arXiv subjects

Blake Bordelon

Publications and source records attributed to Blake Bordelon.

At least 19 recordsLinked to original sources

arXiv2026

Dynamics of learning to integrate in linear recurrent neural networks

Learning recurrent connectivity that supports memory over long intrinsic timescales is a basic problem in the theory of dynamical computation. While continuous attractor and integrator models describe how tuned recurrent circuits can maintain information, less is known about how such slow modes are acquired by gradient-based learning. Here we study this question in an analytically tractable setting: we build a mathematical theory of the learning dynamics of linear RNNs trained to integrate white noise. We show that when the initial recurrent weights are small, the dynamics of learning are described by a low-dimensional system that tracks a single outlier eigenvalue of the recurrent weights. This reveals the precise manner in which the long timescale associated with white noise integration is learned. We extend our analyses to RNNs learning a damped oscillatory filter, and find low-dimensional effective dynamical equations for the evolution of a conjugate pair of outlier eigenvalues. Taken together, our analyses build a rich mathematical framework for studying dynamical learning problems relevant to both machine learning and neuroscience.

q-bio.NC↗

arXiv2026

Hyperparameter Transfer with Mixture-of-Expert Layers

Mixture-of-Experts (MoE) layers have emerged as an important tool in scaling up modern neural networks by decoupling total trainable parameters from activated parameters in the forward pass for each token. However, sparse MoEs add complexity to training due to (i) new trainable parameters (router weights) that, like all other parameter groups, require hyperparameter (HP) tuning; (ii) new architecture scale dimensions (number of and size of experts) that must be chosen and potentially taken large. To make HP selection cheap and reliable, we propose a new parameterization for transformer models with MoE layers when scaling model width, depth, number of experts, and expert (hidden) size. Our parameterization is justified by a novel dynamical mean-field theory (DMFT) analysis. When varying different model dimensions trained at a fixed token budget, we find empirically that our parameterization enables reliable HP transfer across models from 51M to over 2B total parameters. We further take HPs identified from sweeping small models on a short token horizon to train larger models on longer horizons and report performant model behaviors.

cs.LG↗

arXiv2026

Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer

We study the evolution of hidden-weight spectra in wide neural networks trained by (stochastic) gradient descent. We develop a two-level dynamical mean-field theory (DMFT) that jointly tracks bulk and outlier spectral dynamics for spiked ensembles whose spike directions remain statistically dependent on the random bulk. We apply this framework to two settings: (1) infinite-width nonlinear networks in mean-field/$μ$P scaling and (2) deep linear networks in the proportional high-dimensional limit, where width, input dimension, and sample size diverge with fixed ratios. Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks, $μ$P yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS). In contrast, NTK parameterization exhibits strongly width-dependent outlier dynamics, despite converging to a stable large-width limit. We show that this bulk+outlier picture is descriptive of simple tasks with small output channels, but that tasks involving large numbers of outputs (ImageNet classification or GPT language modeling) are better described by a restructuring of the spectral bulk. We develop a toy model with extensive output channels that recapitulates this phenomenon and show that edge of the spectrum still converges for sufficiently wide networks.

cond-mat.dis-nn↗

arXiv2026

Theory of Optimal Learning Rate Schedules and Scaling Laws for a Random Feature Model

Setting the learning rate (LR) for a deep learning model is a critical part of successful training. Choosing LRs is often done empirically with trial and error. In this work, we explore a solvable model of optimal LR schedules for a powerlaw random feature model trained with stochastic gradient descent (SGD). We consider the optimal schedule $η_T^\star(t)$ where $t$ is the current iterate and $T$ is the training horizon. This schedule is computed both as a numerical optimization problem and also analytically using optimal control theory. Our analysis reveals two regimes which we term the easy phase and hard phase. In the easy phase the optimal schedule is a polynomial decay $η_T^\star(t) \simeq T^{-ξ} (1-t/T)^δ$ where $ξ$ and $δ$ depend on the properties of the features and task. In the hard phase, the optimal schedule resembles warmup-stable-decay with constant initial LR and annealing performed over a vanishing fraction of training steps. We investigate joint optimization of LR and batch size and find batch ramps can improve the wall-clock time in the easy phase. Beyond SGD, we derive optimal schedules for momentum parameter $β(t)$ and show that it improves the loss-scaling exponent in the hard phase. We compare our optimal schedule to various benchmarks including (1) optimal constant learning rates $η_T(t) \sim T^{-ξ}$ (2) optimal power laws $η_T(t) \sim T^{-ξ} t^{-χ}$, finding that our schedule achieves better rates than either of these. Our theory suggests that LR transfer across training horizon depends on the structure of the model and task. For ResNet image classification on CIFAR-5M, the learning curves exhibit hard-phase behavior where optimal base LRs are constant under sufficient annealing. GPT-2 style transformers trained in language modeling exhibit easy-phase behavior where optimal LRs shift even under annealing.

