Statistical Learning Theory in Lean 4: Empirical Processes from Scratch

We present the first comprehensive Lean 4 formalization of statistical learning theory (SLT) grounded in empirical process theory. Our end-to-end formal infrastructure implement the missing contents in latest Lean 4 Mathlib library, including a complete development of Gaussian Lipschitz concentration, the first formalization of Dudley's entropy integral theorem for sub-Gaussian processes, and an application to least-squares (sparse) regression with a sharp rate. The project was carried out using a human-AI collaborative workflow, in which humans design proof strategies and AI agents execute tactical proof construction, leading to the human-verified Lean 4 toolbox for SLT. Beyond implementation, the formalization process exposes and resolves implicit assumptions and missing details in standard SLT textbooks, enforcing a granular, line-by-line understanding of the theory. This work establishes a reusable formal foundation and opens the door for future developments in machine learning theory. The code is available at https://github.com/YuanheZ/lean-stat-learning-theory

Provable Learning of Random Hierarchy Models and Hierarchical Shallow-to-Deep Chaining

The empirical success of deep learning is often attributed to deep networks' ability to exploit hierarchical structure in data, constructing increasingly complex features across layers. Yet despite substantial progress in deep learning theory, most optimization results sill focus on networks with only two or three layers, leaving the theoretical understanding of hierarchical learning in genuinely deep models limited. This leads to a natural question: can we prove that deep networks, trained by gradient-based methods, can efficiently exploit hierarchical structure? In this work, we consider Random Hierarchy Models -- a hierarchical context-free grammar introduced by arXiv:2307.02129 and conjectured to separate deep and shallow networks. We prove that, under mild conditions, a deep convolutional network can be efficiently trained to learn this function class. Our proof builds on a general observation: if intermediate layers can receive clean signal from the labels and the relevant features are weakly identifiable, then layerwise training each individual layer suffices to hierarchically learn the target function.

Coverage Improvement and Fast Convergence of On-policy Preference Learning

Online on-policy preference learning algorithms for language model alignment such as online direct policy optimization (DPO) can significantly outperform their offline counterparts. We provide a theoretical explanation for this phenomenon by analyzing how the sampling policy's coverage evolves throughout on-policy training. We propose and rigorously justify the \emph{coverage improvement principle}: with sufficient batch size, each update moves into a region around the target where coverage is uniformly better, making subsequent data increasingly informative and enabling rapid convergence. In the contextual bandit setting with Bradley-Terry preferences and linear softmax policy class, we show that on-policy DPO converges exponentially in the number of iterations for batch size exceeding a generalized coverage threshold. In contrast, any learner restricted to offline samples from the initial policy suffers a slower minimax rate, leading to a sharp separation in total sample complexity. Motivated by this analysis, we further propose a simple hybrid sampler based on a novel \emph{preferential} G-optimal design, which removes dependence on coverage and guarantees convergence in just two rounds. Finally, we develop principled on-policy schemes for reward distillation in the general function class setting, and show faster noiseless rates under an alternative deviation-based notion of coverage. Experimentally, we confirm that on-policy DPO and our proposed reward distillation algorithms outperform their off-policy counterparts and enjoy stable, monotonic performance gains across iterations.

Neural Networks Learn Generic Multi-Index Models Near Information-Theoretic Limit

In deep learning, a central issue is to understand how neural networks efficiently learn high-dimensional features. To this end, we explore the gradient descent learning of a general Gaussian Multi-index model $f(\boldsymbol{x})=g(\boldsymbol{U}\boldsymbol{x})$ with hidden subspace $\boldsymbol{U}\in \mathbb{R}^{r\times d}$, which is the canonical setup to study representation learning. We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error using $\widetilde{\mathcal{O}}(d)$ samples and $\widetilde{\mathcal{O}}(d^2)$ time. The sample and time complexity both align with the information-theoretic limit up to leading order and are therefore optimal. During the first stage of gradient descent learning, the proof proceeds via showing that the inner weights can perform a power-iteration process. This process implicitly mimics a spectral start for the whole span of the hidden subspace and eventually eliminates finite-sample noise and recovers this span. It surprisingly indicates that optimal results can only be achieved if the first layer is trained for more than $\mathcal{O}(1)$ steps. This work demonstrates the ability of neural networks to effectively learn hierarchical functions with respect to both sample and time efficiency.

