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

Setting the learning rate for a deep learning model is a critical part of successful training, yet choosing this hyperparameter is often done empirically with trial and error. In this work, we explore a solvable model of optimal learning rate 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 total training horizon. This schedule is computed both numerically and analytically (when possible) using optimal control methods. 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 (in $T$) initial learning rate and annealing performed over a vanishing (in $T$) fraction of training steps. We investigate joint optimization of learning rate and batch size, identifying a degenerate optimality condition. Our model also predicts the compute-optimal scaling laws (where model size and training steps are chosen optimally) in both easy and hard regimes. Going beyond SGD, we consider optimal schedules for the momentum $β(t)$, where speedups in the hard phase are possible. We compare our optimal schedule to various benchmarks in our task 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 learning rate transfer across training horizon depends on the structure of the model and task. We explore these ideas in simple experimental pretraining setups.

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.

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.

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 unique 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.

Dynamically Learning to Integrate in Recurrent Neural Networks

Learning to remember over long timescales is fundamentally challenging for recurrent neural networks (RNNs). While much prior work has explored why RNNs struggle to learn long timescales and how to mitigate this, we still lack a clear understanding of the dynamics involved when RNNs learn long timescales via gradient descent. Here 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 rich 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 salient for both machine learning and neuroscience.

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.

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 random feature models. Our results include previously known asymptotics as well as novel ones.

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.

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.

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.