Pretrained transformer efficiently learns low-dimensional target functions in-context

Transformers can efficiently learn in-context from example demonstrations. Most existing theoretical analyses studied the in-context learning (ICL) ability of transformers for linear function classes, where it is typically shown that the minimizer of the pretraining loss implements one gradient descent step on the least squares objective. However, this simplified linear setting arguably does not demonstrate the statistical efficiency of ICL, since the pretrained transformer does not outperform directly solving linear regression on the test prompt. In this paper, we study ICL of a nonlinear function class via transformer with nonlinear MLP layer: given a class of \textit{single-index} target functions $f_*(\boldsymbol{x}) = \sigma_*(\langle\boldsymbol{x},\boldsymbol{\beta}\rangle)$, where the index features $\boldsymbol{\beta}\in\mathbb{R}^d$ are drawn from a $r$-dimensional subspace, we show that a nonlinear transformer optimized by gradient descent (with a pretraining sample complexity that depends on the \textit{information exponent} of the link functions $\sigma_*$) learns $f_*$ in-context with a prompt length that only depends on the dimension of the distribution of target functions $r$; in contrast, any algorithm that directly learns $f_*$ on test prompt yields a statistical complexity that scales with the ambient dimension $d$. Our result highlights the adaptivity of the pretrained transformer to low-dimensional structures of the function class, which enables sample-efficient ICL that outperforms estimators that only have access to the in-context data.

Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations

We study the computational and sample complexity of learning a target function $f_*:\mathbb{R}^d\to\mathbb{R}$ with additive structure, that is, $f_*(x) = \frac{1}{\sqrt{M}}\sum_{m=1}^M f_m(\langle x, v_m\rangle)$, where $f_1,f_2,...,f_M:\mathbb{R}\to\mathbb{R}$ are nonlinear link functions of single-index models (ridge functions) with diverse and near-orthogonal index features $\{v_m\}_{m=1}^M$, and the number of additive tasks $M$ grows with the dimensionality $M\asymp d^\gamma$ for $\gamma\ge 0$. This problem setting is motivated by the classical additive model literature, the recent representation learning theory of two-layer neural network, and large-scale pretraining where the model simultaneously acquires a large number of "skills" that are often localized in distinct parts of the trained network. We prove that a large subset of polynomial $f_*$ can be efficiently learned by gradient descent training of a two-layer neural network, with a polynomial statistical and computational complexity that depends on the number of tasks $M$ and the information exponent of $f_m$, despite the unknown link function and $M$ growing with the dimensionality. We complement this learnability guarantee with computational hardness result by establishing statistical query (SQ) lower bounds for both the correlational SQ and full SQ algorithms.

Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit

We study the problem of gradient descent learning of a single-index target function $f_*(\boldsymbol{x}) = \textstyle\sigma_*\left(\langle\boldsymbol{x},\boldsymbol{\theta}\rangle\right)$ under isotropic Gaussian data in $\mathbb{R}^d$, where the link function $\sigma_*:\mathbb{R}\to\mathbb{R}$ is an unknown degree $q$ polynomial with information exponent $p$ (defined as the lowest degree in the Hermite expansion). Prior works showed that gradient-based training of neural networks can learn this target with $n\gtrsim d^{\Theta(p)}$ samples, and such statistical complexity is predicted to be necessary by the correlational statistical query lower bound. Surprisingly, we prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ of arbitrary polynomial link function with a sample and runtime complexity of $n \asymp T \asymp C(q) \cdot d\mathrm{polylog} d$, where constant $C(q)$ only depends on the degree of $\sigma_*$, regardless of information exponent; this dimension dependence matches the information theoretic limit up to polylogarithmic factors. Core to our analysis is the reuse of minibatch in the gradient computation, which gives rise to higher-order information beyond correlational queries.

Flow matching achieves minimax optimal convergence

Flow matching (FM) has gained significant attention as a simulation-free generative model. Unlike diffusion models, which are based on stochastic differential equations, FM employs a simpler approach by solving an ordinary differential equation with an initial condition from a normal distribution, thus streamlining the sample generation process. This paper discusses the convergence properties of FM in terms of the $p$-Wasserstein distance, a measure of distributional discrepancy. We establish that FM can achieve the minmax optimal convergence rate for $1 \leq p \leq 2$, presenting the first theoretical evidence that FM can reach convergence rates comparable to those of diffusion models. Our analysis extends existing frameworks by examining a broader class of mean and variance functions for the vector fields and identifies specific conditions necessary to attain these optimal rates.

Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems

In this paper, we extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We propose mean-field Langevin averaged gradient (MFL-AG), a single-loop algorithm that implements gradient descent ascent in the distribution spaces with a novel weighted averaging, and establish average-iterate convergence to the mixed Nash equilibrium. We also study both time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result which accounts for the dependency of the particle interactions on all previous distributions. Furthermore, we propose mean-field Langevin anchored best response (MFL-ABR), a symmetric double-loop algorithm based on best response dynamics with linear last-iterate convergence. Finally, we study applications to zero-sum Markov games and conduct simulations demonstrating long-term optimality.

Primal and Dual Analysis of Entropic Fictitious Play for Finite-sum Problems

The entropic fictitious play (EFP) is a recently proposed algorithm that minimizes the sum of a convex functional and entropy in the space of measures -- such an objective naturally arises in the optimization of a two-layer neural network in the mean-field regime. In this work, we provide a concise primal-dual analysis of EFP in the setting where the learning problem exhibits a finite-sum structure. We establish quantitative global convergence guarantees for both the continuous-time and discrete-time dynamics based on properties of a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our primal-dual framework entails a memory-efficient particle-based implementation of the EFP update, and also suggests a connection to gradient boosting methods. We illustrate the efficiency of our novel implementation in experiments including neural network optimization and image synthesis.

Diffusion Models are Minimax Optimal Distribution Estimators

While efficient distribution learning is no doubt behind the groundbreaking success of diffusion modeling, its theoretical guarantees are quite limited. In this paper, we provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling for well-known function spaces. The highlight of this paper is that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates in the total variation distance and in the Wasserstein distance of order one. Furthermore, we extend our theory to demonstrate how diffusion models adapt to low-dimensional data distributions. We expect these results advance theoretical understandings of diffusion modeling and its ability to generate verisimilar outputs.