.

Learning rules—prescriptions for updating model parameters to improve performance—are typically assumed rather than derived. Why do some learning rules work better than others, and under what assumptions can a given rule be considered optimal? We propose a theoretical framework that casts learning rules as policies for navigating (partially observable) loss landscapes, and identifies optimal rules as solutions to an associated optimal control problem. A range of well-known rules emerge naturally within this framework under different assumptions: gradient descent from short-horizon optimization, momentum from longer-horizon planning, natural gradients from accounting for parameter space geometry, non-gradient rules from partial controllability, and adaptive optimizers like Adam from online Bayesian inference of loss landscape shape. We further show that continual learning strategies like weight resetting can be understood as optimal responses to task uncertainty. By unifying these phenomena under a single objective, our framework clarifies the computational structure of learning and offers a principled foundation for designing adaptive algorithms.