The problem of model collapse has presented new challenges in iterative training of generative models, where such training with synthetic data leads to an overall degradation of performance. This paper looks at the problem from a statistical viewpoint, illustrating that one can actually hope for improvement when models are trained on data contaminated with synthetic samples, as long as there is some amount of fresh information from the true target distribution. In particular, we consider iterative training on samples sourced from a mixture of the true target and synthetic distributions. We analyze the entire iterative evolution in a next-token prediction language model, capturing how the interplay between the mixture weights and the sample size controls the overall long-term performance. With non-trivial mixture weight of the true distribution, even if it decays over time, simply training the model in a contamination-agnostic manner with appropriate sample sizes can avoid collapse and even recover the true target distribution under certain conditions. Simulation studies support our findings and also show that such behavior is more general for other classes of models.

Recent years have witnessed substantial progress in understanding the behavior of EM for mixture models that are correctly specified.

B.2.2 ExperimentSetup We cannot evaluate the model performance the same as in simulation studies since we no longer have the ground truth knowledge.

However, there are still two unsolved problems.

Most machine learning tasks are inherently multi-objective. This means that the learner has to come up with a model that performs well across a number of base objectives $\cL_{1}, \ldots, \cL_{p}$, as opposed to a single one. Since optimizing with respect to multiple objectives at the same time is often computationally expensive, the base objectives are often combined in an ensemble $\sum_{k=1}^{p}\lambda_{k}\cL_{k}$, thereby reducing the problem to scalar optimization. The mixture weights $\lambda_{k}$ are set to uniform or some other fixed distribution, based on the learner's preferences. We argue that learning with a fixed distribution on the mixture weights runs the risk of overfitting to some individual objectives and significantly harming others, despite performing well on an entire ensemble. Moreover, in reality, the true preferences of a learner across multiple objectives are often unknown or hard to express as a specific distribution. Instead, we propose a new framework of \emph{Agnostic Learning with Multiple Objectives} ($\almo$), where a model is optimized for \emph{any} weights in the mixture of base objectives. We present data-dependent Rademacher complexity guarantees for learning in the $\almo$ framework, which are used to guide a scalable optimization algorithm and the corresponding regularization.

Pre-training datasets are typically collected from web content and lack inherent domain divisions. For instance, widely used datasets like Common Crawl do not include explicit domain labels, while manually curating labeled datasets such as The Pile is labor-intensive. Consequently, identifying an optimal pre-training data mixture remains a challenging problem, despite its significant benefits for pre-training performance. To address these challenges, we propose CLustering-based Iterative Data Mixture Bootstrapping (Nemotron-CLIMB), an automated framework that discovers, evaluates, and refines data mixtures in a pre-training setting. Specifically, Nemotron-CLIMB embeds and clusters large-scale datasets in a semantic space and then iteratively searches for optimal mixtures using a smaller proxy model and a predictor. When continuously trained on 400B tokens with this mixture, our 1B model exceeds the state-of-the-art Llama-3.2-1B by 2.0%. Moreover, we observe that optimizing for a specific domain (e.g., Social Sciences) yields a 5% improvement over random sampling. Finally, we introduce Nemotron-ClimbLab, a filtered 1.2-trillion-token corpus with 20 clusters as a research playground, and Nemotron-ClimbMix, a compact yet powerful 400-billion-token dataset designed for efficient pre-training that delivers superior performance under an equal token budget. We analyze the final data mixture, elucidating the characteristics of an optimal data mixture. Our data is available at: https://research.nvidia.com/labs/lpr/climb/