Quantitative Bounds for Length Generalization in Transformers

We study the problem of length generalization (LG) in transformers: the ability of a model trained on shorter sequences to maintain performance when evaluated on much longer, previously unseen inputs. Prior work by Huang et al. (2025) established that transformers eventually achieve length generalization once the training sequence length exceeds some finite threshold, but left open the question of how large it must be. In this work, we provide the first quantitative bounds on the required training length for length generalization to occur. Motivated by previous empirical and theoretical work, we analyze LG in several distinct problem settings: $\ell_\infty$ error control vs. average error control over an input distribution, infinite-precision softmax attention vs. finite-precision attention (which reduces to an argmax) in the transformer, and one- vs. two-layer transformers. In all scenarios, we prove that LG occurs when the internal behavior of the transformer on longer sequences can be "simulated" by its behavior on shorter sequences seen during training. Our bounds give qualitative estimates for the length of training data required for a transformer to generalize, and we verify these insights empirically. These results sharpen our theoretical understanding of the mechanisms underlying extrapolation in transformers, and formalize the intuition that richer training data is required for generalization on more complex tasks.

What One Cannot, Two Can: Two-Layer Transformers Provably Represent Induction Heads on Any-Order Markov Chains

In-context learning (ICL) is a hallmark capability of transformers, through which trained models learn to adapt to new tasks by leveraging information from the input context. Prior work has shown that ICL emerges in transformers due to the presence of special circuits called induction heads. Given the equivalence between induction heads and conditional k-grams, a recent line of work modeling sequential inputs as Markov processes has revealed the fundamental impact of model depth on its ICL capabilities: while a two-layer transformer can efficiently represent a conditional 1-gram model, its single-layer counterpart cannot solve the task unless it is exponentially large. However, for higher order Markov sources, the best known constructions require at least three layers (each with a single attention head) - leaving open the question: can a two-layer single-head transformer represent any kth-order Markov process? In this paper, we precisely address this and theoretically show that a two-layer transformer with one head per layer can indeed represent any conditional k-gram. Thus, our result provides the tightest known characterization of the interplay between transformer depth and Markov order for ICL. Building on this, we further analyze the learning dynamics of our two-layer construction, focusing on a simplified variant for first-order Markov chains, illustrating how effective in-context representations emerge during training. Together, these results deepen our current understanding of transformer-based ICL and illustrate how even shallow architectures can surprisingly exhibit strong ICL capabilities on structured sequence modeling tasks.

The Generative Leap: Sharp Sample Complexity for Efficiently Learning Gaussian Multi-Index Models

In this work we consider generic Gaussian Multi-index models, in which the labels only depend on the (Gaussian) $d$-dimensional inputs through their projection onto a low-dimensional $r = O_d(1)$ subspace, and we study efficient agnostic estimation procedures for this hidden subspace. We introduce the \emph{generative leap} exponent $k^\star$, a natural extension of the generative exponent from [Damian et al.'24] to the multi-index setting. We first show that a sample complexity of $n=\Theta(d^{1 \vee \k/2})$ is necessary in the class of algorithms captured by the Low-Degree-Polynomial framework. We then establish that this sample complexity is also sufficient, by giving an agnostic sequential estimation procedure (that is, requiring no prior knowledge of the multi-index model) based on a spectral U-statistic over appropriate Hermite tensors. We further compute the generative leap exponent for several examples including piecewise linear functions (deep ReLU networks with bias), and general deep neural networks (with $r$-dimensional first hidden layer).

