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A Dynamical Model of Neural Scaling Laws
Feb 02, 2024
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.
Grokking as the Transition from Lazy to Rich Training Dynamics
Oct 09, 2023We propose that the grokking phenomenon, where the train loss of a neural network decreases much earlier than its test loss, can arise due to a neural network transitioning from lazy training dynamics to a rich, feature learning regime. To illustrate this mechanism, we study the simple setting of vanilla gradient descent on a polynomial regression problem with a two layer neural network which exhibits grokking without regularization in a way that cannot be explained by existing theories. We identify sufficient statistics for the test loss of such a network, and tracking these over training reveals that grokking arises in this setting when the network first attempts to fit a kernel regression solution with its initial features, followed by late-time feature learning where a generalizing solution is identified after train loss is already low. We find that the key determinants of grokking are the rate of feature learning -- which can be controlled precisely by parameters that scale the network output -- and the alignment of the initial features with the target function $y(x)$. We argue this delayed generalization arises when (1) the top eigenvectors of the initial neural tangent kernel and the task labels $y(x)$ are misaligned, but (2) the dataset size is large enough so that it is possible for the network to generalize eventually, but not so large that train loss perfectly tracks test loss at all epochs, and (3) the network begins training in the lazy regime so does not learn features immediately. We conclude with evidence that this transition from lazy (linear model) to rich training (feature learning) can control grokking in more general settings, like on MNIST, one-layer Transformers, and student-teacher networks.
Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit
Sep 28, 2023The cost of hyperparameter tuning in deep learning has been rising with model sizes, prompting practitioners to find new tuning methods using a proxy of smaller networks. One such proposal uses $\mu$P parameterized networks, where the optimal hyperparameters for small width networks transfer to networks with arbitrarily large width. However, in this scheme, hyperparameters do not transfer across depths. As a remedy, we study residual networks with a residual branch scale of $1/\sqrt{\text{depth}}$ in combination with the $\mu$P parameterization. We provide experiments demonstrating that residual architectures including convolutional ResNets and Vision Transformers trained with this parameterization exhibit transfer of optimal hyperparameters across width and depth on CIFAR-10 and ImageNet. Furthermore, our empirical findings are supported and motivated by theory. Using recent developments in the dynamical mean field theory (DMFT) description of neural network learning dynamics, we show that this parameterization of ResNets admits a well-defined feature learning joint infinite-width and infinite-depth limit and show convergence of finite-size network dynamics towards this limit.
Dynamics of Temporal Difference Reinforcement Learning
Jul 10, 2023
Reinforcement learning has been successful across several applications in which agents have to learn to act in environments with sparse feedback. However, despite this empirical success there is still a lack of theoretical understanding of how the parameters of reinforcement learning models and the features used to represent states interact to control the dynamics of learning. In this work, we use concepts from statistical physics, to study the typical case learning curves for temporal difference learning of a value function with linear function approximators. Our theory is derived under a Gaussian equivalence hypothesis where averages over the random trajectories are replaced with temporally correlated Gaussian feature averages and we validate our assumptions on small scale Markov Decision Processes. We find that the stochastic semi-gradient noise due to subsampling the space of possible episodes leads to significant plateaus in the value error, unlike in traditional gradient descent dynamics. We study how learning dynamics and plateaus depend on feature structure, learning rate, discount factor, and reward function. We then analyze how strategies like learning rate annealing and reward shaping can favorably alter learning dynamics and plateaus. To conclude, our work introduces new tools to open a new direction towards developing a theory of learning dynamics in reinforcement learning.
Feature-Learning Networks Are Consistent Across Widths At Realistic Scales
May 28, 2023
We study the effect of width on the dynamics of feature-learning neural networks across a variety of architectures and datasets. Early in training, wide neural networks trained on online data have not only identical loss curves but also agree in their point-wise test predictions throughout training. For simple tasks such as CIFAR-5m this holds throughout training for networks of realistic widths. We also show that structural properties of the models, including internal representations, preactivation distributions, edge of stability phenomena, and large learning rate effects are consistent across large widths. This motivates the hypothesis that phenomena seen in realistic models can be captured by infinite-width, feature-learning limits. For harder tasks (such as ImageNet and language modeling), and later training times, finite-width deviations grow systematically. Two distinct effects cause these deviations across widths. First, the network output has initialization-dependent variance scaling inversely with width, which can be removed by ensembling networks. We observe, however, that ensembles of narrower networks perform worse than a single wide network. We call this the bias of narrower width. We conclude with a spectral perspective on the origin of this finite-width bias.
