arXiv subjects

Joan Bruna

Publications and source records attributed to Joan Bruna.

At least 37 records · Page 2Linked to original sources

arXiv2025

Distributional Associations vs In-Context Reasoning: A Study of Feed-forward and Attention Layers

Large language models have been successful at tasks involving basic forms of in-context reasoning, such as generating coherent language, as well as storing vast amounts of knowledge. At the core of the Transformer architecture behind such models are feed-forward and attention layers, which are often associated to knowledge and reasoning, respectively. In this paper, we study this distinction empirically and theoretically in a controlled synthetic setting where certain next-token predictions involve both distributional and in-context information. We find that feed-forward layers tend to learn simple distributional associations such as bigrams, while attention layers focus on in-context reasoning. Our theoretical analysis identifies the noise in the gradients as a key factor behind this discrepancy. Finally, we illustrate how similar disparities emerge in pre-trained models through ablations on the Pythia model family on simple reasoning tasks.

cs.LG↗

arXiv2024

Stochastic Optimal Control Matching

Stochastic optimal control, which has the goal of driving the behavior of noisy systems, is broadly applicable in science, engineering and artificial intelligence. Our work introduces Stochastic Optimal Control Matching (SOCM), a novel Iterative Diffusion Optimization (IDO) technique for stochastic optimal control that stems from the same philosophy as the conditional score matching loss for diffusion models. That is, the control is learned via a least squares problem by trying to fit a matching vector field. The training loss, which is closely connected to the cross-entropy loss, is optimized with respect to both the control function and a family of reparameterization matrices which appear in the matching vector field. The optimization with respect to the reparameterization matrices aims at minimizing the variance of the matching vector field. Experimentally, our algorithm achieves lower error than all the existing IDO techniques for stochastic optimal control for three out of four control problems, in some cases by an order of magnitude. The key idea underlying SOCM is the path-wise reparameterization trick, a novel technique that may be of independent interest. Code at https://github.com/facebookresearch/SOC-matching

math.OC↗

arXiv2024

On the Benefits of Rank in Attention Layers

Attention-based mechanisms are widely used in machine learning, most prominently in transformers. However, hyperparameters such as the rank of the attention matrices and the number of heads are scaled nearly the same way in all realizations of this architecture, without theoretical justification. In this work we show that there are dramatic trade-offs between the rank and number of heads of the attention mechanism. Specifically, we present a simple and natural target function that can be represented using a single full-rank attention head for any context length, but that cannot be approximated by low-rank attention unless the number of heads is exponential in the embedding dimension, even for short context lengths. Moreover, we prove that, for short context lengths, adding depth allows the target to be approximated by low-rank attention. For long contexts, we conjecture that full-rank attention is necessary. Finally, we present experiments with off-the-shelf transformers that validate our theoretical findings.

cs.LG↗

arXiv2024

Posterior Sampling with Denoising Oracles via Tilted Transport

Score-based diffusion models have significantly advanced high-dimensional data generation across various domains, by learning a denoising oracle (or score) from datasets. From a Bayesian perspective, they offer a realistic modeling of data priors and facilitate solving inverse problems through posterior sampling. Although many heuristic methods have been developed recently for this purpose, they lack the quantitative guarantees needed in many scientific applications. In this work, we introduce the \textit{tilted transport} technique, which leverages the quadratic structure of the log-likelihood in linear inverse problems in combination with the prior denoising oracle to transform the original posterior sampling problem into a new `boosted' posterior that is provably easier to sample from. We quantify the conditions under which this boosted posterior is strongly log-concave, highlighting the dependencies on the condition number of the measurement matrix and the signal-to-noise ratio. The resulting posterior sampling scheme is shown to reach the computational threshold predicted for sampling Ising models [Kunisky'23] with a direct analysis, and is further validated on high-dimensional Gaussian mixture models and scalar field $φ^4$ models.

cs.LG↗

arXiv2024

Computational-Statistical Gaps in Gaussian Single-Index Models

Single-Index Models are high-dimensional regression problems with planted structure, whereby labels depend on an unknown one-dimensional projection of the input via a generic, non-linear, and potentially non-deterministic transformation. As such, they encompass a broad class of statistical inference tasks, and provide a rich template to study statistical and computational trade-offs in the high-dimensional regime. While the information-theoretic sample complexity to recover the hidden direction is linear in the dimension $d$, we show that computationally efficient algorithms, both within the Statistical Query (SQ) and the Low-Degree Polynomial (LDP) framework, necessarily require $Ω(d^{k^\star/2})$ samples, where $k^\star$ is a "generative" exponent associated with the model that we explicitly characterize. Moreover, we show that this sample complexity is also sufficient, by establishing matching upper bounds using a partial-trace algorithm. Therefore, our results provide evidence of a sharp computational-to-statistical gap (under both the SQ and LDP class) whenever $k^\star>2$. To complete the study, we provide examples of smooth and Lipschitz deterministic target functions with arbitrarily large generative exponents $k^\star$.

