Zohar Ringel

Noa Rubin, Inbar Seroussi, Zohar Ringel

A key property of deep neural networks (DNNs) is their ability to learn new features during training. This intriguing aspect of deep learning stands out most clearly in recently reported Grokking phenomena. While mainly reflected as a sudden increase in test accuracy, Grokking is also believed to be a beyond lazy-learning/Gaussian Process (GP) phenomenon involving feature learning. Here we apply a recent development in the theory of feature learning, the adaptive kernel approach, to two teacher-student models with cubic-polynomial and modular addition teachers. We provide analytical predictions on feature learning and Grokking properties of these models and demonstrate a mapping between Grokking and the theory of phase transitions. We show that after Grokking, the state of the DNN is analogous to the mixed phase following a first-order phase transition. In this mixed phase, the DNN generates useful internal representations of the teacher that are sharply distinct from those before the transition.

Inbar Seroussi, Alexander A. Alemi, Moritz Helias et al.

State-of-the-art neural networks require extreme computational power to train. It is therefore natural to wonder whether they are optimally trained. Here we apply a recent advancement in stochastic thermodynamics which allows bounding the speed at which one can go from the initial weight distribution to the final distribution of the fully trained network, based on the ratio of their Wasserstein-2 distance and the entropy production rate of the dynamical process connecting them. Considering both gradient-flow and Langevin training dynamics, we provide analytical expressions for these speed limits for linear and linearizable neural networks e.g. Neural Tangent Kernel (NTK). Remarkably, given some plausible scaling assumptions on the NTK spectra and spectral decomposition of the labels -- learning is optimal in a scaling sense. Our results are consistent with small-scale experiments with Convolutional Neural Networks (CNNs) and Fully Connected Neural networks (FCNs) on CIFAR-10, showing a short highly non-optimal regime followed by a longer optimal regime.

Inbar Seroussi, Asaf Miron, Zohar Ringel

Physically informed neural networks (PINNs) are a promising emerging method for solving differential equations. As in many other deep learning approaches, the choice of PINN design and training protocol requires careful craftsmanship. Here, we suggest a comprehensive theoretical framework that sheds light on this important problem. Leveraging an equivalence between infinitely over-parameterized neural networks and Gaussian process regression (GPR), we derive an integro-differential equation that governs PINN prediction in the large data-set limit -- the neurally-informed equation. This equation augments the original one by a kernel term reflecting architecture choices and allows quantifying implicit bias induced by the network via a spectral decomposition of the source term in the original differential equation.

Orit Davidovich, Zohar Ringel

We formally define Algorithmic Capture (i.e., ``grokking'' an algorithm) as the ability of a neural network to generalize to arbitrary problem sizes ($T$) with controllable error and minimal sample adaptation, distinguishing true algorithmic learning from statistical interpolation. By analyzing infinite-width transformers in both the lazy and rich regimes, we derive upper bounds on the inference-time computational complexity of the functions these networks can learn. We show that despite their universal expressivity, transformers possess an inductive bias towards low-complexity algorithms within the Efficient Polynomial Time Heuristic Scheme (EPTHS) class. This bias effectively prevents them from capturing higher-complexity algorithms, while allowing success on simpler tasks like search, copy, and sort.

Noa Rubin, Orit Davidovich, Zohar Ringel

Two pressing topics in the theory of deep learning are the interpretation of feature learning mechanisms and the determination of implicit bias of networks in the rich regime. Current theories of rich feature learning effects revolve around networks with one or two trainable layers or deep linear networks. Furthermore, even under such limiting settings, predictions often appear in the form of high-dimensional non-linear equations, which require computationally intensive numerical solutions. Given the many details that go into defining a deep learning problem, this analytical complexity is a significant and often unavoidable challenge. Here, we propose a powerful heuristic route for predicting the data and width scales at which various patterns of feature learning emerge. This form of scale analysis is considerably simpler than such exact theories and reproduces the scaling exponents of various known results. In addition, we make novel predictions on complex toy architectures, such as three-layer non-linear networks and attention heads, thus extending the scope of first-principle theories of deep learning.

Itay Lavie, Guy Gur-Ari, Zohar Ringel

We study inductive bias in Transformers in the infinitely over-parameterized Gaussian process limit and argue transformers tend to be biased towards more permutation symmetric functions in sequence space. We show that the representation theory of the symmetric group can be used to give quantitative analytical predictions when the dataset is symmetric to permutations between tokens. We present a simplified transformer block and solve the model at the limit, including accurate predictions for the learning curves and network outputs. We show that in common setups, one can derive tight bounds in the form of a scaling law for the learnability as a function of the context length. Finally, we argue WikiText dataset, does indeed possess a degree of permutation symmetry.

Noa Rubin, Kirsten Fischer, Javed Lindner et al.

