Pretrained Large Language Models (LLMs) achieve strong performance across a wide range of tasks, yet exhibit substantial variability in the various layers' training quality with respect to specific downstream applications, limiting their downstream performance. It is therefore critical to estimate layer-wise training quality in a manner that accounts for both model architecture and training data. However, existing approaches predominantly rely on model-centric heuristics (such as spectral statistics, outlier detection, or uniform allocation) while overlooking the influence of data. To address these limitations, we propose LAYERIF, a datadriven framework that leverages Influence Functions to quantify the training quality of individual layers in a principled and task-sensitive manner. By isolating each layer's gradients and measuring the sensitivity of the validation loss to training examples by computing layer-wise influences, we derive data-driven estimates of layer importance. Notably, our method produces task-specific layer importance estimates for the same LLM, revealing how layers specialize for different test-time evaluation tasks. We demonstrate the utility of our scores by leveraging them for two downstream applications: (a) expert allocation in LoRA-MoE architectures and (b) layer-wise sparsity distribution for LLM pruning. Experiments across multiple LLM architectures demonstrate that our model-agnostic, influence-guided allocation leads to consistent gains in task performance.

Optimization modeling is one of the most crucial but technical parts of operations research (OR). To automate the modeling process, existing works have leveraged large language models (LLMs), prompting them to break down tasks into steps for generating variables, constraints, and objectives. However, due to the highly complex mathematical structures inherent in OR problems, standard fixed-step decomposition often fails to achieve high performance. To address this challenge, we introduce OptiTree, a novel tree search approach designed to enhance modeling capabilities for complex problems through adaptive problem decomposition into simpler subproblems. Specifically, we develop a modeling tree that organizes a wide range of OR problems based on their hierarchical problem taxonomy and complexity, with each node representing a problem category and containing relevant high-level modeling thoughts. Given a problem to model, we recurrently search the tree to identify a series of simpler subproblems and synthesize the global modeling thoughts by adaptively integrating the hierarchical thoughts. Experiments show that OptiTree significantly improves the modeling accuracy compared to the state-of-theart, achieving over 10% improvements on the challenging benchmarks.

Large Language Models have excelled in various domains but face efficiency challenges due to the growing Key-Value (KV) cache required for long-sequence inference. Recent efforts aim to reduce KV cache size by evicting vast non-critical cache elements during runtime while preserving generation quality. However, these methods typically allocate compression budgets uniformly across all attention heads, ignoring the unique attention patterns of each head. In this paper, we establish a theoretical loss upper bound between pre-and post-eviction attention output, explaining the optimization target of prior cache eviction methods, while guiding the optimization of adaptive budget allocation. Base on this, we propose Ada-KV, the first head-wise adaptive budget allocation strategy. It offers plug-and-play benefits, enabling seamless integration with prior cache eviction methods. Extensive evaluations on 13 datasets from Ruler and 16 datasets from LongBench, all conducted under both question-aware and question-agnostic scenarios, demonstrate substantial quality improvements over existing methods. Our code is available at https://github.com/FFY0/AdaKV.

Best Arm Identification (BAI) algorithms are deployed in data-sensitive applications, such as adaptive clinical trials or user studies. Driven by the privacy concerns of these applications, we study the problem of fixed-confidence BAI under global Differential Privacy (DP) for Bernoulli distributions. While numerous asymptotically optimal BAI algorithms exist in the non-private setting, a significant gap remains between the best lower and upper bounds in the global DP setting. This work reduces this gap to a small multiplicative constant, for any privacy budget ϵ. First, we provide a tighter lower bound on the expected sample complexity of any δ-correct and ϵ-global DP strategy.

Test-Time Scaling (TTS) improves the performance of Large Language Models (LLMs) by using additional inference-time computation to explore multiple reasoning paths through search. Yet how to allocate a fixed rollout budget most effectively during search remains underexplored, often resulting in inefficient use of compute at test time. To bridge this gap, we formulate test-time search as a resource allocation problem and derive the optimal allocation strategy that maximizes the probability of obtaining a correct solution under a fixed rollout budget. Within this formulation, we reveal a core limitation of existing search methods: solution-level allocation tends to favor reasoning directions with more candidates, leading to theoretically suboptimal and inefficient use of compute. To address this, we propose Direction-Oriented Resource Allocation (DORA), a provably optimal method that mitigates this bias by decoupling direction quality from candidate count and allocating resources at the direction level. To demonstrate DORA's effectiveness, we conduct extensive experiments on challenging mathematical reasoning benchmarks including MATH500, AIME2024, and AIME2025. The empirical results show that DORA consistently outperforms strong baselines with comparable computational cost, achieving state-of-the-art accuracy. We hope our findings contribute to a broader understanding of optimal TTS for LLMs.1

Test-time scaling (TTS) enhances the performance of large language models (LLMs) by allocating additional compute resources during inference. However, existing research primarily investigates TTS in single-stage tasks; while many real-world problems are multi-stage complex tasks, composed of a sequence of heterogeneous subtasks with each subtask requires LLM of specific capability. Therefore, we study a novel problem: the test-time compute-optimal scaling in multi-stage complex tasks, aiming to select suitable models and allocate budgets per subtask to maximize overall performance. TTS in multi-stage tasks introduces two fundamental challenges: (i) The combinatorial search space of model and budget allocations, combined with the high cost of inference, makes brute-force search impractical.