cs.LG↗

arXiv2024

Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations

Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering applications. This work proposes Neural Galerkin schemes based on deep learning that generate training data with active learning for numerically solving high-dimensional partial differential equations. Neural Galerkin schemes build on the Dirac-Frenkel variational principle to train networks by minimizing the residual sequentially over time, which enables adaptively collecting new training data in a self-informed manner that is guided by the dynamics described by the partial differential equations. This is in contrast to other machine learning methods that aim to fit network parameters globally in time without taking into account training data acquisition. Our finding is that the active form of gathering training data of the proposed Neural Galerkin schemes is key for numerically realizing the expressive power of networks in high dimensions. Numerical experiments demonstrate that Neural Galerkin schemes have the potential to enable simulating phenomena and processes with many variables for which traditional and other deep-learning-based solvers fail, especially when features of the solutions evolve locally such as in high-dimensional wave propagation problems and interacting particle systems described by Fokker-Planck and kinetic equations.

math.NA↗

arXiv2023

Symmetry Breaking in Symmetric Tensor Decomposition

In this note, we consider the highly nonconvex optimization problem associated with computing the rank decomposition of symmetric tensors. We formulate the invariance properties of the loss function and show that critical points detected by standard gradient based methods are \emph{symmetry breaking} with respect to the target tensor. The phenomena, seen for different choices of target tensors and norms, make possible the use of recently developed analytic and algebraic tools for studying nonconvex optimization landscapes exhibiting symmetry breaking phenomena of similar nature.

math.OC↗

arXiv2023

On Learning Gaussian Multi-index Models with Gradient Flow

We study gradient flow on the multi-index regression problem for high-dimensional Gaussian data. Multi-index functions consist of a composition of an unknown low-rank linear projection and an arbitrary unknown, low-dimensional link function. As such, they constitute a natural template for feature learning in neural networks. We consider a two-timescale algorithm, whereby the low-dimensional link function is learnt with a non-parametric model infinitely faster than the subspace parametrizing the low-rank projection. By appropriately exploiting the matrix semigroup structure arising over the subspace correlation matrices, we establish global convergence of the resulting Grassmannian population gradient flow dynamics, and provide a quantitative description of its associated `saddle-to-saddle' dynamics. Notably, the timescales associated with each saddle can be explicitly characterized in terms of an appropriate Hermite decomposition of the target link function. In contrast with these positive results, we also show that the related \emph{planted} problem, where the link function is known and fixed, in fact has a rough optimization landscape, in which gradient flow dynamics might get trapped with high probability.

stat.ML↗

arXiv2023

A Neural Collapse Perspective on Feature Evolution in Graph Neural Networks

Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on node-wise classification, illustrated with community detection on stochastic block model graphs, and explore the feature evolution through the lens of the "Neural Collapse" (NC) phenomenon. When training instance-wise deep classifiers (e.g. for image classification) beyond the zero training error point, NC demonstrates a reduction in the deepest features' within-class variability and an increased alignment of their class means to certain symmetric structures. We start with an empirical study that shows that a decrease in within-class variability is also prevalent in the node-wise classification setting, however, not to the extent observed in the instance-wise case. Then, we theoretically study this distinction. Specifically, we show that even an "optimistic" mathematical model requires that the graphs obey a strict structural condition in order to possess a minimizer with exact collapse. Interestingly, this condition is viable also for heterophilic graphs and relates to recent empirical studies on settings with improved GNNs' generalization. Furthermore, by studying the gradient dynamics of the theoretical model, we provide reasoning for the partial collapse observed empirically. Finally, we present a study on the evolution of within- and between-class feature variability across layers of a well-trained GNN and contrast the behavior with spectral methods.

cs.LG↗

arXiv2023

Inverse Dynamics Pretraining Learns Good Representations for Multitask Imitation

In recent years, domains such as natural language processing and image recognition have popularized the paradigm of using large datasets to pretrain representations that can be effectively transferred to downstream tasks. In this work we evaluate how such a paradigm should be done in imitation learning, where both pretraining and finetuning data are trajectories collected by experts interacting with an unknown environment. Namely, we consider a setting where the pretraining corpus consists of multitask demonstrations and the task for each demonstration is set by an unobserved latent context variable. The goal is to use the pretraining corpus to learn a low dimensional representation of the high dimensional (e.g., visual) observation space which can be transferred to a novel context for finetuning on a limited dataset of demonstrations. Among a variety of possible pretraining objectives, we argue that inverse dynamics modeling -- i.e., predicting an action given the observations appearing before and after it in the demonstration -- is well-suited to this setting. We provide empirical evidence of this claim through evaluations on a variety of simulated visuomotor manipulation problems. While previous work has attempted various theoretical explanations regarding the benefit of inverse dynamics modeling, we find that these arguments are insufficient to explain the empirical advantages often observed in our settings, and so we derive a novel analysis using a simple but general environment model.

