arXiv subjects

Arthur Jacot

Publications and source records attributed to Arthur Jacot.

At least 19 recordsLinked to original sources

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

Polynomial Speedup in Diffusion Models with the Multilevel Euler-Maruyama Method

We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators $f^1,\dots,f^k$ to the drift $f$ with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate $f^k$ and many evaluations of the less costly $f^1,\dots,f^{k-1}$. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires $\epsilon^{-\gamma}$ compute to be $\epsilon$-approximated for some $\gamma>2$, then ML-EM $\epsilon$-approximates the solution of the SDE with $\epsilon^{-\gamma}$ compute, improving over the traditional EM rate of $\epsilon^{-\gamma-1}$. In other terms it allows us to solve the SDE at the same cost as a single evaluation of the drift. In the context of diffusion models, the different levels $f^{1},\dots,f^{k}$ are obtained by training UNets of increasing sizes, and ML-EM allows us to perform sampling with the equivalent of a single evaluation of the largest UNet. Our numerical experiments confirm our theory: we obtain up to fourfold speedups for image generation on the CelebA dataset downscaled to 64x64, where we measure a $\gamma\approx2.5$. Given that this is a polynomial speedup, we expect even stronger speedups in practical applications which involve orders of magnitude larger networks.

cs.LG↗

arXiv2025

Deep Learning as a Convex Paradigm of Computation: Minimizing Circuit Size with ResNets

This paper argues that DNNs implement a computational Occam's razor -- finding the `simplest' algorithm that fits the data -- and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $\epsilon$-approximated with a binary circuit of size at most $c\epsilon^{-\gamma}$ becomes convex in the `Harder than Monte Carlo' (HTMC) regime, when $\gamma>2$, allowing for the definition of a HTMC norm on functions. In parallel one can define a complexity measure on the parameters of a ResNets (a weighted $\ell_1$ norm of the parameters), which induce a `ResNet norm' on functions. The HTMC and ResNet norms can then be related by an almost matching sandwich bound. Thus minimizing this ResNet norm is equivalent to finding a circuit that fits the data with an almost minimal number of nodes (within a power of 2 of being optimal). ResNets thus appear as an alternative model for computation of real functions, better adapted to the HTMC regime and its convexity.

stat.ML↗

arXiv2025

Saddle-To-Saddle Dynamics in Deep ReLU Networks: Low-Rank Bias in the First Saddle Escape

When a deep ReLU network is initialized with small weights, gradient descent (GD) is at first dominated by the saddle at the origin in parameter space. We study the so-called escape directions along which GD leaves the origin, which play a similar role as the eigenvectors of the Hessian for strict saddles. We show that the optimal escape direction features a low-rank bias in its deeper layers: the first singular value of the $\ell$-th layer weight matrix is at least $\ell^{\frac{1}{4}}$ larger than any other singular value. We also prove a number of related results about these escape directions. We suggest that deep ReLU networks exhibit saddle-to-saddle dynamics, with GD visiting a sequence of saddles with increasing bottleneck rank (Jacot, 2023).

cs.LG↗

arXiv2024

Shallow diffusion networks provably learn hidden low-dimensional structure

Diffusion-based generative models provide a powerful framework for learning to sample from a complex target distribution. The remarkable empirical success of these models applied to high-dimensional signals, including images and video, stands in stark contrast to classical results highlighting the curse of dimensionality for distribution recovery. In this work, we take a step towards understanding this gap through a careful analysis of learning diffusion models over the Barron space of single layer neural networks. In particular, we show that these shallow models provably adapt to simple forms of low dimensional structure, thereby avoiding the curse of dimensionality. We combine our results with recent analyses of sampling with diffusion models to provide an end-to-end sample complexity bound for learning to sample from structured distributions. Importantly, our results do not require specialized architectures tailored to particular latent structures, and instead rely on the low-index structure of the Barron space to adapt to the underlying distribution.

