Gradient Boosting Decision Trees (GBDTs) dominate tabular machine learning, with modern implementations like XGBoost, LightGBM, and CatBoost being based on Newton boosting: a second-order descent step in the space of decision trees. Despite its empirical success, the global convergence of Newton boosting is poorly understood compared to first-order boosting. In this paper, we introduce Restricted Newton Descent, which studies convex optimization with Newton's method on Hilbert spaces with inexact iterates, based on the concepts of cosine angle and weak gradient edge. Within this framework, we recover Newton boosting with GBDTs and classical finite-dimensional theory as special cases. We first prove that vanilla Newton boosting achieves a linear rate of convergence for smooth, strongly convex losses that satisfy a Hessian-dominance condition. To handle general convex losses with Lipschitz Hessians, we extend a recent gradient regularized Newton scheme to the restricted weak learner setting. This scheme minimally modifies the classical algorithm by introducing an adaptive $\ell_2$-regularization term proportional to the square root of the gradient norm at each iteration. We establish a $\mathcal{O}(\frac{1}{k^2})$ rate for this scheme, thereby obtaining a globally convergent second-order GBDT algorithm with a rate matching that of first-order boosting with Nesterov momentum. In numerical experiments, we show that our scheme converges while vanilla Newton boosting may diverge.

We show that, compared to earlier works, it is less inclined to be stuck in trivial solutions and avoids the need for a strong generative model.

Mitigating the negative impact of noisy labels has been a perennial issue in supervised learning. Robust loss functions have emerged as a prevalent solution to this problem. In this work, we introduce the V ariation Ratio as a novel property related to the robustness of loss functions, and propose a new family of robust loss functions, termed V ariation-Bounded Loss (VBL), which is characterized by a bounded variation ratio. We provide theoretical analyses of the variation ratio, proving that a smaller variation ratio would lead to better robustness. Furthermore, we reveal that the variation ratio provides a feasible method to relax the symmetric condition and offers a more concise path to achieve the asymmetric condition. Based on the variation ratio, we reformulate several commonly used loss functions into a variation-bounded form for practical applications.

The increasing reliance on generative AI models is rapidly increasing the volume of synthetic data, with some projections suggesting that most available new data for training could be machine-generated by 2030 Gartner, Inc. (2022). This shift to a mainly synthetic content presents a critical challenge: repeated training in synthetic data leads to a phenomenon known as model collapse, where model performance degrades over generations of training, eventually rendering the models ineffective. While the causes of model collapse are increasingly understood, effective mitigation strategies remain scarce. We address this challenge by leveraging a key insight: auto-regressive models tend to generate text sequences to which they assign high confidence (i.e., high log-likelihood). Based on this observation, we introduce the Truncated-Cross-Entropy (TCE) loss function. Our experiments demonstrate that models trained with TCE not only learn effectively but also exhibit significantly increased resilience, tolerating over 2.3 more synthetic data before the onset of collapse. In addition, we provide an open-source benchmark for collapse dynamics in mixed-data settings. Our results demonstrate that confidence-aware training objectives can substantially delay collapse onset, offering a practical and generalizable tool for model robustness under synthetic-data exposure. Generative models have become the foundation for modern AI applications in several modalities, including text, image, code, and audio. Large Language Models (LLMs) such as ChatGPT (OpenAI et al., 2024), LLaMA (Grattafiori et al., 2024) and Gemma (Team et al., 2025), as well as image generators DALL-E (Ramesh et al., 2021) and Imagen (Saharia et al., 2022), all rely on large datasets scraped from the Web. As these models are continuously updated to reflect recent knowledge and linguistic patterns, the need for ever larger and frequently refreshed training corpora has grown substantially. However, this demand is colliding with a shift in the data landscape: synthetic content is increasingly populating the Internet, contaminating the very datasets used for model training. This shift raises fundamental concerns.