However, sampling is not the only source of uncertainty. In practice, distributions change between locations and across time.

Reinforcement learning for large language models (LLMs) faces a fundamental tension: high-throughput inference engines and numerically-precise training systems produce different probability distributions from the same parameters, creating a training-inference mismatch. We prove this mismatch has an asymmetric effect: the bound on log-probability mismatch scales as $(1-p)$ where $p$ is the token probability. For high-probability tokens, this bound vanishes, contributing negligibly to sequence-level mismatch. For low-probability tokens in the tail, the bound remains large, and moreover, when sampled, these tokens exhibit systematically biased mismatches that accumulate over sequences, destabilizing gradient estimation. Rather than applying post-hoc corrections, we propose constraining the RL objective to a dynamically-pruned ``safe'' vocabulary that excludes the extreme tail. By pruning such tokens, we trade large, systematically biased mismatches for a small, bounded optimization bias. Empirically, our method achieves stable training; theoretically, we bound the optimization bias introduced by vocabulary pruning.

Recent studies show that Large Language Models (LLMs) with safety alignment can be jail-broken by fine-tuning on a dataset mixed with harmful data. For the first time in the literature, we show that the jail-break effect can be mitigated by separating two states in the fine-tuning stage to respectively optimize over the alignment and user datasets. Unfortunately, our subsequent study shows that this simple Bi-State Optimization (BSO) solution experiences convergence instability when steps invested in its alignment state is too small, leading to downgraded alignment performance. By statistical analysis, we show that the \textit{excess drift} towards the switching iterates of the two states could be a probable reason for the instability. To remedy this issue, we propose \textbf{L}azy(\textbf{i}) \textbf{s}afety \textbf{a}lignment (\textbf{Lisa}), which introduces a proximal term to constraint the drift of each state. Theoretically, the benefit of the proximal term is supported by the convergence analysis, wherein we show that a sufficient large proximal factor is necessary to guarantee Lisa's convergence. Empirically, our results on four downstream fine-tuning tasks show that Lisa with a proximal term can significantly increase alignment performance while maintaining the LLM's accuracy on the user tasks. Code is available at https://github.com/git-disl/Lisa.

Common statistical measures of uncertainty such as $p$-values and confidence intervals quantify the uncertainty due to sampling, that is, the uncertainty due to not observing the full population. However, sampling is not the only source of uncertainty. In practice, distributions change between locations and across time. This makes it difficult to gather knowledge that transfers across data sets. We propose a measure of instability that quantifies the distributional instability of a statistical parameter with respect to Kullback-Leibler divergence, that is, the sensitivity of the parameter under general distributional perturbations within a Kullback-Leibler divergence ball. In addition, we quantify the instability of parameters with respect to directional or variable-specific shifts. Measuring instability with respect to directional shifts can be used to detect under which kind of distribution shifts a statistical conclusion might be reversed. We discuss how such knowledge can inform data collection for transfer learning of statistical parameters under shifted distributions. We evaluate the performance of the proposed measure on real data and show that it can elucidate the distributional instability of a parameter with respect to certain shifts and can be used to improve estimation accuracy under shifted distributions.

In diffusion models, UNet is the most popular network backbone, since its long skip connects (LSCs) to connect distant network blocks can aggregate long-distant information and alleviate vanishing gradient. Unfortunately, UNet often suffers from unstable training in diffusion models which can be alleviated by scaling its LSC coefficients smaller. However, theoretical understandings of the instability of UNet in diffusion models and also the performance improvement of LSC scaling remain absent yet. To solve this issue, we theoretically show that the coefficients of LSCs in UNet have big effects on the stableness of the forward and backward propagation and robustness of UNet. Specifically, the hidden feature and gradient of UNet at any layer can oscillate and their oscillation ranges are actually large which explains the instability of UNet training. Moreover, UNet is also provably sensitive to perturbed input, and predicts an output distant from the desired output, yielding oscillatory loss and thus oscillatory gradient. Besides, we also observe the theoretical benefits of the LSC coefficient scaling of UNet in the stableness of hidden features and gradient and also robustness. Finally, inspired by our theory, we propose an effective coefficient scaling framework ScaleLong that scales the coefficients of LSC in UNet and better improve the training stability of UNet. Experimental results on CIFAR10, CelebA, ImageNet and COCO show that our methods are superior to stabilize training, and yield about 1.5x training acceleration on different diffusion models with UNet or UViT backbones.