Haiyun He

Haiyun He, Gholamali Aminian, Yuheng Bu et al.

We provide an exact characterization of the expected generalization error (gen-error) for semi-supervised learning (SSL) with pseudo-labeling via the Gibbs algorithm. The gen-error is expressed in terms of the symmetrized KL information between the output hypothesis, the pseudo-labeled dataset, and the labeled dataset. Distribution-free upper and lower bounds on the gen-error can also be obtained. Our findings offer new insights that the generalization performance of SSL with pseudo-labeling is affected not only by the information between the output hypothesis and input training data but also by the information {\em shared} between the {\em labeled} and {\em pseudo-labeled} data samples. This serves as a guideline to choose an appropriate pseudo-labeling method from a given family of methods. To deepen our understanding, we further explore two examples -- mean estimation and logistic regression. In particular, we analyze how the ratio of the number of unlabeled to labeled data $λ$ affects the gen-error under both scenarios. As $λ$ increases, the gen-error for mean estimation decreases and then saturates at a value larger than when all the samples are labeled, and the gap can be quantified {\em exactly} with our analysis, and is dependent on the \emph{cross-covariance} between the labeled and pseudo-labeled data samples. For logistic regression, the gen-error and the variance component of the excess risk also decrease as $λ$ increases.

Bingying Li, Haiyun He

Knowledge distillation is widely used to improve generalization in practice, yet its theoretical understanding remains elusive. In the standard distillation setting, a teacher model provides soft predictions to guide the training of a student model. We model teacher and student training as coupled stochastic processes and introduce a distillation divergence, defined as the Kullback-Leibler divergence between these two stochastic kernels. Within this framework, we derive two generalization bounds for the student model relative to the teacher's generalization gap: an upper bound under a sub-Gaussian assumption via algorithmic stability, and a lower bound under a central condition with sharper dependence on the distillation divergence. We further develop a loss-sharpness-aware bound with an explicit tightness regime, showing that the teacher's local flatness can strictly tighten the bound. Additionally, in a linear Gaussian case study, the distillation divergence admits an interpretable decomposition into bias, variance, and rank-bottleneck costs, yielding practical guidance for distillation design.

Haiyun He, Yepeng Liu, Zhuoer Shen et al.

We study multi-bit watermarking for data generated by stochastic processes, where a hidden message is embedded during sampling and must be decodable by an authorized detector that possesses side information unavailable to unauthorized observers. In high-stakes deployments, a practical watermark must simultaneously control false alarms, preserve generation quality without distorting the output distribution, and support reliable multi-bit decoding. Satisfying all three goals at once inevitably creates fundamental trade-offs. We formulate watermark embedding as a distributional information-embedding problem and watermark detection as a multiple-hypothesis testing problem under distortion and rate constraints, leading to four fundamental metrics: false-alarm probability, detection error probability, distortion, and information rate. Within this information-theoretic framework, we derive matched converse and achievability bounds that characterize the optimal trade-offs and provide scheme-agnostic benchmarks for any watermarking method. For stationary ergodic stochastic processes, we further obtain matched asymptotic limits and connect them to the finite-sample regime. Finally, we present a reference watermarking construction satisfying our assumptions and empirically illustrating the predicted trade-offs.

Zhenxin Ai, Haiyun He

Watermarking for large language models (LLMs) is a promising approach for detecting LLM-generated text and enabling responsible deployment. However, existing watermarking methods are often vulnerable to semantic-invariant attacks, such as paraphrasing. We propose PASA, a principled, robust, and distortion-free watermarking algorithm that embeds and detects a watermark at the semantic level. PASA operates on semantic clusters in a latent embedding space and constructs a distributional dependency between token and auxiliary sequences via shared randomness synchronized by a secret key and semantic history. This design is grounded in our theoretical framework that characterizes a jointly optimal embedding-detection pair, achieving the fundamental trade-offs among detection accuracy, robustness, and distortion. Evaluations across multiple LLMs and semantic-invariant attacks demonstrate that PASA remains robust even under strong paraphrasing attacks while preserving high text quality, outperforming standard vocabulary-space baselines. Ablation studies further validate the effectiveness of our hyperparameter choices. Webpage: https://ai-kunkun.github.io/PASA_page/.

