High-quality data is essential for training large generative models, yet the vast reservoir of real data available online has become nearly depleted. Consequently, models increasingly generate their own data for further training, forming Self-consuming Training Loops (STLs). However, the empirical results have been strikingly inconsistent: some models degrade or even collapse, while others successfully avoid these failures, leaving a significant gap in theoretical understanding to explain this discrepancy. This paper introduces the intriguing notion of *recursive stability* and presents the first theoretical generalization analysis, revealing how both model architecture and the proportion between real and synthetic data influence the success of STLs. We further extend this analysis to transformers in in-context learning, showing that even a constant-sized proportion of real data ensures convergence, while also providing insights into optimal synthetic data sizing.

Large generative models are increasingly trained on synthetic data from earlier generations, raising concerns about *model collapse*, a progressive performance decline consistently observed in empirical studies. However, theoretical understanding of recursive training dynamics and their failure modes remains limited. In this work, we theoretically show that recursive training inherently leads to exponential error growth unless mitigated by sufficient real data. Addressing the growing scarcity of real data, we introduce a self-verification mechanism enabling models to filter their outputs based on internal confidence scores without external validation. Through rigorous analysis, we derive finite-sample error bounds demonstrating that self-verification alone can prevent collapse, even in fully synthetic training regimes. Our theoretical framework extends to large language models (LLMs), characterizing the conditions under which recursive training can maintain stability without performance degradation.

This paper presents the first theoretical guarantee for Bayesian bilevel optimization (BBO) that we term for the prevalent bilevel framework combining Bayesian optimization at the outer level to tune hyperparameters, and the inner-level stochastic gradient descent (SGD) for training the model. We prove sublinear regret bounds suggesting simultaneous convergence of the inner-level model parameters and outer-level hyperparameters to optimal configurations for generalization capability. A pivotal, technical novelty in the proofs is modeling the excess risk of the SGD-trained parameters as evaluation noise during Bayesian optimization. Our theory implies the inner unit horizon, defined as the number of SGD iterations, shapes the convergence behavior of BBO. This suggests practical guidance on configuring the inner unit horizon to enhance training efficiency and model performance.

This paper tackles the emerging challenge of training generative models within a self-consuming loop, wherein successive generations of models are recursively trained on mixtures of real and synthetic data from previous generations. We construct a theoretical framework to rigorously evaluate how this training procedure impacts the data distributions learned by future models, including parametric and non-parametric models. Specifically, we derive bounds on the total variation (TV) distance between the synthetic data distributions produced by future models and the original real data distribution under various mixed training scenarios for diffusion models with a one-hidden-layer neural network score function. Our analysis demonstrates that this distance can be effectively controlled under the condition that mixed training dataset sizes or proportions of real data are large enough. Interestingly, we further unveil a phase transition induced by expanding synthetic data amounts, proving theoretically that while the TV distance exhibits an initial ascent, it declines beyond a threshold point. Finally, we present results for kernel density estimation, delivering nuanced insights such as the impact of mixed data training on error propagation.

This paper studies how to tune the stepping schedule in diffusion models, which is mostly fixed in current practice, lacking theoretical foundations and assurance of optimal performance at the chosen discretization points. In this paper, we advocate the use of adaptive time-stepping schedules and design two algorithms with an optimized sampling error bound $EB$: (1) for continuous diffusion, we treat $EB$ as the loss function to discretization points and run gradient descent to adjust them; and (2) for discrete diffusion, we propose a greedy algorithm that adjusts only one discretization point to its best position in each iteration. We conducted extensive experiments that show (1) improved generation ability in well-trained models, and (2) premature though usable generation ability in under-trained models. The code is submitted and will be released publicly.