Feature learning in neural networks is crucial for their expressive power and inductive biases, motivating various theoretical approaches. Some approaches describe network behavior after training through a change in kernel scale from initialization, resulting in a generalization power comparable to a Gaussian process. Conversely, in other approaches training results in the adaptation of the kernel to the data, involving directional changes to the kernel. The relationship and respective strengths of these two views have so far remained unresolved. This work presents a theoretical framework of multi-scale adaptive feature learning bridging these two views. Using methods from statistical mechanics, we derive analytical expressions for network output statistics which are valid across scaling regimes and in the continuum between them. A systematic expansion of the network's probability distribution reveals that mean-field scaling requires only a saddle-point approximation, while standard scaling necessitates additional correction terms. Remarkably, we find across regimes that kernel adaptation can be reduced to an effective kernel rescaling when predicting the mean network output in the special case of a linear network. However, for linear and non-linear networks, the multi-scale adaptive approach captures directional feature learning effects, providing richer insights than what could be recovered from a rescaling of the kernel alone.

Zohar Ringel, Noa Rubin, Edo Mor et al.

Deep learning algorithms have made incredible strides in the past decade, yet due to their complexity, the science of deep learning remains in its early stages. Being an experimentally driven field, it is natural to seek a theory of deep learning within the physics paradigm. As deep learning is largely about learning functions and distributions over functions, statistical field theory, a rich and versatile toolbox for tackling complex distributions over functions (fields) is an obvious choice of formalism. Research efforts carried out in the past few years have demonstrated the ability of field theory to provide useful insights on generalization, implicit bias, and feature learning effects. Here we provide a pedagogical review of this emerging line of research.

Itay Lavie, Zohar Ringel

Kernel ridge regression (KRR) and Gaussian processes (GPs) are fundamental tools in statistics and machine learning, with recent applications to highly over-parameterized deep neural networks. The ability of these tools to learn a target function is directly related to the eigenvalues of their kernel sampled on the input data distribution. Targets that have support on higher eigenvalues are more learnable. However, solving such eigenvalue problems on real-world data remains a challenge. Here, we consider cross-dataset learnability and show that one may use eigenvalues and eigenfunctions associated with highly idealized data measures to reveal spectral bias on complex datasets and bound learnability on real-world data. This allows us to leverage various symmetries that realistic kernels manifest to unravel their spectral bias.

Jessica N. Howard, Ro Jefferson, Anindita Maiti et al.

Separating relevant and irrelevant information is key to any modeling process or scientific inquiry. Theoretical physics offers a powerful tool for achieving this in the form of the renormalization group (RG). Here we demonstrate a practical approach to performing Wilsonian RG in the context of Gaussian Process (GP) Regression. We systematically integrate out the unlearnable modes of the GP kernel, thereby obtaining an RG flow of the GP in which the data sets the IR scale. In simple cases, this results in a universal flow of the ridge parameter, which becomes input-dependent in the richer scenario in which non-Gaussianities are included. In addition to being analytically tractable, this approach goes beyond structural analogies between RG and neural networks by providing a natural connection between RG flow and learnable vs. unlearnable modes. Studying such flows may improve our understanding of feature learning in deep neural networks, and enable us to identify potential universality classes in these models.

Inbar Seroussi, Gadi Naveh, Zohar Ringel

Deep neural networks (DNNs) are powerful tools for compressing and distilling information. Their scale and complexity, often involving billions of inter-dependent parameters, render direct microscopic analysis difficult. Under such circumstances, a common strategy is to identify slow variables that average the erratic behavior of the fast microscopic variables. Here, we identify a similar separation of scales occurring in fully trained finitely over-parameterized deep convolutional neural networks (CNNs) and fully connected networks (FCNs). Specifically, we show that DNN layers couple only through the second moment (kernels) of their activations and pre-activations. Moreover, the latter fluctuates in a nearly Gaussian manner. For infinite width DNNs, these kernels are inert, while for finite ones they adapt to the data and yield a tractable data-aware Gaussian Process. The resulting thermodynamic theory of deep learning yields accurate predictions in various settings. In addition, it provides new ways of analyzing and understanding DNNs in general.

Gadi Naveh, Zohar Ringel

Deep neural networks (DNNs) in the infinite width/channel limit have received much attention recently, as they provide a clear analytical window to deep learning via mappings to Gaussian Processes (GPs). Despite its theoretical appeal, this viewpoint lacks a crucial ingredient of deep learning in finite DNNs, laying at the heart of their success -- feature learning. Here we consider DNNs trained with noisy gradient descent on a large training set and derive a self consistent Gaussian Process theory accounting for strong finite-DNN and feature learning effects. Applying this to a toy model of a two-layer linear convolutional neural network (CNN) shows good agreement with experiments. We further identify, both analytical and numerically, a sharp transition between a feature learning regime and a lazy learning regime in this model. Strong finite-DNN effects are also derived for a non-linear two-layer fully connected network. Our self consistent theory provides a rich and versatile analytical framework for studying feature learning and other non-lazy effects in finite DNNs.

Gadi Naveh, Oded Ben-David, Haim Sompolinsky et al.