Haiyun He, Ziv Goldfeld

Deep neural networks (DNNs) exhibit an exceptional capacity for generalization in practical applications. This work aims to capture the effect and benefits of depth for supervised learning via information-theoretic generalization bounds. We first derive two hierarchical bounds on the generalization error in terms of the Kullback-Leibler (KL) divergence or the 1-Wasserstein distance between the train and test distributions of the network internal representations. The KL divergence bound shrinks as the layer index increases, while the Wasserstein bound implies the existence of a layer that serves as a generalization funnel, which attains a minimal 1-Wasserstein distance. Analytic expressions for both bounds are derived under the setting of binary Gaussian classification with linear DNNs. To quantify the contraction of the relevant information measures when moving deeper into the network, we analyze the strong data processing inequality (SDPI) coefficient between consecutive layers of three regularized DNN models: $\mathsf{Dropout}$, $\mathsf{DropConnect}$, and Gaussian noise injection. This enables refining our generalization bounds to capture the contraction as a function of the network architecture parameters. Specializing our results to DNNs with a finite parameter space and the Gibbs algorithm reveals that deeper yet narrower network architectures generalize better in those examples, although how broadly this statement applies remains a question.

Haiyun He, Yepeng Liu, Ziqiao Wang et al.

This paper introduces a novel problem, distributional information embedding, motivated by the practical demands of multi-bit watermarking for large language models (LLMs). Unlike traditional information embedding, which embeds information into a pre-existing host signal, LLM watermarking actively controls the text generation process--adjusting the token distribution--to embed a detectable signal. We develop an information-theoretic framework to analyze this distributional information embedding problem, characterizing the fundamental trade-offs among three critical performance metrics: text quality, detectability, and information rate. In the asymptotic regime, we demonstrate that the maximum achievable rate with vanishing error corresponds to the entropy of the LLM's output distribution and increases with higher allowable distortion. We also characterize the optimal watermarking scheme to achieve this rate. Extending the analysis to the finite-token case with non-i.i.d. tokens, we identify schemes that maximize detection probability while adhering to constraints on false alarm and distortion.

Haiyun He, Hanshu Yan, Vincent Y. F. Tan

Using information-theoretic principles, we consider the generalization error (gen-error) of iterative semi-supervised learning (SSL) algorithms that iteratively generate pseudo-labels for a large amount of unlabelled data to progressively refine the model parameters. In contrast to most previous works that {\em bound} the gen-error, we provide an {\em exact} expression for the gen-error and particularize it to the binary Gaussian mixture model. Our theoretical results suggest that when the class conditional variances are not too large, the gen-error decreases with the number of iterations, but quickly saturates. On the flip side, if the class conditional variances (and so amount of overlap between the classes) are large, the gen-error increases with the number of iterations. To mitigate this undesirable effect, we show that regularization can reduce the gen-error. The theoretical results are corroborated by extensive experiments on the MNIST and CIFAR datasets in which we notice that for easy-to-distinguish classes, the gen-error improves after several pseudo-labelling iterations, but saturates afterwards, and for more difficult-to-distinguish classes, regularization improves the generalization performance.

Haiyun He, Qiaosheng Zhang, Vincent Y. F. Tan

This paper investigates a novel offline change-point detection problem from an information-theoretic perspective. In contrast to most related works, we assume that the knowledge of the underlying pre- and post-change distributions are not known and can only be learned from the training sequences which are available. We further require the probability of the \emph{estimation error} to decay either exponentially or sub-exponentially fast (corresponding respectively to the large and moderate deviations regimes in information theory parlance). Based on the training sequences as well as the test sequence consisting of a single change-point, we design a change-point estimator and further show that this estimator is optimal by establishing matching (strong) converses. This leads to a full characterization of the optimal confidence width (i.e., half the width of the confidence interval within which the true change-point is located at with high probability) as a function of the undetected error, under both the large and moderate deviations regimes.