We study matching markets with ties, where workers on one side of the market may have tied preferences over jobs, determined by their matching utilities. Unlike classical two-sided markets with strict preferences, no single stable matching exists that is utility-maximizing for all workers. To address this challenge, we introduce the Optimal Stable Share (OSS)-ratio, which measures the ratio of a worker's maximum achievable utility in any stable matching to their utility in a given matching. We prove that distributions over only stable matchings can incur linear utility losses, i.e., an Ω(N) OSS-ratio, where N is the number of workers. To overcome this, we design an algorithm that efficiently computes a distribution over (possibly non-stable) matchings, achieving an asymptotically tight O(logN) OSS-ratio. When exact utilities are unknown, our second algorithm guarantees workers a logarithmic approximation of their optimal utility under bounded instability. Finally, we extend our offline approximation results to a bandit learning setting where utilities are only observed for matched pairs. In this setting, we consider worker-optimal stable regret, design an adaptive algorithm that smoothly interpolates between markets with strict preferences and those with statistical ties, and establish a lower bound revealing the fundamental trade-off between strict and tied preference regimes.

We consider the privacy amplification properties of a sampling scheme in which a user's data is used in k steps chosen randomly and uniformly from a sequence (or set) of t steps. This sampling scheme has been recently applied in the context of differentially private optimization [Chua et al., 2024a, Choquette-Choo et al., 2025] and is also motivated by communication-efficient high-dimensional private aggregation [Asi et al., 2025]. Existing analyses of this scheme either rely on privacy amplification by shuffling which leads to overly conservative bounds or require Monte Carlo simulations that are computationally prohibitive in most practical scenarios. We give the first theoretical guarantees and numerical estimation algorithms for this sampling scheme. In particular, we demonstrate that the privacy guarantees of random k-out-of-t allocation can be upper bounded by the privacy guarantees of the well-studied independent (or Poisson) subsampling in which each step uses the user's data with probability (1+o(1))k/t. Further, we provide two additional analysis techniques that lead to numerical improvements in several parameter regimes. Altogether, our bounds give efficiently-computable and nearly tight numerical results for random allocation applied to Gaussian noise addition.

Motivated by applications such as cloud platforms allocating GPUs to users or governments deploying mobile health units across competing regions, we study the constrained dynamic allocation of a reusable resource to a group of strategic agents. Our objective is to simultaneously (i) maximize social welfare, (ii) satisfy multidimensional long-term cost constraints, and (iii) incentivize truthful reporting. We begin by numerically evaluating primal-dual methods widely used in constrained online optimization and find them to be highly fragile in strategic settings - agents can easily manipulate their reports to distort future dual updates for future gain. To address this vulnerability, we develop an incentive-aware framework that makes primal-dual methods robust to strategic behavior. Our primal-side design combines epoch-based lazy updates - discouraging agents from distorting dual updates - with dual-adjust pricing and randomized exploration techniques that extract approximately truthful signals for learning. On the dual side, we design a novel online learning subroutine to resolve a circular dependency between actions and predictions; this makes our mechanism achieve eO( T)social welfare regret (where T is the number of allocation rounds), satisfies all cost constraints, and ensures incentive alignment. This eO( T) performance matches that of non-strategic allocation approaches while additionally exhibiting robustness to strategic agents.

The growing interest in employing large language models (LLMs) for decision-making in social and economic contexts has raised questions about their potential to function as agents in these domains. A significant number of societal problems involve the distribution of resources, where fairness, along with economic efficiency, play a critical role in the desirability of outcomes. In this paper, we examine whether LLM responses adhere to fundamental fairness concepts such as equitability, envy-freeness, and Rawlsian maximin, and investigate their alignment with human preferences. We evaluate the performance of several LLMs, providing a comparative benchmark of their ability to reflect these measures. Our results demonstrate a lack of alignment between current LLM responses and human distributional preferences. Moreover, LLMs are unable to utilize money as a transferable resource to mitigate inequality. Nonetheless, we demonstrate a stark contrast when (some) LLMs are tasked with selecting from a predefined menu of options rather than generating one. In addition, we analyze the robustness of LLM responses to variations in semantic factors (e.g., intentions or personas) or non-semantic prompting changes (e.g., templates or orderings). Finally, we highlight potential strategies aimed at enhancing the alignment of LLM behavior with well-established fairness concepts.