cs.LG↗

arXiv2023

On Single Index Models beyond Gaussian Data

Sparse high-dimensional functions have arisen as a rich framework to study the behavior of gradient-descent methods using shallow neural networks, showcasing their ability to perform feature learning beyond linear models. Amongst those functions, the simplest are single-index models $f(x) = ϕ( x \cdot θ^*)$, where the labels are generated by an arbitrary non-linear scalar link function $ϕ$ applied to an unknown one-dimensional projection $θ^*$ of the input data. By focusing on Gaussian data, several recent works have built a remarkable picture, where the so-called information exponent (related to the regularity of the link function) controls the required sample complexity. In essence, these tools exploit the stability and spherical symmetry of Gaussian distributions. In this work, building from the framework of \cite{arous2020online}, we explore extensions of this picture beyond the Gaussian setting, where both stability or symmetry might be violated. Focusing on the planted setting where $ϕ$ is known, our main results establish that Stochastic Gradient Descent can efficiently recover the unknown direction $θ^*$ in the high-dimensional regime, under assumptions that extend previous works \cite{yehudai2020learning,wu2022learning}.

cs.LG↗

arXiv2023

Symmetric Single Index Learning

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging. In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.

cs.LG↗

arXiv2023

Beyond the Edge of Stability via Two-step Gradient Updates

Gradient Descent (GD) is a powerful workhorse of modern machine learning thanks to its scalability and efficiency in high-dimensional spaces. Its ability to find local minimisers is only guaranteed for losses with Lipschitz gradients, where it can be seen as a `bona-fide' discretisation of an underlying gradient flow. Yet, many ML setups involving overparametrised models do not fall into this problem class, which has motivated research beyond the so-called ``Edge of Stability'' (EoS), where the step-size crosses the admissibility threshold inversely proportional to the Lipschitz constant above. Perhaps surprisingly, GD has been empirically observed to still converge regardless of local instability and oscillatory behavior. The incipient theoretical analysis of this phenomena has mainly focused in the overparametrised regime, where the effect of choosing a large learning rate may be associated to a `Sharpness-Minimisation' implicit regularisation within the manifold of minimisers, under appropriate asymptotic limits. In contrast, in this work we directly examine the conditions for such unstable convergence, focusing on simple, yet representative, learning problems, via analysis of two-step gradient updates. Specifically, we characterize a local condition involving third-order derivatives that guarantees existence and convergence to fixed points of the two-step updates, and leverage such property in a teacher-student setting, under population loss. Finally, starting from Matrix Factorization, we provide observations of period-2 orbit of GD in high-dimensional settings with intuition of its dynamics, along with exploration into more general settings.

cs.LG↗

arXiv2023

Reliable coarse-grained turbulent simulations through combined offline learning and neural emulation

Integration of machine learning (ML) models of unresolved dynamics into numerical simulations of fluid dynamics has been demonstrated to improve the accuracy of coarse resolution simulations. However, when trained in a purely offline mode, integrating ML models into the numerical scheme can lead to instabilities. In the context of a 2D, quasi-geostrophic turbulent system, we demonstrate that including an additional network in the loss function, which emulates the state of the system into the future, produces offline-trained ML models that capture important subgrid processes, with improved stability properties.

physics.flu-dyn↗

arXiv2023

Towards Antisymmetric Neural Ansatz Separation

We study separations between two fundamental models (or \emph{Ansätze}) of antisymmetric functions, that is, functions $f$ of the form $f(x_{σ(1)}, \ldots, x_{σ(N)}) = \text{sign}(σ)f(x_1, \ldots, x_N)$, where $σ$ is any permutation. These arise in the context of quantum chemistry, and are the basic modeling tool for wavefunctions of Fermionic systems. Specifically, we consider two popular antisymmetric Ansätze: the Slater representation, which leverages the alternating structure of determinants, and the Jastrow ansatz, which augments Slater determinants with a product by an arbitrary symmetric function. We construct an antisymmetric function in $N$ dimensions that can be efficiently expressed in Jastrow form, yet provably cannot be approximated by Slater determinants unless there are exponentially (in $N^2$) many terms. This represents the first explicit quantitative separation between these two Ansätze.

cs.LG↗

arXiv2023

Conditionally Strongly Log-Concave Generative Models

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the $φ^4$ model and weak lensing convergence maps with higher resolution than in previous works.

stat.ML↗

arXiv2023

On the equivalence between graph isomorphism testing and function approximation with GNNs

Graph Neural Networks (GNNs) have achieved much success on graph-structured data. In light of this, there have been increasing interests in studying their expressive power. One line of work studies the capability of GNNs to approximate permutation-invariant functions on graphs, and another focuses on the their power as tests for graph isomorphism. Our work connects these two perspectives and proves their equivalence. We further develop a framework of the expressive power of GNNs that incorporates both of these viewpoints using the language of sigma-algebra, through which we compare the expressive power of different types of GNNs together with other graph isomorphism tests. In particular, we prove that the second-order Invariant Graph Network fails to distinguish non-isomorphic regular graphs with the same degree. Then, we extend it to a new architecture, Ring-GNN, which succeeds in distinguishing these graphs and achieves good performances on real-world datasets.

cs.LG↗

arXiv2023

On Non-Linear operators for Geometric Deep Learning

This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_ω(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_ω(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.

cs.LG↗