Bilinear Sequence Regression: A Model for Learning from Long Sequences of High-dimensional Tokens

Current progress in artificial intelligence is centered around so-called large language models that consist of neural networks processing long sequences of high-dimensional vectors called tokens. Statistical physics provides powerful tools to study the functioning of learning with neural networks and has played a recognized role in the development of modern machine learning. The statistical physics approach relies on simplified and analytically tractable models of data. However, simple tractable models for long sequences of high-dimensional tokens are largely underexplored. Inspired by the crucial role models such as the single-layer teacher-student perceptron (aka generalized linear regression) played in the theory of fully connected neural networks, in this paper, we introduce and study the bilinear sequence regression (BSR) as one of the most basic models for sequences of tokens. We note that modern architectures naturally subsume the BSR model due to the skip connections. Building on recent methodological progress, we compute the Bayes-optimal generalization error for the model in the limit of long sequences of high-dimensional tokens, and provide a message-passing algorithm that matches this performance. We quantify the improvement that optimal learning brings with respect to vectorizing the sequence of tokens and learning via simple linear regression. We also unveil surprising properties of the gradient descent algorithms in the BSR model.

Building Conformal Prediction Intervals with Approximate Message Passing

Conformal prediction has emerged as a powerful tool for building prediction intervals that are valid in a distribution-free way. However, its evaluation may be computationally costly, especially in the high-dimensional setting where the dimensionality and sample sizes are both large and of comparable magnitudes. To address this challenge in the context of generalized linear regression, we propose a novel algorithm based on Approximate Message Passing (AMP) to accelerate the computation of prediction intervals using full conformal prediction, by approximating the computation of conformity scores. Our work bridges a gap between modern uncertainty quantification techniques and tools for high-dimensional problems involving the AMP algorithm. We evaluate our method on both synthetic and real data, and show that it produces prediction intervals that are close to the baseline methods, while being orders of magnitude faster. Additionally, in the high-dimensional limit and under assumptions on the data distribution, the conformity scores computed by AMP converge to the one computed exactly, which allows theoretical study and benchmarking of conformal methods in high dimensions.

Bayes-optimal learning of an extensive-width neural network from quadratically many samples

We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.

Could ChatGPT get an Engineering Degree? Evaluating Higher Education Vulnerability to AI Assistants

AI assistants are being increasingly used by students enrolled in higher education institutions. While these tools provide opportunities for improved teaching and education, they also pose significant challenges for assessment and learning outcomes. We conceptualize these challenges through the lens of vulnerability, the potential for university assessments and learning outcomes to be impacted by student use of generative AI. We investigate the potential scale of this vulnerability by measuring the degree to which AI assistants can complete assessment questions in standard university-level STEM courses. Specifically, we compile a novel dataset of textual assessment questions from 50 courses at EPFL and evaluate whether two AI assistants, GPT-3.5 and GPT-4 can adequately answer these questions. We use eight prompting strategies to produce responses and find that GPT-4 answers an average of 65.8% of questions correctly, and can even produce the correct answer across at least one prompting strategy for 85.1% of questions. When grouping courses in our dataset by degree program, these systems already pass non-project assessments of large numbers of core courses in various degree programs, posing risks to higher education accreditation that will be amplified as these models improve. Our results call for revising program-level assessment design in higher education in light of advances in generative AI.

Understanding Counting in Small Transformers: The Interplay between Attention and Feed-Forward Layers

We provide a comprehensive analysis of simple transformer models trained on the histogram task, where the goal is to count the occurrences of each item in the input sequence from a fixed alphabet. Despite its apparent simplicity, this task exhibits a rich phenomenology that allows us to characterize how different architectural components contribute towards the emergence of distinct algorithmic solutions. In particular, we showcase the existence of two qualitatively different mechanisms that implement a solution, relation- and inventory-based counting. Which solution a model can implement depends non-trivially on the precise choice of the attention mechanism, activation function, memorization capacity and the presence of a beginning-of-sequence token. By introspecting learned models on the counting task, we find evidence for the formation of both mechanisms. From a broader perspective, our analysis offers a framework to understand how the interaction of different architectural components of transformer models shapes diverse algorithmic solutions and approximations.

Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions

This work explores multi-modal inference in a high-dimensional simplified model, analytically quantifying the performance gain of multi-modal inference over that of analyzing modalities in isolation. We present the Bayes-optimal performance and weak recovery thresholds in a model where the objective is to recover the latent structures from two noisy data matrices with correlated spikes. The paper derives the approximate message passing (AMP) algorithm for this model and characterizes its performance in the high-dimensional limit via the associated state evolution. The analysis holds for a broad range of priors and noise channels, which can differ across modalities. The linearization of AMP is compared numerically to the widely used partial least squares (PLS) and canonical correlation analysis (CCA) methods, which are both observed to suffer from a sub-optimal recovery threshold.