cond-mat.dis-nn↗

arXiv2026

There Will Be a Scientific Theory of Deep Learning

In this paper, we make the case that a scientific theory of deep learning is emerging. By this we mean a theory which characterizes important properties and statistics of the training process, hidden representations, final weights, and performance of neural networks. We pull together major strands of ongoing research in deep learning theory and identify five growing bodies of work that point toward such a theory: (a) solvable idealized settings that provide intuition for learning dynamics in realistic systems; (b) tractable limits that reveal insights into fundamental learning phenomena; (c) simple mathematical laws that capture important macroscopic observables; (d) theories of hyperparameters that disentangle them from the rest of the training process, leaving simpler systems behind; and (e) universal behaviors shared across systems and settings which clarify which phenomena call for explanation. Taken together, these bodies of work share certain broad traits: they are concerned with the dynamics of the training process; they primarily seek to describe coarse aggregate statistics; and they emphasize falsifiable quantitative predictions. We argue that the emerging theory is best thought of as a mechanics of the learning process, and suggest the name learning mechanics. We discuss the relationship between this mechanics perspective and other approaches for building a theory of deep learning, including the statistical and information-theoretic perspectives. In particular, we anticipate a symbiotic relationship between learning mechanics and mechanistic interpretability. We also review and address common arguments that fundamental theory will not be possible or is not important. We conclude with a portrait of important open directions in learning mechanics and advice for beginners. We host further introductory materials, perspectives, and open questions at learningmechanics.pub.

stat.ML↗

arXiv2026

Transfer Learning in Infinite Width Feature Learning Networks

We develop a theory of transfer learning in infinitely wide neural networks under gradient flow that quantifies when pretraining on a source task improves generalization on a target task. We analyze both (i) fine-tuning, when the downstream predictor is trained on top of source-induced features and (ii) a jointly rich setting, where both pretraining and downstream tasks can operate in a feature learning regime, but the downstream model is initialized with the features obtained after pre-training. In this setup, the summary statistics of randomly initialized networks after a rich pre-training are adaptive kernels which depend on both source data and labels. For (i), we analyze the performance of a readout for different pretraining data regimes. For (ii), the summary statistics after learning the target task are still adaptive kernels with features from both source and target tasks. We test our theory on linear and polynomial regression tasks as well as real datasets. Our theory allows interpretable conclusions on performance, which depend on the amount of data on both tasks, the alignment between tasks, and the feature learning strength.

cs.LG↗

arXiv2026

Don't be lazy: CompleteP enables compute-efficient deep transformers

We study compute efficiency of LLM training when using different parameterizations, i.e., rules for adjusting model and optimizer hyperparameters (HPs) as model size changes. Some parameterizations fail to transfer optimal base HPs (such as learning rate) across changes in model depth, requiring practitioners to either re-tune these HPs as they scale up (expensive), or accept sub-optimal training when re-tuning is prohibitive. Even when they achieve HP transfer, we develop theory to show parameterizations may still exist in the lazy learning regime where layers learn only features close to their linearization, preventing effective use of depth and nonlinearity. Finally, we identify and adopt the parameterization we call CompleteP that achieves both depth-wise HP transfer and non-lazy learning in all layers. CompleteP enables a wider range of model width/depth ratios to remain compute-efficient, unlocking shapes better suited for different hardware settings and operational contexts. Moreover, CompleteP enables 12-34% compute efficiency improvements over the prior state-of-the-art. All experiments were run on Cerebras CS-3 systems. A minimal implementation is available at https://github.com/EleutherAI/nanoGPT-mup/tree/completep.

cs.LG↗

arXiv2026

Disordered Dynamics in High Dimensions: Connections to Random Matrices and Machine Learning

We provide an overview of high dimensional dynamical systems driven by random matrices, focusing on applications to simple models of learning and generalization in machine learning theory. Using both cavity method arguments and path integrals, we review how the behavior of a coupled infinite dimensional system can be characterized as a stochastic process for each single site of the system. We provide a pedagogical treatment of dynamical mean field theory (DMFT), a framework that can be flexibly applied to these settings. The DMFT single site stochastic process is fully characterized by a set of (two-time) correlation and response functions. For linear time-invariant systems, we illustrate connections between random matrix resolvents and the DMFT response. We demonstrate applications of these ideas to machine learning models such as gradient flow, stochastic gradient descent on random feature models and deep linear networks in the feature learning regime trained on random data. We demonstrate how bias and variance decompositions (analysis of ensembling/bagging etc) can be computed by averaging over subsets of the DMFT noise variables. From our formalism we also investigate how linear systems driven with random non-Hermitian matrices (such as random feature models) can exhibit non-monotonic loss curves with training time, while Hermitian matrices with the matching spectra do not, highlighting a different mechanism for non-monotonicity than small eigenvalues causing instability to label noise. Lastly, we provide asymptotic descriptions of the training and test loss dynamics for randomly initialized deep linear neural networks trained in the feature learning regime with high-dimensional random data. In this case, the time translation invariance structure is lost and the hidden layer weights are characterized as spiked random matrices.