A recent line of works studied wide deep neural networks (DNNs) by approximating them as Gaussian Processes (GPs). A DNN trained with gradient flow was shown to map to a GP governed by the Neural Tangent Kernel (NTK), whereas earlier works showed that a DNN with an i.i.d. prior over its weights maps to the so-called Neural Network Gaussian Process (NNGP). Here we consider a DNN training protocol, involving noise, weight decay and finite width, whose outcome corresponds to a certain non-Gaussian stochastic process. An analytical framework is then introduced to analyze this non-Gaussian process, whose deviation from a GP is controlled by the finite width. Our contribution is three-fold: (i) In the infinite width limit, we establish a correspondence between DNNs trained with noisy gradients and the NNGP, not the NTK. (ii) We provide a general analytical form for the finite width correction (FWC) for DNNs with arbitrary activation functions and depth and use it to predict the outputs of empirical finite networks with high accuracy. Analyzing the FWC behavior as a function of $n$, the training set size, we find that it is negligible for both the very small $n$ regime, and, surprisingly, for the large $n$ regime (where the GP error scales as $O(1/n)$). (iii) We flesh out algebraically how these FWCs can improve the performance of finite convolutional neural networks (CNNs) relative to their GP counterparts on image classification tasks.

Omry Cohen, Or Malka, Zohar Ringel

In the past decade, deep neural networks (DNNs) came to the fore as the leading machine learning algorithms for a variety of tasks. Their raise was founded on market needs and engineering craftsmanship, the latter based more on trial and error than on theory. While still far behind the application forefront, the theoretical study of DNNs has recently made important advancements in analyzing the highly over-parameterized regime where some exact results have been obtained. Leveraging these ideas and adopting a more physics-like approach, here we construct a versatile field-theory formalism for supervised deep learning, involving renormalization group, Feynman diagrams and replicas. In particular we show that our approach leads to highly accurate predictions of learning curves of truly deep DNNs trained on polynomial regression tasks and that these predictions can be used for efficient hyper-parameter optimization. In addition, they explain how DNNs generalize well despite being highly over-parameterized, this due to an entropic bias to simple functions which, for the case of fully-connected DNNs with data sampled on the hypersphere, are low order polynomials in the input vector. Being a complex interacting system of artificial neurons, we believe that such tools and methodologies borrowed from condensed matter physics would prove essential for obtaining an accurate quantitative understanding of deep learning.

Oded Ben-David, Zohar Ringel

A fundamental question in deep learning concerns the role played by individual layers in a deep neural network (DNN) and the transferable properties of the data representations which they learn. To the extent that layers have clear roles, one should be able to optimize them separately using layer-wise loss functions. Such loss functions would describe what is the set of good data representations at each depth of the network and provide a target for layer-wise greedy optimization (LEGO). Here we derive a novel correspondence between Gaussian Processes and SGD trained deep neural networks. Leveraging this correspondence, we derive the Deep Gaussian Layer-wise loss functions (DGLs) which, we believe, are the first supervised layer-wise loss functions which are both explicit and competitive in terms of accuracy. Being highly structured and symmetric, the DGLs provide a promising analytic route to understanding the internal representations generated by DNNs.

Zohar Ringel, Rodrigo de Bem

In this paper we approach two relevant deep learning topics: i) tackling of graph structured input data and ii) a better understanding and analysis of deep networks and related learning algorithms. With this in mind we focus on the topological classification of reachability in a particular subset of planar graphs (Mazes). Doing so, we are able to model the topology of data while staying in Euclidean space, thus allowing its processing with standard CNN architectures. We suggest a suitable architecture for this problem and show that it can express a perfect solution to the classification task. The shape of the cost function around this solution is also derived and, remarkably, does not depend on the size of the maze in the large maze limit. Responsible for this behavior are rare events in the dataset which strongly regulate the shape of the cost function near this global minimum. We further identify an obstacle to learning in the form of poorly performing local minima in which the network chooses to ignore some of the inputs. We further support our claims with training experiments and numerical analysis of the cost function on networks with up to $128$ layers.

Maciej Koch-Janusz, Zohar Ringel

Physical systems differring in their microscopic details often display strikingly similar behaviour when probed at macroscopic scales. Those universal properties, largely determining their physical characteristics, are revealed by the powerful renormalization group (RG) procedure, which systematically retains "slow" degrees of freedom and integrates out the rest. However, the important degrees of freedom may be difficult to identify. Here we demonstrate a machine learning algorithm capable of identifying the relevant degrees of freedom and executing RG steps iteratively without any prior knowledge about the system. We introduce an artificial neural network based on a model-independent, information-theoretic characterization of a real-space RG procedure, performing this task. We apply the algorithm to classical statistical physics problems in one and two dimensions. We demonstrate RG flow and extract the Ising critical exponent. Our results demonstrate that machine learning techniques can extract abstract physical concepts and consequently become an integral part of theory- and model-building.