Large Language Models (LLMs) have demonstrated remarkable capabilities across numerous tasks, yet principled explanations for their underlying mechanisms and several phenomena, such as scaling laws, hallucinations, and related behaviors, remain elusive. In this work, we revisit the classical relationship between compression and prediction, grounded in Kolmogorov complexity and Shannon information theory, to provide deeper insights into LLM behaviors. By leveraging the Kolmogorov Structure Function and interpreting LLM compression as a two-part coding process, we offer a detailed view of how LLMs acquire and store information across increasing model and data scales - from pervasive syntactic patterns to progressively rarer knowledge elements. Motivated by this theoretical perspective and natural assumptions inspired by Heap's and Zipf's laws, we introduce a simplified yet representative hierarchical data-generation framework called the Syntax-Knowledge model. Under the Bayesian setting, we show that prediction and compression within this model naturally lead to diverse learning and scaling behaviors of LLMs. In particular, our theoretical analysis offers intuitive and principled explanations for both data and model scaling laws, the dynamics of knowledge acquisition during training and fine-tuning, factual knowledge hallucinations in LLMs.

Geometry optimization of atomic structures is a common and crucial task in computational chemistry and materials design. Following the learning to optimize paradigm, we propose a new multi-agent reinforcement learning method called Multi-Agent Crystal Structure optimization (MACS) to address periodic crystal structure optimization. MACS treats geometry optimization as a partially observable Markov game in which atoms are agents that adjust their positions to collectively discover a stable configuration. We train MACS across various compositions of reported crystalline materials to obtain a policy that successfully optimizes structures from the training compositions as well as structures of larger sizes and unseen compositions, confirming its excellent scalability and zero-shot transferability. We benchmark our approach against a broad range of state-of-theart optimization methods and demonstrate that MACS optimizes periodic crystal structures significantly faster, with fewer energy calculations, and the lowest failure rate. Code is available at https://github.com/lrcfmd/macs.

Reinforcement Learning from Human Feedback (RLHF) has shown remarkable success in aligning Large Language Models (LLMs) with human preferences. Traditional RLHF methods rely on a fixed dataset, which often suffers from limited coverage. To this end, online RLHF has emerged as a promising direction, enabling iterative data collection and refinement. Despite its potential, this paradigm faces a key bottleneck: the requirement to continuously integrate new data into the dataset and re-optimize the model from scratch at each iteration, resulting in computational and storage costs that grow linearly with the number of iterations. In this work, we address this challenge by proposing a one-pass reward modeling method that eliminates the need to store historical data and achieves constant-time updates per iteration. Specifically, we first formalize RLHF as a contextual preference bandit and develop a new algorithm based on online mirror descent with a tailored local norm, replacing the standard maximum likelihood estimation for reward modeling. We then apply it to various online RLHF settings, including passive data collection, active data collection, and deployment-time adaptation. We provide theoretical guarantees showing that our method enhances both statistical and computational efficiency.

Prompting is one of the main ways to adapt a pretrained model to target tasks. Besides manually constructing prompts, many prompt optimization methods have been proposed in the literature. Method development is mainly empirically driven, with less emphasis on a conceptual understanding of prompting. In this paper we discuss how optimal prompting can be understood through a Bayesian view, which also implies some fundamental limitations of prompting that can only be overcome by tuning weights. The paper explains in detail how meta-trained neural networks behave as Bayesian predictors over the pretraining distribution, whose hallmark feature is rapid in-context adaptation. Optimal prompting can be studied formally as conditioning these Bayesian predictors, yielding criteria for target tasks where optimal prompting is and is not possible. We support the theory with educational experiments on LSTMs and Transformers, where we compare different versions of prefix-tuning and different weight-tuning methods. We also confirm that soft prefixes, which are sequences of real-valued vectors outside the token alphabet, can lead to very effective prompts for trained and even untrained networks by manipulating activations in ways that are not achievable by hard tokens. This adds an important mechanistic aspect beyond the conceptual Bayesian theory.

Efficient online decision-making in contextual bandits is challenging, as methods without informative priors often suffer from computational or statistical inefficiencies. In this work, we leverage pre-trained diffusion models as expressive priors to capture complex action dependencies and develop a practical algorithm that efficiently approximates posteriors under such priors, enabling both fast updates and sampling. Empirical results demonstrate the effectiveness and versatility of our approach across diverse contextual bandit settings.

