When Less Is More: Binary Feedback Can Outperform Ordinal Comparisons in Ranking Recovery

Paired comparison data, where users evaluate items in pairs, play a central role in ranking and preference learning tasks. While ordinal comparison data intuitively offer richer information than binary comparisons, this paper challenges that conventional wisdom. We propose a general parametric framework for modeling ordinal paired comparisons without ties. The model adopts a generalized additive structure, featuring a link function that quantifies the preference difference between two items and a pattern function that governs the distribution over ordinal response levels. This framework encompasses classical binary comparison models as special cases, by treating binary responses as binarized versions of ordinal data. Within this framework, we show that binarizing ordinal data can significantly improve the accuracy of ranking recovery. Specifically, we prove that under the counting algorithm, the ranking error associated with binary comparisons exhibits a faster exponential convergence rate than that of ordinal data. Furthermore, we characterize a substantial performance gap between binary and ordinal data in terms of a signal-to-noise ratio (SNR) determined by the pattern function. We identify the pattern function that minimizes the SNR and maximizes the benefit of binarization. Extensive simulations and a real application on the MovieLens dataset further corroborate our theoretical findings.

A Probabilistic Perspective on Model Collapse

In recent years, model collapse has become a critical issue in language model training, making it essential to understand the underlying mechanisms driving this phenomenon. In this paper, we investigate recursive parametric model training from a probabilistic perspective, aiming to characterize the conditions under which model collapse occurs and, crucially, how it can be mitigated. We conceptualize the recursive training process as a random walk of the model estimate, highlighting how the sample size influences the step size and how the estimation procedure determines the direction and potential bias of the random walk. Under mild conditions, we rigorously show that progressively increasing the sample size at each training step is necessary to prevent model collapse. In particular, when the estimation is unbiased, the required growth rate follows a superlinear pattern. This rate needs to be accelerated even further in the presence of substantial estimation bias. Building on this probabilistic framework, we also investigate the probability that recursive training on synthetic data yields models that outperform those trained solely on real data. Moreover, we extend these results to general parametric model family in an asymptotic regime. Finally, we validate our theoretical results through extensive simulations and a real-world dataset.

Golden Ratio Mixing of Real and Synthetic Data for Stabilizing Generative Model Training

Recent studies identified an intriguing phenomenon in recursive generative model training known as model collapse, where models trained on data generated by previous models exhibit severe performance degradation. Addressing this issue and developing more effective training strategies have become central challenges in generative model research. In this paper, we investigate this phenomenon theoretically within a novel framework, where generative models are iteratively trained on a combination of newly collected real data and synthetic data from the previous training step. To develop an optimal training strategy for integrating real and synthetic data, we evaluate the performance of a weighted training scheme in various scenarios, including Gaussian distribution estimation and linear regression. We theoretically characterize the impact of the mixing proportion and weighting scheme of synthetic data on the final model's performance. Our key finding is that, across different settings, the optimal weighting scheme under different proportions of synthetic data asymptotically follows a unified expression, revealing a fundamental trade-off between leveraging synthetic data and generative model performance. Notably, in some cases, the optimal weight assigned to real data corresponds precisely to the reciprocal of the golden ratio. Finally, we validate our theoretical results on extensive simulated datasets and a real tabular dataset.

TimeAutoDiff: Combining Autoencoder and Diffusion model for time series tabular data synthesizing

In this paper, we leverage the power of latent diffusion models to generate synthetic time series tabular data. Along with the temporal and feature correlations, the heterogeneous nature of the feature in the table has been one of the main obstacles in time series tabular data modeling. We tackle this problem by combining the ideas of the variational auto-encoder (VAE) and the denoising diffusion probabilistic model (DDPM). Our model named as \texttt{TimeAutoDiff} has several key advantages including (1) Generality: the ability to handle the broad spectrum of time series tabular data from single to multi-sequence datasets; (2) Good fidelity and utility guarantees: numerical experiments on six publicly available datasets demonstrating significant improvements over state-of-the-art models in generating time series tabular data, across four metrics measuring fidelity and utility; (3) Fast sampling speed: entire time series data generation as opposed to the sequential data sampling schemes implemented in the existing diffusion-based models, eventually leading to significant improvements in sampling speed, (4) Entity conditional generation: the first implementation of conditional generation of multi-sequence time series tabular data with heterogenous features in the literature, enabling scenario exploration across multiple scientific and engineering domains. Codes are in preparation for release to the public, but available upon request.

