arXiv subjects

Jason D. Lee

Publications and source records attributed to Jason D. Lee.

At least 37 records · Page 2Linked to original sources

arXiv2026

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.

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arXiv2025

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.

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arXiv2025

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: $f_*(\boldsymbol{x}) = \sum_{p=1}^P a_p\cdot σ(\langle\boldsymbol{x},\boldsymbol{v}_p^*\rangle)$, $\boldsymbol{x} \sim \mathcal{N}(0,\boldsymbol{I}_d)$, where the activation $σ:\mathbb{R}\to\mathbb{R}$ 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 $a_p\asymp p^{-β}$ where $β\in\mathbb{R}_{\ge 0}$. 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.

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arXiv2025

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.

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arXiv2025

Reward Collapse in Aligning Large Language Models

The extraordinary capabilities of large language models (LLMs) such as ChatGPT and GPT-4 are in part unleashed by aligning them with reward models that are trained on human preferences, which are often represented as rankings of responses to prompts. In this paper, we document the phenomenon of \textit{reward collapse}, an empirical observation where the prevailing ranking-based approach results in an \textit{identical} reward distribution \textit{regardless} of the prompts during the terminal phase of training. This outcome is undesirable as open-ended prompts like ``write a short story about your best friend'' should yield a continuous range of rewards for their completions, while specific prompts like ``what is the capital of New Zealand'' should generate either high or low rewards. Our theoretical investigation reveals that reward collapse is primarily due to the insufficiency of the ranking-based objective function to incorporate prompt-related information during optimization. This insight allows us to derive closed-form expressions for the reward distribution associated with a set of utility functions in an asymptotic regime. To overcome reward collapse, we introduce a prompt-aware optimization scheme that provably admits a prompt-dependent reward distribution within the interpolating regime. Our experimental results suggest that our proposed prompt-aware utility functions significantly alleviate reward collapse during the training of reward models.

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arXiv2025

On the Statistical Query Complexity of Learning Semiautomata: a Random Walk Approach

Semiautomata form a rich class of sequence-processing algorithms with applications in natural language processing, robotics, computational biology, and data mining. We establish the first Statistical Query hardness result for semiautomata under the uniform distribution over input words and initial states. We show that Statistical Query hardness can be established when both the alphabet size and input length are polynomial in the number of states. Unlike the case of deterministic finite automata, where hardness typically arises through the hardness of the language they recognize (e.g., parity), our result is derived solely from the internal state-transition structure of semiautomata. Our analysis reduces the task of distinguishing the final states of two semiautomata to studying the behavior of a random walk on the group $S_{N} \times S_{N}$. By applying tools from Fourier analysis and the representation theory of the symmetric group, we obtain tight spectral gap bounds, demonstrating that after a polynomial number of steps in the number of states, distinct semiautomata become nearly uncorrelated, yielding the desired hardness result.

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arXiv2025

Learning Orthogonal Multi-Index Models: A Fine-Grained Information Exponent Analysis

The information exponent ([BAGJ21]) and its extensions -- which are equivalent to the lowest degree in the Hermite expansion of the link function (after a potential label transform) for Gaussian single-index models -- have played an important role in predicting the sample complexity of online stochastic gradient descent (SGD) in various learning tasks. In this work, we demonstrate that, for multi-index models, focusing solely on the lowest degree can miss key structural details of the model and result in suboptimal rates. Specifically, we consider the task of learning target functions of form $f_*(\mathbf{x}) = \sum_{k=1}^{P} ϕ(\mathbf{v}_k^* \cdot \mathbf{x})$, where $P \ll d$, the ground-truth directions $\{ \mathbf{v}_k^* \}_{k=1}^P$ are orthonormal, and the information exponent of $ϕ$ is $L$. Based on the theory of information exponent, when $L = 2$, only the relevant subspace (not the exact directions) can be recovered due to the rotational invariance of the second-order terms, and when $L > 2$, recovering the directions using online SGD require $\tilde{O}(P d^{L-1})$ samples. In this work, we show that by considering both second- and higher-order terms, we can first learn the relevant space using the second-order terms, and then the exact directions using the higher-order terms, and the overall sample and complexity of online SGD is $\tilde{O}( d P^{L-1} )$.

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arXiv2025

Understanding Optimization in Deep Learning with Central Flows

Traditional theories of optimization cannot describe the dynamics of optimization in deep learning, even in the simple setting of deterministic training. The challenge is that optimizers typically operate in a complex, oscillatory regime called the "edge of stability." In this paper, we develop theory that can describe the dynamics of optimization in this regime. Our key insight is that while the *exact* trajectory of an oscillatory optimizer may be challenging to analyze, the *time-averaged* (i.e. smoothed) trajectory is often much more tractable. To analyze an optimizer, we derive a differential equation called a "central flow" that characterizes this time-averaged trajectory. We empirically show that these central flows can predict long-term optimization trajectories for generic neural networks with a high degree of numerical accuracy. By interpreting these central flows, we are able to understand how gradient descent makes progress even as the loss sometimes goes up; how adaptive optimizers "adapt" to the local loss landscape; and how adaptive optimizers implicitly navigate towards regions where they can take larger steps. Our results suggest that central flows can be a valuable theoretical tool for reasoning about optimization in deep learning.

