We develop an analytical framework for understanding how the generated distribution evolves during diffusion model training. Leveraging a Gaussian-equivalence principle, we solve the full-batch gradient-flow dynamics of linear and convolutional denoisers and integrate the resulting probability-flow ODE, yielding analytic expressions for the generated distribution. The theory exposes a universal inverse-variance spectral law: the time for an eigen-or Fourier mode to match its target variance scales as τ λ 1, so high-variance (coarse) structure is mastered orders of magnitude sooner than low-variance (fine) detail. Extending the analysis to deep linear networks and circulant full-width convolutions shows that weight sharing merely multiplies learning rates--accelerating but not eliminating the bias--whereas local convolution introduces a qualitatively different bias. Experiments on Gaussian and natural-image datasets confirm the spectral law persists in deep MLP-based UNet.

Recurrent neural networks (RNNs) trained on neuroscience-inspired tasks offer powerful models of brain computation. However, typical training paradigms rely on open-loop, supervised settings, whereas real-world learning unfolds in closed-loop environments. Here, we develop a mathematical theory describing the learning dynamics of linear RNNs trained in closed-loop contexts. We first demonstrate that two otherwise identical RNNs, trained in either closed-or open-loop modes, follow markedly different learning trajectories. To probe this divergence, we analytically characterize the closed-loop case, revealing distinct stages aligned with the evolution of the training loss. Specifically, we show that the learning dynamics of closed-loop RNNs, in contrast to open-loop ones, are governed by an interplay between two competing objectives: short-term policy improvement and long-term stability of the agent-environment interaction. Finally, we apply our framework to a realistic motor control task, highlighting its broader applicability. Taken together, our results underscore the importance of modeling closed-loop dynamics in a biologically plausible setting.

Understanding how animals learn is a central challenge in neuroscience, with growing relevance to the development of animal-or human-aligned artificial intelligence. However, existing approaches tend to assume fixed parametric forms for the learning rule (e.g., Q-learning, policy gradient), which may not accurately describe the complex forms of learning employed by animals in realistic settings. Here we address this gap by developing a framework to infer learning rules directly from behavioral data collected during task learning. We assume that animals follow a decision policy parameterized by a generalized linear model (GLM), and we model their learning rule--the mapping from task covariates to per-trial weight updates--using a deep neural network (DNN). This formulation allows flexible, data-driven inference of learning rules while maintaining an interpretable form of the decision policy itself.

In this paper we establish efficient and uncoupled learning dynamics so that, when employed by all players in a general-sum multiplayer game, the swap regret of each player after T repetitions of the game is bounded by O(logT), improving over the prior best bounds of O(log4(T)). At the same time, we guarantee optimal O( T) swap regret in the adversarial regime as well. To obtain these results, our primary contribution is to show that when all players follow our dynamics with a time-invariant learning rate, the second-order path lengths of the dynamics up to time T are bounded by O(logT), a fundamental property which could have further implications beyond near-optimally bounding the (swap) regret. Our proposed learning dynamics combine in a novel way optimistic regularized learning with the use of self-concordant barriers. Further, our analysis is remarkably simple, bypassing the cumbersome framework of higher-order smoothness recently developed by Daskalakis, Fishelson, and Golowich (NeurIPS'21).

In this work, we provide a comparative analysis of the learning dynamics for the Euclidean and cosine-based asymmetric losses in the eigenspace of the closed-form predictor DirectPred.

A recent line of work has established uncoupled learning dynamics such that, when employed by all players in a game, each player's regret after T repetitions grows polylogarithmically in T, an exponential improvement over the traditional guarantees within the no-regret framework. However, so far these results have only been limited to certain classes of games with structured strategy spaces--such as normal-form and extensive-form games. The question as to whether O(polylogT) regret bounds can be obtained for general convex and compact strategy sets--which occur in many fundamental models in economics and multiagent systems--while retaining efficient strategy updates is an importantquestion.

The framework of multi-agent learning explores the dynamics of how an agent's

Now, let us derive the differential equation that gives a solution to the variational problem. This condition yields the Euler-Lagrange equation, d dt @ L @ q = @ L @q . Here, we derive the Noether's learning dynamics by applying Noether's theorem to the A general form of the Noether's theorem relates the dynamics of Noether By evaluating the right hand side of Eq. 23, we get e Now, we harness the covariant property of the Lagrangian formulation, i.e., it preserves the form Plugging this expression obtained from the steady-state condition of Eq.27 Here, we ignore the inertia term in Eq. 16, assuming that the mass (learning rate) is finite but small All the experiments were run using the PyTorch code base. We used Tiny ImageNet dataset to generate all the empirical figures in this work. The key hyperparameters we used are listed with each figure.