Dynamics of Finite Width Kernel and Prediction Fluctuations in Mean Field Neural Networks
Apr 06, 2023
We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Unlike many prior analyses, our results, while perturbative in width, are non-perturbative in the strength of feature learning. Starting from a dynamical mean field theory (DMFT) description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the $\mathcal{O}(1/\sqrt{\text{width}})$ fluctuations of the DMFT order parameters over random initialization of the network weights. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final NTK and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the SNR of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For CNNs trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width.
The Onset of Variance-Limited Behavior for Networks in the Lazy and Rich Regimes
Dec 23, 2022
For small training set sizes $P$, the generalization error of wide neural networks is well-approximated by the error of an infinite width neural network (NN), either in the kernel or mean-field/feature-learning regime. However, after a critical sample size $P^*$, we empirically find the finite-width network generalization becomes worse than that of the infinite width network. In this work, we empirically study the transition from infinite-width behavior to this variance limited regime as a function of sample size $P$ and network width $N$. We find that finite-size effects can become relevant for very small dataset sizes on the order of $P^* \sim \sqrt{N}$ for polynomial regression with ReLU networks. We discuss the source of these effects using an argument based on the variance of the NN's final neural tangent kernel (NTK). This transition can be pushed to larger $P$ by enhancing feature learning or by ensemble averaging the networks. We find that the learning curve for regression with the final NTK is an accurate approximation of the NN learning curve. Using this, we provide a toy model which also exhibits $P^* \sim \sqrt{N}$ scaling and has $P$-dependent benefits from feature learning.
The Influence of Learning Rule on Representation Dynamics in Wide Neural Networks
Oct 05, 2022
It is unclear how changing the learning rule of a deep neural network alters its learning dynamics and representations. To gain insight into the relationship between learned features, function approximation, and the learning rule, we analyze infinite-width deep networks trained with gradient descent (GD) and biologically-plausible alternatives including feedback alignment (FA), direct feedback alignment (DFA), and error modulated Hebbian learning (Hebb), as well as gated linear networks (GLN). We show that, for each of these learning rules, the evolution of the output function at infinite width is governed by a time varying effective neural tangent kernel (eNTK). In the lazy training limit, this eNTK is static and does not evolve, while in the rich mean-field regime this kernel's evolution can be determined self-consistently with dynamical mean field theory (DMFT). This DMFT enables comparisons of the feature and prediction dynamics induced by each of these learning rules. In the lazy limit, we find that DFA and Hebb can only learn using the last layer features, while full FA can utilize earlier layers with a scale determined by the initial correlation between feedforward and feedback weight matrices. In the rich regime, DFA and FA utilize a temporally evolving and depth-dependent NTK. Counterintuitively, we find that FA networks trained in the rich regime exhibit more feature learning if initialized with smaller correlation between the forward and backward pass weights. GLNs admit a very simple formula for their lazy limit kernel and preserve conditional Gaussianity of their preactivations under gating functions. Error modulated Hebb rules show very small task-relevant alignment of their kernels and perform most task relevant learning in the last layer.
Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks
May 19, 2022
We analyze feature learning in infinite width neural networks trained with gradient flow through a self-consistent dynamical field theory. We construct a collection of deterministic dynamical order parameters which are inner-product kernels for hidden unit activations and gradients in each layer at pairs of time points, providing a reduced description of network activity through training. These kernel order parameters collectively define the hidden layer activation distribution, the evolution of the neural tangent kernel, and consequently output predictions. For deep linear networks, these kernels satisfy a set of algebraic matrix equations. For nonlinear networks, we provide an alternating sampling procedure to self-consistently solve for the kernel order parameters. We provide comparisons of the self-consistent solution to various approximation schemes including the static NTK approximation, gradient independence assumption, and leading order perturbation theory, showing that each of these approximations can break down in regimes where general self-consistent solutions still provide an accurate description. Lastly, we provide experiments in more realistic settings which demonstrate that the loss and kernel dynamics of CNNs at fixed feature learning strength is preserved across different widths on a CIFAR classification task.
Neural Networks as Kernel Learners: The Silent Alignment Effect
Oct 29, 2021
Neural networks in the lazy training regime converge to kernel machines. Can neural networks in the rich feature learning regime learn a kernel machine with a data-dependent kernel? We demonstrate that this can indeed happen due to a phenomenon we term silent alignment, which requires that the tangent kernel of a network evolves in eigenstructure while small and before the loss appreciably decreases, and grows only in overall scale afterwards. We show that such an effect takes place in homogenous neural networks with small initialization and whitened data. We provide an analytical treatment of this effect in the linear network case. In general, we find that the kernel develops a low-rank contribution in the early phase of training, and then evolves in overall scale, yielding a function equivalent to a kernel regression solution with the final network's tangent kernel. The early spectral learning of the kernel depends on both depth and on relative learning rates in each layer. We also demonstrate that non-whitened data can weaken the silent alignment effect.