Towards Understanding the Universality of Transformers for Next-Token Prediction

Causal Transformers are trained to predict the next token for a given context. While it is widely accepted that self-attention is crucial for encoding the causal structure of sequences, the precise underlying mechanism behind this in-context autoregressive learning ability remains unclear. In this paper, we take a step towards understanding this phenomenon by studying the approximation ability of Transformers for next-token prediction. Specifically, we explore the capacity of causal Transformers to predict the next token $x_{t+1}$ given an autoregressive sequence $(x_1, \dots, x_t)$ as a prompt, where $ x_{t+1} = f(x_t) $, and $ f $ is a context-dependent function that varies with each sequence. On the theoretical side, we focus on specific instances, namely when $ f $ is linear or when $ (x_t)_{t \geq 1} $ is periodic. We explicitly construct a Transformer (with linear, exponential, or softmax attention) that learns the mapping $f$ in-context through a causal kernel descent method. The causal kernel descent method we propose provably estimates $x_{t+1} $ based solely on past and current observations $ (x_1, \dots, x_t) $, with connections to the Kaczmarz algorithm in Hilbert spaces. We present experimental results that validate our theoretical findings and suggest their applicability to more general mappings $f$.

Transformers are Universal In-context Learners

Transformers are deep architectures that define "in-context mappings" which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for vision transformers). This work studies in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically and uniformly address the expressivity of these architectures, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens (discrete for a finite number of tokens). The related notion of smoothness corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLP layers between multi-head attention layers is also explicitly controlled.

Keep the Momentum: Conservation Laws beyond Euclidean Gradient Flows

Conservation laws are well-established in the context of Euclidean gradient flow dynamics, notably for linear or ReLU neural network training. Yet, their existence and principles for non-Euclidean geometries and momentum-based dynamics remain largely unknown. In this paper, we characterize "all" conservation laws in this general setting. In stark contrast to the case of gradient flows, we prove that the conservation laws for momentum-based dynamics exhibit temporal dependence. Additionally, we often observe a "conservation loss" when transitioning from gradient flow to momentum dynamics. Specifically, for linear networks, our framework allows us to identify all momentum conservation laws, which are less numerous than in the gradient flow case except in sufficiently over-parameterized regimes. With ReLU networks, no conservation law remains. This phenomenon also manifests in non-Euclidean metrics, used e.g. for Nonnegative Matrix Factorization (NMF): all conservation laws can be determined in the gradient flow context, yet none persists in the momentum case.

Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transport

We study the convergence of gradient flow for the training of deep neural networks. If Residual Neural Networks are a popular example of very deep architectures, their training constitutes a challenging optimization problem due notably to the non-convexity and the non-coercivity of the objective. Yet, in applications, those tasks are successfully solved by simple optimization algorithms such as gradient descent. To better understand this phenomenon, we focus here on a ``mean-field'' model of infinitely deep and arbitrarily wide ResNet, parameterized by probability measures over the product set of layers and parameters and with constant marginal on the set of layers. Indeed, in the case of shallow neural networks, mean field models have proven to benefit from simplified loss-landscapes and good theoretical guarantees when trained with gradient flow for the Wasserstein metric on the set of probability measures. Motivated by this approach, we propose to train our model with gradient flow w.r.t. the conditional Optimal Transport distance: a restriction of the classical Wasserstein distance which enforces our marginal condition. Relying on the theory of gradient flows in metric spaces we first show the well-posedness of the gradient flow equation and its consistency with the training of ResNets at finite width. Performing a local Polyak-\L{}ojasiewicz analysis, we then show convergence of the gradient flow for well-chosen initializations: if the number of features is finite but sufficiently large and the risk is sufficiently small at initialization, the gradient flow converges towards a global minimizer. This is the first result of this type for infinitely deep and arbitrarily wide ResNets.

Enhancing Hypergradients Estimation: A Study of Preconditioning and Reparameterization

Bilevel optimization aims to optimize an outer objective function that depends on the solution to an inner optimization problem. It is routinely used in Machine Learning, notably for hyperparameter tuning. The conventional method to compute the so-called hypergradient of the outer problem is to use the Implicit Function Theorem (IFT). As a function of the error of the inner problem resolution, we study the error of the IFT method. We analyze two strategies to reduce this error: preconditioning the IFT formula and reparameterizing the inner problem. We give a detailed account of the impact of these two modifications on the error, highlighting the role played by higher-order derivatives of the functionals at stake. Our theoretical findings explain when super efficiency, namely reaching an error on the hypergradient that depends quadratically on the error on the inner problem, is achievable and compare the two approaches when this is impossible. Numerical evaluations on hyperparameter tuning for regression problems substantiate our theoretical findings.

How do Transformers perform In-Context Autoregressive Learning?