cs.LG↗

arXiv2024

Wide Neural Networks Trained with Weight Decay Provably Exhibit Neural Collapse

Deep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a highly symmetric geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical research aimed at proving the emergence of neural collapse, mostly focusing on the unconstrained features model. Here, the features of the penultimate layer are free variables, which makes the model data-agnostic and, hence, puts into question its ability to capture DNN training. Our work addresses the issue, moving away from unconstrained features and studying DNNs that end with at least two linear layers. We first prove generic guarantees on neural collapse that assume (i) low training error and balancedness of the linear layers (for within-class variability collapse), and (ii) bounded conditioning of the features before the linear part (for orthogonality of class-means, as well as their alignment with weight matrices). We then show that such assumptions hold for gradient descent training with weight decay: (i) for networks with a wide first layer, we prove low training error and balancedness, and (ii) for solutions that are either nearly optimal or stable under large learning rates, we additionally prove the bounded conditioning. Taken together, our results are the first to show neural collapse in the end-to-end training of DNNs.

cs.LG↗

arXiv2024

How DNNs break the Curse of Dimensionality: Compositionality and Symmetry Learning

We show that deep neural networks (DNNs) can efficiently learn any composition of functions with bounded $F_{1}$-norm, which allows DNNs to break the curse of dimensionality in ways that shallow networks cannot. More specifically, we derive a generalization bound that combines a covering number argument for compositionality, and the $F_{1}$-norm (or the related Barron norm) for large width adaptivity. We show that the global minimizer of the regularized loss of DNNs can fit for example the composition of two functions $f^{*}=h\circ g$ from a small number of observations, assuming $g$ is smooth/regular and reduces the dimensionality (e.g. $g$ could be the quotient map of the symmetries of $f^{*}$), so that $h$ can be learned in spite of its low regularity. The measures of regularity we consider is the Sobolev norm with different levels of differentiability, which is well adapted to the $F_{1}$ norm. We compute scaling laws empirically and observe phase transitions depending on whether $g$ or $h$ is harder to learn, as predicted by our theory.

stat.ML↗

arXiv2024

Hamiltonian Mechanics of Feature Learning: Bottleneck Structure in Leaky ResNets

We study Leaky ResNets, which interpolate between ResNets and Fully-Connected nets depending on an 'effective depth' hyper-parameter $\tilde{L}$. In the infinite depth limit, we study 'representation geodesics' $A_{p}$: continuous paths in representation space (similar to NeuralODEs) from input $p=0$ to output $p=1$ that minimize the parameter norm of the network. We give a Lagrangian and Hamiltonian reformulation, which highlight the importance of two terms: a kinetic energy which favors small layer derivatives $\partial_{p}A_{p}$ and a potential energy that favors low-dimensional representations, as measured by the 'Cost of Identity'. The balance between these two forces offers an intuitive understanding of feature learning in ResNets. We leverage this intuition to explain the emergence of a bottleneck structure, as observed in previous work: for large $\tilde{L}$ the potential energy dominates and leads to a separation of timescales, where the representation jumps rapidly from the high dimensional inputs to a low-dimensional representation, move slowly inside the space of low-dimensional representations, before jumping back to the potentially high-dimensional outputs. Inspired by this phenomenon, we train with an adaptive layer step-size to adapt to the separation of timescales.

stat.ML↗

arXiv2024

Mixed Dynamics In Linear Networks: Unifying the Lazy and Active Regimes

The training dynamics of linear networks are well studied in two distinct setups: the lazy regime and balanced/active regime, depending on the initialization and width of the network. We provide a surprisingly simple unifying formula for the evolution of the learned matrix that contains as special cases both lazy and balanced regimes but also a mixed regime in between the two. In the mixed regime, a part of the network is lazy while the other is balanced. More precisely the network is lazy along singular values that are below a certain threshold and balanced along those that are above the same threshold. At initialization, all singular values are lazy, allowing for the network to align itself with the task, so that later in time, when some of the singular value cross the threshold and become active they will converge rapidly (convergence in the balanced regime is notoriously difficult in the absence of alignment). The mixed regime is the `best of both worlds': it converges from any random initialization (in contrast to balanced dynamics which require special initialization), and has a low rank bias (absent in the lazy dynamics). This allows us to prove an almost complete phase diagram of training behavior as a function of the variance at initialization and the width, for a MSE training task.

cs.LG↗

arXiv2024

Which Frequencies do CNNs Need? Emergent Bottleneck Structure in Feature Learning