cond-mat.dis-nn↗

arXiv2025

Two-Point Deterministic Equivalence for Stochastic Gradient Dynamics in Linear Models

We derive a novel deterministic equivalence for the two-point function of a random matrix resolvent. Using this result, we give a unified derivation of the performance of a wide variety of high-dimensional linear models trained with stochastic gradient descent. This includes high-dimensional linear regression, kernel regression, and linear random feature models. Our results include previously known asymptotics as well as novel ones.

cond-mat.dis-nn↗

arXiv2025

Theory of Scaling Laws for In-Context Regression: Depth, Width, Context and Time

We study in-context learning (ICL) of linear regression in a deep linear self-attention model, characterizing how performance depends on various computational and statistical resources (width, depth, number of training steps, batch size and data per context). In a joint limit where data dimension, context length, and residual stream width scale proportionally, we analyze the limiting asymptotics for three ICL settings: (1) isotropic covariates and tasks (ISO), (2) fixed and structured covariance (FS), and (3) where covariances are randomly rotated and structured (RRS). For ISO and FS settings, we find that depth only aids ICL performance if context length is limited. Alternatively, in the RRS setting where covariances change across contexts, increasing the depth leads to significant improvements in ICL, even at infinite context length. This provides a new solvable toy model of neural scaling laws which depends on both width and depth of a transformer and predicts an optimal transformer shape as a function of compute. This toy model enables computation of exact asymptotics for the risk as well as derivation of powerlaws under source/capacity conditions for the ICL tasks.

stat.ML↗

arXiv2025

Adaptive kernel predictors from feature-learning infinite limits of neural networks

Previous influential work showed that infinite width limits of neural networks in the lazy training regime are described by kernel machines. Here, we show that neural networks trained in the rich, feature learning infinite-width regime in two different settings are also described by kernel machines, but with data-dependent kernels. For both cases, we provide explicit expressions for the kernel predictors and prescriptions to numerically calculate them. To derive the first predictor, we study the large-width limit of feature-learning Bayesian networks, showing how feature learning leads to task-relevant adaptation of layer kernels and preactivation densities. The saddle point equations governing this limit result in a min-max optimization problem that defines the kernel predictor. To derive the second predictor, we study gradient flow training of randomly initialized networks trained with weight decay in the infinite-width limit using dynamical mean field theory (DMFT). The fixed point equations of the arising DMFT defines the task-adapted internal representations and the kernel predictor. We compare our kernel predictors to kernels derived from lazy regime and demonstrate that our adaptive kernels achieve lower test loss on benchmark datasets.

cs.LG↗

arXiv2025

Summary statistics of learning link changing neural representations to behavior

How can we make sense of large-scale recordings of neural activity across learning? Theories of neural network learning with their origins in statistical physics offer a potential answer: for a given task, there are often a small set of summary statistics that are sufficient to predict performance as the network learns. Here, we review recent advances in how summary statistics can be used to build theoretical understanding of neural network learning. We then argue for how this perspective can inform the analysis of neural data, enabling better understanding of learning in biological and artificial neural networks.

q-bio.NC↗

arXiv2025

Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer

We theoretically characterize gradient descent dynamics in deep linear networks trained at large width from random initialization and on large quantities of random data. Our theory captures the ``wider is better" effect of mean-field/maximum-update parameterized networks as well as hyperparameter transfer effects, which can be contrasted with the neural-tangent parameterization where optimal learning rates shift with model width. We provide asymptotic descriptions of both non-residual and residual neural networks, the latter of which enables an infinite depth limit when branches are scaled as $1/\sqrt{\text{depth}}$. We also compare training with one-pass stochastic gradient descent to the dynamics when training data are repeated at each iteration. Lastly, we show that this model recovers the accelerated power law training dynamics for power law structured data in the rich regime observed in recent works.