Large language models (LLMs) have recently transformed from text-based assistants to autonomous agents capable of planning, reasoning, and iteratively improving their actions. While numerical reward signals and verifiers can effectively rank candidate actions, they often provide limited contextual guidance.

Prompt caching is critical for reducing latency and cost in LLM inference--OpenAI and Anthropic report up to 50-90% cost savings through prompt reuse. Despite its widespread success, little is known about what constitutes an optimal prompt caching policy, particularly when optimizing tail latency--a metric of central importance to practitioners. The widely used Least Recently Used (LRU) policy can perform arbitrarily poor on this metric, as it is oblivious to the heterogeneity of conversation lengths. To address this gap, we propose Tail-Optimized LRU, a simple two-line modification that reallocates KV cache capacity to prioritize high-latency conversations by evicting cache entries that are unlikely to affect future turns. Though the implementation is simple, we prove its optimality under a natural stochastic model of conversation dynamics, providing the first theoretical justification for LRU in this setting--a result that may be of independent interest to the caching community. Experimentally, on real conversation data WildChat [Zhao et al., 2024], Tail-Optimized LRU achieves up to 27.5% reduction in P90 tail Time to First Token latency and 23.9% in P95 tail latency compared to LRU, along with up to 38.9% decrease in SLO violations of 200ms. We believe this provides a practical and theoretically grounded option for practitioners seeking to optimize tail latency in real-world LLM deployments.

Directed acyclic graph (DAG) learning is a central task in structure discovery and causal inference. Although the field has witnessed remarkable advances over the past few years, it remains statistically and computationally challenging to learn a single (point estimate) DAG from data, let alone provide uncertainty quantification. We address the difficult task of quantifying graph uncertainty by developing a Bayesian variational inference framework based on novel, provably valid distributions that have support directly on the space of sparse DAGs. These distributions, which we use to define our prior and variational posterior, are induced by a projection operation that maps an arbitrary continuous distribution onto the space of sparse weighted acyclic adjacency matrices. While this projection is combinatorial, it can be solved efficiently using recent continuous reformulations of acyclicity constraints. We empirically demonstrate that our method, ProDAG, can outperform state-of-the-art alternatives in both accuracy and uncertainty quantification.

Monitoring AIs at runtime can help us detect and stop harmful actions. In this paper, we study how to efficiently combine multiple runtime monitors into a single monitoring protocol. The protocol's objective is to maximize the probability of applying a safety intervention on misaligned outputs (i.e., maximize recall). Since running monitors and applying safety interventions are costly, the protocol also needs to adhere to an average-case budget constraint. Taking the monitors' performance and cost as given, we develop an algorithm to find the best protocol. The algorithm exhaustively searches over when and which monitors to call, and allocates safety interventions based on the Neyman-Pearson lemma. By focusing on likelihood ratios and strategically trading off spending on monitors against spending on interventions, we more than double our recall rate compared to a naive baseline in a code review setting. We also show that combining two monitors can Pareto dominate using either monitor alone. Our framework provides a principled methodology for combining existing monitors to detect undesirable behavior in cost-sensitive settings.

Molecular synthetic space projecting is a critical technique in de novo molecular design, which aims to rectify molecules without synthesizability guarantee by converting them into synthetic postfix notations. However, the vast synthesizable chemical space and the discrete data modalities involved pose significant challenges to postfix notation conversion benchmarking. In this paper, we exploit conditional probability transitions in discrete state space and introduce MoleBridge, a deep generative model built on the Markov bridge approach for designing postfix notations of molecular synthesis pathways. MoleBridge consists of two iterative optimizations: i) Autoregressive extending of notation tokens from molecular graphs, and ii) generation of discrete reaction postfix notations through Markov bridge, where noisy token blocks are progressively denoised over multi-step iterations. For the challenging second iteration, which demands sensitivity to incorrect generative probability paths within intricate chemical spaces, we employ a thinking and denoising separation approach to denoise. Empirically, we find that MoleBridge is capable of accurately predicting synthesis pathways while exhibiting excellent performance in a variety of application scenarios.