We describe the emergence of a Convolution Bottleneck (CBN) structure in CNNs, where the network uses its first few layers to transform the input representation into a representation that is supported only along a few frequencies and channels, before using the last few layers to map back to the outputs. We define the CBN rank, which describes the number and type of frequencies that are kept inside the bottleneck, and partially prove that the parameter norm required to represent a function $f$ scales as depth times the CBN rank $f$. We also show that the parameter norm depends at next order on the regularity of $f$. We show that any network with almost optimal parameter norm will exhibit a CBN structure in both the weights and - under the assumption that the network is stable under large learning rate - the activations, which motivates the common practice of down-sampling; and we verify that the CBN results still hold with down-sampling. Finally we use the CBN structure to interpret the functions learned by CNNs on a number of tasks.

cs.LG↗

arXiv2023

Bottleneck Structure in Learned Features: Low-Dimension vs Regularity Tradeoff

Previous work has shown that DNNs with large depth $L$ and $L_{2}$-regularization are biased towards learning low-dimensional representations of the inputs, which can be interpreted as minimizing a notion of rank $R^{(0)}(f)$ of the learned function $f$, conjectured to be the Bottleneck rank. We compute finite depth corrections to this result, revealing a measure $R^{(1)}$ of regularity which bounds the pseudo-determinant of the Jacobian $\left|Jf(x)\right|_{+}$ and is subadditive under composition and addition. This formalizes a balance between learning low-dimensional representations and minimizing complexity/irregularity in the feature maps, allowing the network to learn the `right' inner dimension. Finally, we prove the conjectured bottleneck structure in the learned features as $L\to\infty$: for large depths, almost all hidden representations are approximately $R^{(0)}(f)$-dimensional, and almost all weight matrices $W_{\ell}$ have $R^{(0)}(f)$ singular values close to 1 while the others are $O(L^{-\frac{1}{2}})$. Interestingly, the use of large learning rates is required to guarantee an order $O(L)$ NTK which in turns guarantees infinite depth convergence of the representations of almost all layers.

cs.LG↗

arXiv2023

Implicit bias of SGD in $L_{2}$-regularized linear DNNs: One-way jumps from high to low rank

The $L_{2}$-regularized loss of Deep Linear Networks (DLNs) with more than one hidden layers has multiple local minima, corresponding to matrices with different ranks. In tasks such as matrix completion, the goal is to converge to the local minimum with the smallest rank that still fits the training data. While rank-underestimating minima can be avoided since they do not fit the data, GD might get stuck at rank-overestimating minima. We show that with SGD, there is always a probability to jump from a higher rank minimum to a lower rank one, but the probability of jumping back is zero. More precisely, we define a sequence of sets $B_{1}\subset B_{2}\subset\cdots\subset B_{R}$ so that $B_{r}$ contains all minima of rank $r$ or less (and not more) that are absorbing for small enough ridge parameters $\lambda$ and learning rates $\eta$: SGD has prob. 0 of leaving $B_{r}$, and from any starting point there is a non-zero prob. for SGD to go in $B_{r}$.

cs.LG↗

arXiv2022

Implicit Bias of Large Depth Networks: a Notion of Rank for Nonlinear Functions

We show that the representation cost of fully connected neural networks with homogeneous nonlinearities - which describes the implicit bias in function space of networks with $L_2$-regularization or with losses such as the cross-entropy - converges as the depth of the network goes to infinity to a notion of rank over nonlinear functions. We then inquire under which conditions the global minima of the loss recover the `true' rank of the data: we show that for too large depths the global minimum will be approximately rank 1 (underestimating the rank); we then argue that there is a range of depths which grows with the number of datapoints where the true rank is recovered. Finally, we discuss the effect of the rank of a classifier on the topology of the resulting class boundaries and show that autoencoders with optimal nonlinear rank are naturally denoising.