cs.LG↗

arXiv2025

How Feature Learning Can Improve Neural Scaling Laws

We develop a solvable model of neural scaling laws beyond the kernel limit. Theoretical analysis of this model shows how performance scales with model size, training time, and the total amount of available data. We identify three scaling regimes corresponding to varying task difficulties: hard, easy, and super easy tasks. For easy and super-easy target functions, which lie in the reproducing kernel Hilbert space (RKHS) defined by the initial infinite-width Neural Tangent Kernel (NTK), the scaling exponents remain unchanged between feature learning and kernel regime models. For hard tasks, defined as those outside the RKHS of the initial NTK, we demonstrate both analytically and empirically that feature learning can improve scaling with training time and compute, nearly doubling the exponent for hard tasks. This leads to a different compute optimal strategy to scale parameters and training time in the feature learning regime. We support our finding that feature learning improves the scaling law for hard tasks but not for easy and super-easy tasks with experiments of nonlinear MLPs fitting functions with power-law Fourier spectra on the circle and CNNs learning vision tasks.

stat.ML↗

arXiv2024

Do Mice Grok? Glimpses of Hidden Progress During Overtraining in Sensory Cortex

Does learning of task-relevant representations stop when behavior stops changing? Motivated by recent theoretical advances in machine learning and the intuitive observation that human experts continue to learn from practice even after mastery, we hypothesize that task-specific representation learning can continue, even when behavior plateaus. In a novel reanalysis of recently published neural data, we find evidence for such learning in posterior piriform cortex of mice following continued training on a task, long after behavior saturates at near-ceiling performance ("overtraining"). This learning is marked by an increase in decoding accuracy from piriform neural populations and improved performance on held-out generalization tests. We demonstrate that class representations in cortex continue to separate during overtraining, so that examples that were incorrectly classified at the beginning of overtraining can abruptly be correctly classified later on, despite no changes in behavior during that time. We hypothesize this hidden yet rich learning takes the form of approximate margin maximization; we validate this and other predictions in the neural data, as well as build and interpret a simple synthetic model that recapitulates these phenomena. We conclude by showing how this model of late-time feature learning implies an explanation for the empirical puzzle of overtraining reversal in animal learning, where task-specific representations are more robust to particular task changes because the learned features can be reused.

cs.LG↗

arXiv2024

Scaling Laws for Precision

Low precision training and inference affect both the quality and cost of language models, but current scaling laws do not account for this. In this work, we devise "precision-aware" scaling laws for both training and inference. We propose that training in lower precision reduces the model's "effective parameter count," allowing us to predict the additional loss incurred from training in low precision and post-train quantization. For inference, we find that the degradation introduced by post-training quantization increases as models are trained on more data, eventually making additional pretraining data actively harmful. For training, our scaling laws allow us to predict the loss of a model with different parts in different precisions, and suggest that training larger models in lower precision may be compute optimal. We unify the scaling laws for post and pretraining quantization to arrive at a single functional form that predicts degradation from training and inference in varied precisions. We fit on over 465 pretraining runs and validate our predictions on model sizes up to 1.7B parameters trained on up to 26B tokens.

cs.LG↗

arXiv2024

Infinite Limits of Multi-head Transformer Dynamics

In this work, we analyze various scaling limits of the training dynamics of transformer models in the feature learning regime. We identify the set of parameterizations that admit well-defined infinite width and depth limits, allowing the attention layers to update throughout training--a relevant notion of feature learning in these models. We then use tools from dynamical mean field theory (DMFT) to analyze various infinite limits (infinite key/query dimension, infinite heads, and infinite depth) which have different statistical descriptions depending on which infinite limit is taken and how attention layers are scaled. We provide numerical evidence of convergence to the limits and discuss how the parameterization qualitatively influences learned features.

stat.ML↗

arXiv2024

A Dynamical Model of Neural Scaling Laws

On a variety of tasks, the performance of neural networks predictably improves with training time, dataset size and model size across many orders of magnitude. This phenomenon is known as a neural scaling law. Of fundamental importance is the compute-optimal scaling law, which reports the performance as a function of units of compute when choosing model sizes optimally. We analyze a random feature model trained with gradient descent as a solvable model of network training and generalization. This reproduces many observations about neural scaling laws. First, our model makes a prediction about why the scaling of performance with training time and with model size have different power law exponents. Consequently, the theory predicts an asymmetric compute-optimal scaling rule where the number of training steps are increased faster than model parameters, consistent with recent empirical observations. Second, it has been observed that early in training, networks converge to their infinite-width dynamics at a rate $1/\textit{width}$ but at late time exhibit a rate $\textit{width}^{-c}$, where $c$ depends on the structure of the architecture and task. We show that our model exhibits this behavior. Lastly, our theory shows how the gap between training and test loss can gradually build up over time due to repeated reuse of data.

stat.ML↗