Learning Compositional Functions with Transformers from Easy-to-Hard Data

Transformer-based language models have demonstrated impressive capabilities across a range of complex reasoning tasks. Prior theoretical work exploring the expressive power of transformers has shown that they can efficiently perform multi-step reasoning tasks involving parallelizable computations. However, the learnability of such constructions, particularly the conditions on the data distribution that enable efficient learning via gradient-based optimization, remains an open question. Towards answering this question, in this work we study the learnability of the $k$-fold composition task, which requires computing an interleaved composition of $k$ input permutations and $k$ hidden permutations, and can be expressed by a transformer with $O(\log k)$ layers. On the negative front, we prove a Statistical Query (SQ) lower bound showing that any SQ learner that makes only polynomially-many queries to an SQ oracle for the $k$-fold composition task distribution must have sample size exponential in $k$, thus establishing a statistical-computational gap. On the other hand, we show that this function class can be efficiently learned, with runtime and sample complexity polynomial in $k$, by gradient descent on an $O(\log k)$-depth transformer via two different curriculum learning strategies: one in which data consists of $k'$-fold composition functions with $k' \le k$ presented in increasing difficulty, and another in which all such data is presented simultaneously. Our work sheds light on the necessity and sufficiency of having both easy and hard examples in the data distribution for transformers to learn complex compositional tasks.

Accelerating RL for LLM Reasoning with Optimal Advantage Regression

Reinforcement learning (RL) has emerged as a powerful tool for fine-tuning large language models (LLMs) to improve complex reasoning abilities. However, state-of-the-art policy optimization methods often suffer from high computational overhead and memory consumption, primarily due to the need for multiple generations per prompt and the reliance on critic networks or advantage estimates of the current policy. In this paper, we propose $A$*-PO, a novel two-stage policy optimization framework that directly approximates the optimal advantage function and enables efficient training of LLMs for reasoning tasks. In the first stage, we leverage offline sampling from a reference policy to estimate the optimal value function $V$*, eliminating the need for costly online value estimation. In the second stage, we perform on-policy updates using a simple least-squares regression loss with only a single generation per prompt. Theoretically, we establish performance guarantees and prove that the KL-regularized RL objective can be optimized without requiring complex exploration strategies. Empirically, $A$*-PO achieves competitive performance across a wide range of mathematical reasoning benchmarks, while reducing training time by up to 2$\times$ and peak memory usage by over 30% compared to PPO, GRPO, and REBEL. Implementation of $A$*-PO can be found at https://github.com/ZhaolinGao/A-PO.

Emergence and scaling laws in SGD learning of shallow neural networks

We study the complexity of online stochastic gradient descent (SGD) for learning a two-layer neural network with $P$ neurons on isotropic Gaussian data: $f_*(\boldsymbol{x}) = \sum_{p=1}^P a_p\cdot \sigma(\langle\boldsymbol{x},\boldsymbol{v}_p^*\rangle)$, $\boldsymbol{x} \sim \mathcal{N}(0,\boldsymbol{I}_d)$, where the activation $\sigma:\mathbb{R}\to\mathbb{R}$ is an even function with information exponent $k_*>2$ (defined as the lowest degree in the Hermite expansion), $\{\boldsymbol{v}^*_p\}_{p\in[P]}\subset \mathbb{R}^d$ are orthonormal signal directions, and the non-negative second-layer coefficients satisfy $\sum_{p} a_p^2=1$. We focus on the challenging ``extensive-width'' regime $P\gg 1$ and permit diverging condition number in the second-layer, covering as a special case the power-law scaling $a_p\asymp p^{-\beta}$ where $\beta\in\mathbb{R}_{\ge 0}$. We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective, and explicitly identify sharp transition times to recover each signal direction. In the power-law setting, we characterize scaling law exponents for the MSE loss with respect to the number of training samples and SGD steps, as well as the number of parameters in the student neural network. Our analysis entails that while the learning of individual teacher neurons exhibits abrupt transitions, the juxtaposition of $P\gg 1$ emergent learning curves at different timescales leads to a smooth scaling law in the cumulative objective.