stat.ML↗

arXiv2022

Feature Learning in $L_{2}$-regularized DNNs: Attraction/Repulsion and Sparsity

We study the loss surface of DNNs with $L_{2}$ regularization. We show that the loss in terms of the parameters can be reformulated into a loss in terms of the layerwise activations $Z_{\ell}$ of the training set. This reformulation reveals the dynamics behind feature learning: each hidden representations $Z_{\ell}$ are optimal w.r.t. to an attraction/repulsion problem and interpolate between the input and output representations, keeping as little information from the input as necessary to construct the activation of the next layer. For positively homogeneous non-linearities, the loss can be further reformulated in terms of the covariances of the hidden representations, which takes the form of a partially convex optimization over a convex cone. This second reformulation allows us to prove a sparsity result for homogeneous DNNs: any local minimum of the $L_{2}$-regularized loss can be achieved with at most $N(N+1)$ neurons in each hidden layer (where $N$ is the size of the training set). We show that this bound is tight by giving an example of a local minimum that requires $N^{2}/4$ hidden neurons. But we also observe numerically that in more traditional settings much less than $N^{2}$ neurons are required to reach the minima.

stat.ML↗

arXiv2021

Understanding Layer-wise Contributions in Deep Neural Networks through Spectral Analysis

Spectral analysis is a powerful tool, decomposing any function into simpler parts. In machine learning, Mercer's theorem generalizes this idea, providing for any kernel and input distribution a natural basis of functions of increasing frequency. More recently, several works have extended this analysis to deep neural networks through the framework of Neural Tangent Kernel. In this work, we analyze the layer-wise spectral bias of Deep Neural Networks and relate it to the contributions of different layers in the reduction of generalization error for a given target function. We utilize the properties of Hermite polynomials and Spherical Harmonics to prove that initial layers exhibit a larger bias towards high-frequency functions defined on the unit sphere. We further provide empirical results validating our theory in high dimensional datasets for Deep Neural Networks.

cs.LG↗

arXiv2021

Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

The dynamics of Deep Linear Networks (DLNs) is dramatically affected by the variance $\sigma^2$ of the parameters at initialization $\theta_0$. For DLNs of width $w$, we show a phase transition w.r.t. the scaling $\gamma$ of the variance $\sigma^2=w^{-\gamma}$ as $w\to\infty$: for large variance ($\gamma<1$), $\theta_0$ is very close to a global minimum but far from any saddle point, and for small variance ($\gamma>1$), $\theta_0$ is close to a saddle point and far from any global minimum. While the first case corresponds to the well-studied NTK regime, the second case is less understood. This motivates the study of the case $\gamma \to +\infty$, where we conjecture a Saddle-to-Saddle dynamics: throughout training, gradient descent visits the neighborhoods of a sequence of saddles, each corresponding to linear maps of increasing rank, until reaching a sparse global minimum. We support this conjecture with a theorem for the dynamics between the first two saddles, as well as some numerical experiments.

stat.ML↗

arXiv2021

DNN-Based Topology Optimisation: Spatial Invariance and Neural Tangent Kernel

We study the Solid Isotropic Material Penalisation (SIMP) method with a density field generated by a fully-connected neural network, taking the coordinates as inputs. In the large width limit, we show that the use of DNNs leads to a filtering effect similar to traditional filtering techniques for SIMP, with a filter described by the Neural Tangent Kernel (NTK). This filter is however not invariant under translation, leading to visual artifacts and non-optimal shapes. We propose two embeddings of the input coordinates, which lead to (approximate) spatial invariance of the NTK and of the filter. We empirically confirm our theoretical observations and study how the filter size is affected by the architecture of the network. Our solution can easily be applied to any other coordinates-based generation method.

stat.ML↗

arXiv2021

Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances

We study how permutation symmetries in overparameterized multi-layer neural networks generate `symmetry-induced' critical points. Assuming a network with $ L $ layers of minimal widths $ r_1^*, \ldots, r_{L-1}^* $ reaches a zero-loss minimum at $ r_1^*! \cdots r_{L-1}^*! $ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $ r^*+ h =: m $ we explicitly describe the manifold of global minima: it consists of $ T(r^*, m) $ affine subspaces of dimension at least $ h $ that are connected to one another. For a network of width $m$, we identify the number $G(r,m)$ of affine subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width $r<r^*$. Via a combinatorial analysis, we derive closed-form formulas for $ T $ and $ G $ and show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold in the mildly overparameterized regime (small $ h $) and vice versa in the vastly overparameterized regime ($h \gg r^*$). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.

cs.LG↗