Fundamental limits of weak learnability in high-dimensional multi-index models

Multi-index models -- functions which only depend on the covariates through a non-linear transformation of their projection on a subspace -- are a useful benchmark for investigating feature learning with neural networks. This paper examines the theoretical boundaries of learnability in this hypothesis class, focusing particularly on the minimum sample complexity required for weakly recovering their low-dimensional structure with first-order iterative algorithms, in the high-dimensional regime where the number of samples is $n=\alpha d$ is proportional to the covariate dimension $d$. Our findings unfold in three parts: (i) first, we identify under which conditions a \textit{trivial subspace} can be learned with a single step of a first-order algorithm for any $\alpha\!>\!0$; (ii) second, in the case where the trivial subspace is empty, we provide necessary and sufficient conditions for the existence of an {\it easy subspace} consisting of directions that can be learned only above a certain sample complexity $\alpha\!>\!\alpha_c$. The critical threshold $\alpha_{c}$ marks the presence of a computational phase transition, in the sense that no efficient iterative algorithm can succeed for $\alpha\!<\!\alpha_c$. In a limited but interesting set of really hard directions -- akin to the parity problem -- $\alpha_c$ is found to diverge. Finally, (iii) we demonstrate that interactions between different directions can result in an intricate hierarchical learning phenomenon, where some directions can be learned sequentially when coupled to easier ones. Our analytical approach is built on the optimality of approximate message-passing algorithms among first-order iterative methods, delineating the fundamental learnability limit across a broad spectrum of algorithms, including neural networks trained with gradient descent.

Fundamental limits of Non-Linear Low-Rank Matrix Estimation

We consider the task of estimating a low-rank matrix from non-linear and noisy observations. We prove a strong universality result showing that Bayes-optimal performances are characterized by an equivalent Gaussian model with an effective prior, whose parameters are entirely determined by an expansion of the non-linear function. In particular, we show that to reconstruct the signal accurately, one requires a signal-to-noise ratio growing as $N^{\frac 12 (1-1/k_F)}$, where $k_F$ is the first non-zero Fisher information coefficient of the function. We provide asymptotic characterization for the minimal achievable mean squared error (MMSE) and an approximate message-passing algorithm that reaches the MMSE under conditions analogous to the linear version of the problem. We also provide asymptotic errors achieved by methods such as principal component analysis combined with Bayesian denoising, and compare them with Bayes-optimal MMSE.

Analysis of Bootstrap and Subsampling in High-dimensional Regularized Regression

We investigate popular resampling methods for estimating the uncertainty of statistical models, such as subsampling, bootstrap and the jackknife, and their performance in high-dimensional supervised regression tasks. We provide a tight asymptotic description of the biases and variances estimated by these methods in the context of generalized linear models, such as ridge and logistic regression, taking the limit where the number of samples $n$ and dimension $d$ of the covariates grow at a comparable fixed rate $\alpha\!=\! n/d$. Our findings are three-fold: i) resampling methods are fraught with problems in high dimensions and exhibit the double-descent-like behavior typical of these situations; ii) only when $\alpha$ is large enough do they provide consistent and reliable error estimations (we give convergence rates); iii) in the over-parametrized regime $\alpha\!<\!1$ relevant to modern machine learning practice, their predictions are not consistent, even with optimal regularization.

Asymptotics of feature learning in two-layer networks after one gradient-step

In this manuscript we investigate the problem of how two-layer neural networks learn features from data, and improve over the kernel regime, after being trained with a single gradient descent step. Leveraging a connection from (Ba et al., 2022) with a non-linear spiked matrix model and recent progress on Gaussian universality (Dandi et al., 2023), we provide an exact asymptotic description of the generalization error in the high-dimensional limit where the number of samples $n$, the width $p$ and the input dimension $d$ grow at a proportional rate. We characterize exactly how adapting to the data is crucial for the network to efficiently learn non-linear functions in the direction of the gradient -- where at initialization it can only express linear functions in this regime. To our knowledge, our results provides the first tight description of the impact of feature learning in the generalization of two-layer neural networks in the large learning rate regime $\eta=\Theta_{d}(d)$, beyond perturbative finite width corrections of the conjugate and neural tangent kernels.