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arXiv2025

Rethinking Addressing in Language Models via Contexualized Equivariant Positional Encoding

Transformers rely on both content-based and position-based addressing mechanisms to make predictions, but existing positional encoding techniques often diminish the effectiveness of position-based addressing. Many current methods enforce rigid patterns in attention maps, limiting the ability to model long-range dependencies and adapt to diverse tasks. Additionally, most positional encodings are learned as general biases, lacking the specialization required for different instances within a dataset. To address this, we propose con\textbf{T}extualized equivari\textbf{A}nt \textbf{P}osition \textbf{E}ncoding (\textbf{TAPE}), a novel framework that enhances positional embeddings by incorporating sequence content across layers. TAPE introduces dynamic, context-aware positional encodings, overcoming the constraints of traditional fixed patterns. We show that TAPE can provably facilitate LLM reasoning ability by emulating a broader class of algorithms. By enforcing permutation and orthogonal equivariance, TAPE ensures the stability of positional encodings during updates, improving long-context ability. Our method can be easily integrated into pre-trained transformers, offering parameter-efficient fine-tuning with minimal overhead. Extensive experiments show that TAPE achieves superior performance in language modeling, arithmetic reasoning, and long-context retrieval tasks compared to existing positional embedding techniques. Code is available at https://github.com/VITA-Group/TAPE.

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arXiv2025

Task Diversity Shortens the ICL Plateau

In-context learning (ICL) describes a language model's ability to generate outputs based on a set of input demonstrations and a subsequent query. To understand this remarkable capability, researchers have studied simplified, stylized models. These studies have consistently observed long loss plateaus, during which models exhibit minimal improvement, followed by a sudden, rapid surge of learning. In this work, we reveal that training on multiple diverse ICL tasks simultaneously shortens the loss plateaus, making each task easier to learn. This finding is surprising as it contradicts the natural intuition that the combined complexity of multiple ICL tasks would lengthen the learning process, not shorten it. Our result suggests that the recent success in large-scale training of language models may be attributed not only to the richness of the data at scale but also to the easier optimization (training) induced by the diversity of natural language training data.

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arXiv2025

Optimal Multi-Distribution Learning

Multi-distribution learning (MDL), which seeks to learn a shared model that minimizes the worst-case risk across $k$ distinct data distributions, has emerged as a unified framework in response to the evolving demand for robustness, fairness, multi-group collaboration, etc. Achieving data-efficient MDL necessitates adaptive sampling, also called on-demand sampling, throughout the learning process. However, there exist substantial gaps between the state-of-the-art upper and lower bounds on the optimal sample complexity. Focusing on a hypothesis class of Vapnik-Chervonenkis (VC) dimension d, we propose a novel algorithm that yields an varepsilon-optimal randomized hypothesis with a sample complexity on the order of (d+k)/varepsilon^2 (modulo some logarithmic factor), matching the best-known lower bound. Our algorithmic ideas and theory are further extended to accommodate Rademacher classes. The proposed algorithms are oracle-efficient, which access the hypothesis class solely through an empirical risk minimization oracle. Additionally, we establish the necessity of randomization, revealing a large sample size barrier when only deterministic hypotheses are permitted. These findings resolve three open problems presented in COLT 2023 (i.e., citet[Problems 1, 3 and 4]{awasthi2023sample}).

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arXiv2025

Scaling Laws in Linear Regression: Compute, Parameters, and Data

Empirically, large-scale deep learning models often satisfy a neural scaling law: the test error of the trained model improves polynomially as the model size and data size grow. However, conventional wisdom suggests the test error consists of approximation, bias, and variance errors, where the variance error increases with model size. This disagrees with the general form of neural scaling laws, which predict that increasing model size monotonically improves performance. We study the theory of scaling laws in an infinite dimensional linear regression setup. Specifically, we consider a model with $M$ parameters as a linear function of sketched covariates. The model is trained by one-pass stochastic gradient descent (SGD) using $N$ data. Assuming the optimal parameter satisfies a Gaussian prior and the data covariance matrix has a power-law spectrum of degree $a>1$, we show that the reducible part of the test error is $Θ(M^{-(a-1)} + N^{-(a-1)/a})$. The variance error, which increases with $M$, is dominated by the other errors due to the implicit regularization of SGD, thus disappearing from the bound. Our theory is consistent with the empirical neural scaling laws and verified by numerical simulation.