Discriminative Estimation of Total Variation Distance: A Fidelity Auditor for Generative Data

With the proliferation of generative AI and the increasing volume of generative data (also called as synthetic data), assessing the fidelity of generative data has become a critical concern. In this paper, we propose a discriminative approach to estimate the total variation (TV) distance between two distributions as an effective measure of generative data fidelity. Our method quantitatively characterizes the relation between the Bayes risk in classifying two distributions and their TV distance. Therefore, the estimation of total variation distance reduces to that of the Bayes risk. In particular, this paper establishes theoretical results regarding the convergence rate of the estimation error of TV distance between two Gaussian distributions. We demonstrate that, with a specific choice of hypothesis class in classification, a fast convergence rate in estimating the TV distance can be achieved. Specifically, the estimation accuracy of the TV distance is proven to inherently depend on the separation of two Gaussian distributions: smaller estimation errors are achieved when the two Gaussian distributions are farther apart. This phenomenon is also validated empirically through extensive simulations. In the end, we apply this discriminative estimation method to rank fidelity of synthetic image data using the MNIST dataset.

Rate-Optimal Rank Aggregation with Private Pairwise Rankings

In various real-world scenarios like recommender systems and political surveys, pairwise rankings are commonly collected and utilized for rank aggregation to obtain an overall ranking of items. However, preference rankings can reveal individuals' personal preferences, underscoring the need to protect them before releasing for downstream analysis. In this paper, we address the challenge of preserving privacy while ensuring the utility of rank aggregation based on pairwise rankings generated from the Bradley-Terry-Luce (BTL) model. Using the randomized response mechanism to perturb raw pairwise rankings is a common privacy protection strategy used in practice, but a critical challenge arises because the privatized rankings no longer adhere to the BTL model, resulting in significant bias in downstream rank aggregation tasks. Motivated from this, we propose a debiased randomized response mechanism to protect the raw pairwise rankings, ensuring consistent estimation of true preferences and rankings in downstream rank aggregation. Theoretically, we offer insights into the relationship between overall privacy guarantees and estimation errors from private ranking data, and establish minimax rates for estimation errors. This enables the determination of optimal privacy guarantees that balance consistency in rank aggregation with robust privacy protection. We also investigate convergence rates of expected ranking errors for partial and full ranking recovery, quantifying how privacy protection influences the specification of top-$K$ item sets and complete rankings. Our findings are validated through extensive simulations and a real application.

Utility Theory of Synthetic Data Generation

Evaluating the utility of synthetic data is critical for measuring the effectiveness and efficiency of synthetic algorithms. Existing results focus on empirical evaluations of the utility of synthetic data, whereas the theoretical understanding of how utility is affected by synthetic data algorithms remains largely unexplored. This paper establishes utility theory from a statistical perspective, aiming to quantitatively assess the utility of synthetic algorithms based on a general metric. The metric is defined as the absolute difference in generalization between models trained on synthetic and original datasets. We establish analytical bounds for this utility metric to investigate critical conditions for the metric to converge. An intriguing result is that the synthetic feature distribution is not necessarily identical to the original one for the convergence of the utility metric as long as the model specification in downstream learning tasks is correct. Another important utility metric is model comparison based on synthetic data. Specifically, we establish sufficient conditions for synthetic data algorithms so that the ranking of generalization performances of models trained on the synthetic data is consistent with that from the original data. Finally, we conduct extensive experiments using non-parametric models and deep neural networks to validate our theoretical findings.

Ranking Differential Privacy

Rankings are widely collected in various real-life scenarios, leading to the leakage of personal information such as users' preferences on videos or news. To protect rankings, existing works mainly develop privacy protection on a single ranking within a set of ranking or pairwise comparisons of a ranking under the $\epsilon$-differential privacy. This paper proposes a novel notion called $\epsilon$-ranking differential privacy for protecting ranks. We establish the connection between the Mallows model (Mallows, 1957) and the proposed $\epsilon$-ranking differential privacy. This allows us to develop a multistage ranking algorithm to generate synthetic rankings while satisfying the developed $\epsilon$-ranking differential privacy. Theoretical results regarding the utility of synthetic rankings in the downstream tasks, including the inference attack and the personalized ranking tasks, are established. For the inference attack, we quantify how $\epsilon$ affects the estimation of the true ranking based on synthetic rankings. For the personalized ranking task, we consider varying privacy preferences among users and quantify how their privacy preferences affect the consistency in estimating the optimal ranking function. Extensive numerical experiments are carried out to verify the theoretical results and demonstrate the effectiveness of the proposed synthetic ranking algorithm.