Transformers have achieved state-of-the-art performance in language modeling tasks. However, the reasons behind their tremendous success are still unclear. In this paper, towards a better understanding, we train a Transformer model on a simple next token prediction task, where sequences are generated as a first-order autoregressive process $s_{t+1} = W s_t$. We show how a trained Transformer predicts the next token by first learning $W$ in-context, then applying a prediction mapping. We call the resulting procedure in-context autoregressive learning. More precisely, focusing on commuting orthogonal matrices $W$, we first show that a trained one-layer linear Transformer implements one step of gradient descent for the minimization of an inner objective function, when considering augmented tokens. When the tokens are not augmented, we characterize the global minima of a one-layer diagonal linear multi-head Transformer. Importantly, we exhibit orthogonality between heads and show that positional encoding captures trigonometric relations in the data. On the experimental side, we consider the general case of non-commuting orthogonal matrices and generalize our theoretical findings.

Understanding the Regularity of Self-Attention with Optimal Transport

Transformers and their multi-head attention mechanism have completely changed the machine learning landscape in just a few years, by outperforming state-of-art models in a wide range of domains. Still, little is known about their robustness from a theoretical perspective. We tackle this problem by studying the local Lipschitz constant of self-attention, that provides an attack-agnostic way of measuring the robustness of a neural network. We adopt a measure-theoretic framework, by viewing inputs as probability measures equipped with the Wasserstein distance. This allows us to generalize attention to inputs of infinite length, and to derive an upper bound and a lower bound on the Lipschitz constant of self-attention on compact sets. The lower bound significantly improves prior results, and grows more than exponentially with the radius of the compact set, which rules out the possibility of obtaining robustness guarantees without any additional constraint on the input space. Our results also point out that measures with a high local Lipschitz constant are typically made of a few diracs, with a very unbalanced distribution of mass. Finally, we analyze the stability of self-attention under perturbations that change the number of tokens, which appears to be a natural question in the measure-theoretic framework. In particular, we show that for some inputs, attacks that duplicate tokens before perturbing them are more efficient than attacks that simply move tokens. We call this phenomenon mass splitting.

Structured Transforms Across Spaces with Cost-Regularized Optimal Transport

Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points. When these measures live in the same metric space, the ground cost often defaults to its distance. When instantiated across two different spaces, however, choosing that cost in the absence of aligned data is a conundrum. As a result, practitioners often resort to solving instead a quadratic Gromow-Wasserstein (GW) problem. We exploit in this work a parallel between GW and cost-regularized OT, the regularized minimization of a linear OT objective parameterized by a ground cost. We use this cost-regularized formulation to match measures across two different Euclidean spaces, where the cost is evaluated between transformed source points and target points. We show that several quadratic OT problems fall in this category, and consider enforcing structure in linear transform (e.g. sparsity), by introducing structure-inducing regularizers. We provide a proximal algorithm to extract such transforms from unaligned data, and demonstrate its applicability to single-cell spatial transcriptomics/multiomics matching tasks.

Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows

Understanding the geometric properties of gradient descent dynamics is a key ingredient in deciphering the recent success of very large machine learning models. A striking observation is that trained over-parameterized models retain some properties of the optimization initialization. This "implicit bias" is believed to be responsible for some favorable properties of the trained models and could explain their good generalization properties. The purpose of this article is threefold. First, we rigorously expose the definition and basic properties of "conservation laws", which are maximal sets of independent quantities conserved during gradient flows of a given model (e.g. of a ReLU network with a given architecture) with any training data and any loss. Then we explain how to find the exact number of these quantities by performing finite-dimensional algebraic manipulations on the Lie algebra generated by the Jacobian of the model. Finally, we provide algorithms (implemented in SageMath) to: a) compute a family of polynomial laws; b) compute the number of (not necessarily polynomial) conservation laws. We provide showcase examples that we fully work out theoretically. Besides, applying the two algorithms confirms for a number of ReLU network architectures that all known laws are recovered by the algorithm, and that there are no other laws. Such computational tools pave the way to understanding desirable properties of optimization initialization in large machine learning models.

Test like you Train in Implicit Deep Learning

Implicit deep learning has recently gained popularity with applications ranging from meta-learning to Deep Equilibrium Networks (DEQs). In its general formulation, it relies on expressing some components of deep learning pipelines implicitly, typically via a root equation called the inner problem. In practice, the solution of the inner problem is approximated during training with an iterative procedure, usually with a fixed number of inner iterations. During inference, the inner problem needs to be solved with new data. A popular belief is that increasing the number of inner iterations compared to the one used during training yields better performance. In this paper, we question such an assumption and provide a detailed theoretical analysis in a simple setting. We demonstrate that overparametrization plays a key role: increasing the number of iterations at test time cannot improve performance for overparametrized networks. We validate our theory on an array of implicit deep-learning problems. DEQs, which are typically overparametrized, do not benefit from increasing the number of iterations at inference while meta-learning, which is typically not overparametrized, benefits from it.