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arXiv2025

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 $n=Θ(d^{1 \vee \k/2})$ 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).

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arXiv2025

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.

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arXiv2025

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.

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arXiv2025

Settling the Sample Complexity of Online Reinforcement Learning

A central issue lying at the heart of online reinforcement learning (RL) is data efficiency. While a number of recent works achieved asymptotically minimal regret in online RL, the optimality of these results is only guaranteed in a ``large-sample'' regime, imposing enormous burn-in cost in order for their algorithms to operate optimally. How to achieve minimax-optimal regret without incurring any burn-in cost has been an open problem in RL theory. We settle this problem for the context of finite-horizon inhomogeneous Markov decision processes. Specifically, we prove that a modified version of Monotonic Value Propagation (MVP), a model-based algorithm proposed by \cite{zhang2020reinforcement}, achieves a regret on the order of (modulo log factors) \begin{equation*} \min\big\{ \sqrt{SAH^3K}, \,HK \big\}, \end{equation*} where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, and $K$ is the total number of episodes. This regret matches the minimax lower bound for the entire range of sample size $K\geq 1$, essentially eliminating any burn-in requirement. It also translates to a PAC sample complexity (i.e., the number of episodes needed to yield $\varepsilon$-accuracy) of $\frac{SAH^3}{\varepsilon^2}$ up to log factor, which is minimax-optimal for the full $\varepsilon$-range. Further, we extend our theory to unveil the influences of problem-dependent quantities like the optimal value/cost and certain variances. The key technical innovation lies in the development of a new regret decomposition strategy and a novel analysis paradigm to decouple complicated statistical dependency -- a long-standing challenge facing the analysis of online RL in the sample-hungry regime.

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arXiv2025

Regressing the Relative Future: Efficient Policy Optimization for Multi-turn RLHF

Large Language Models (LLMs) have achieved remarkable success at tasks like summarization that involve a single turn of interaction. However, they can still struggle with multi-turn tasks like dialogue that require long-term planning. Previous works on multi-turn dialogue extend single-turn reinforcement learning from human feedback (RLHF) methods to the multi-turn setting by treating all prior dialogue turns as a long context. Such approaches suffer from covariate shift: the conversations in the training set have previous turns generated by some reference policy, which means that low training error may not necessarily correspond to good performance when the learner is actually in the conversation loop. In response, we introduce REgressing the RELative FUture (REFUEL), an efficient policy optimization approach designed to address multi-turn RLHF in LLMs. REFUEL employs a single model to estimate $Q$-values and trains on self-generated data, addressing the covariate shift issue. REFUEL frames the multi-turn RLHF problem as a sequence of regression tasks on iteratively collected datasets, enabling ease of implementation. Theoretically, we prove that REFUEL can match the performance of any policy covered by the training set. Empirically, we evaluate our algorithm by using Llama-3.1-70B-it to simulate a user in conversation with our model. REFUEL consistently outperforms state-of-the-art methods such as DPO and REBEL across various settings. Furthermore, despite having only 8 billion parameters, Llama-3-8B-it fine-tuned with REFUEL outperforms Llama-3.1-70B-it on long multi-turn dialogues. Implementation of REFUEL can be found at https://github.com/ZhaolinGao/REFUEL/, and models trained by REFUEL can be found at https://huggingface.co/Cornell-AGI.

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arXiv2025

Provable Guarantees for Nonlinear Feature Learning in Three-Layer Neural Networks

One of the central questions in the theory of deep learning is to understand how neural networks learn hierarchical features. The ability of deep networks to extract salient features is crucial to both their outstanding generalization ability and the modern deep learning paradigm of pretraining and finetuneing. However, this feature learning process remains poorly understood from a theoretical perspective, with existing analyses largely restricted to two-layer networks. In this work we show that three-layer neural networks have provably richer feature learning capabilities than two-layer networks. We analyze the features learned by a three-layer network trained with layer-wise gradient descent, and present a general purpose theorem which upper bounds the sample complexity and width needed to achieve low test error when the target has specific hierarchical structure. We instantiate our framework in specific statistical learning settings -- single-index models and functions of quadratic features -- and show that in the latter setting three-layer networks obtain a sample complexity improvement over all existing guarantees for two-layer networks. Crucially, this sample complexity improvement relies on the ability of three-layer networks to efficiently learn nonlinear features. We then establish a concrete optimization-based depth separation by constructing a function which is efficiently learnable via gradient descent on a three-layer network, yet cannot be learned efficiently by a two-layer network. Our work makes progress towards understanding the provable benefit of three-layer neural networks over two-layer networks in the feature learning regime.

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