The Fisher Information Matrix (FIM) is the canonical local measure of the curvature of a statistical model's log-likelihood surface, and its dominant eigenvalue λmax quantifies the worst-case sensitivity of the model's output distribution to infinitesimal parameter perturbation [1, 2]. The spectral properties of the FIM of neural networks have been studied directly in the random matrix theory literature. Pennington and Worah [4] derive the limiting spectral density of the FIM of a single-hidden-layer network in the high-dimensional asymptotic regime, building on the broader programme of analysing neural network Hessian and kernel spectra via random matrix methods [5, 6], with subsequent work extending these techniques to deeper architectures and non-asymptotic regimes [7, 8]. These results characterize the typical (bulk and edge) spectral behaviour of the FIM for a fixed network and a random or structured input ensemble. This paper studies a complementary question, posed as a perturbation problem rather than an asymptotic-spectrum problem: how does the dominant eigenvalue of a fixed, evaluated empirical FIM change under two specific structured perturbations of the underlying distribution? The first perturbation is a change in the conditioning input away from a reference (in-distribution) ensemble. The second is a structured additive perturbation of the model's own parameters by finite-precision quantization noise -- a perturbation of independent mathematical interest, since it falls outside the i.i.d.-input asymptotic regime treated in the random matrix literature cited above, and instead concerns a fixed network whose parameters, not its input distribution, are perturbed by a noise process with a specific, analytically tractable structure (Definition 4.1). To our knowledge, this parameterperturbation question for the FIM's dominant eigenvalue, under either source of departure, has not been previously formalized.

Contrastive decoding strategies are widely used to reduce object hallucinations in multimodal large language models (MLLMs). These methods work by constructing contrastive samples to induce hallucinations and then suppressing them in the output distribution. However, this paper demonstrates that such approaches fail to effectively mitigate the hallucination problem. The performance improvements observed on POPE Benchmark are largely driven by two misleading factors: (1) crude, unidirectional adjustments to the model's output distribution and (2) the adaptive plausibility constraint, which reduces the sampling strategy to greedy search. To further illustrate these issues, we introduce a series of spurious improvement methods and evaluate their performance against contrastive decoding techniques. Experimental results reveal that the observed performance gains in contrastive decoding are entirely unrelated to its intended goal of mitigating hallucinations. Our findings challenge common assumptions about the effectiveness of contrastive decoding strategies and pave the way for developing genuinely effective solutions to hallucinations in MLLMs.

Diffusion LLMs have emerged as a promising alternative to conventional autoregressive LLMs, offering substantial potential for improving runtime efficiency. However, existing diffusion models fail to provably enforce user-specified formal constraints, such as regular expressions, which makes them unreliable for tasks that require structured outputs, such as fixed-schema JSON generation. Unlike autoregressive models, which generate tokens sequentially, diffusion LLMs predict a block of tokens in parallel. This parallelism makes traditional constrained decoding algorithms, designed to enforce constraints with sequential token prediction, ineffective at preserving the true output distribution. To address this limitation, we propose DINGO, a dynamic programming-based constrained decoding strategy that is both efficient and provably distribution-preserving. DINGO enables sampling of output strings with the highest probability under the model's predicted distribution while strictly adhering to any user-specified regular expression. On standard symbolic math and JSON generation benchmarks, DINGO achieves up to a 68%points of improvement over unconstrained inference. The code is available at DINGO.

Despite the remarkable capabilities of large language models (LLMs) across diverse applications, they remain vulnerable to generating content that violates safety regulations and policies. To mitigate these risks, LLMs undergo safety alignment; however, they can still be effectively jailbroken. Off-the-shelf guardrail models are commonly deployed to monitor generations, but these models primarily focus on detection rather than ensuring safe decoding of LLM outputs. Moreover, existing efforts lack rigorous safety guarantees, which are crucial for the universal deployment of LLMs and certifiable compliance with regulatory standards. In this paper, we propose a Claim-based Stream Decoding (CSD) algorithm coupled with a statistical risk guarantee framework using conformal analysis.

Adapting billion-parameter language models to a downstream task is still costly, even with parameter-efficient fine-tuning (PEFT). We re-cast task adaptation as output-distribution alignment: the objective is to steer the output distribution toward the task distribution directly during decoding rather than indirectly through weight updates. Building on this view, we introduce Steering Vector Decoding (SVDecode), a lightweight, PEFT-compatible, and theoretically grounded method. We start with a short warm-start fine-tune and extract a task-aware steering vector from the Kullback-Leibler (KL) divergence gradient between the output distribution of the warm-started and pre-trained models. This steering vector is then used to guide the decoding process to steer the model's output distribution towards the task distribution. We theoretically prove that SVDecode is first-order equivalent to the gradient step of full fine-tuning and derive a globally optimal solution for the strength of the steering vector. Across three tasks and nine benchmarks, SVDecode paired with four standard PEFT methods improves multiple-choice accuracy by up to 5 percentage points and open-ended truthfulness by 2 percentage points, with similar gains (1-2 percentage points) on commonsense datasets without adding trainable parameters beyond the PEFT adapter. SVDecode thus offers a lightweight, theoretically grounded path to stronger task adaptation for large language models.

Sampling multiple outputs from a Large Language Model (LLM) and selecting the most frequent (Self-consistency) or highest-scoring (Best-of-N) candidate is a popular approach to achieve higher accuracy in tasks with discrete final answers. Best-of-N (BoN) selects the output with the highest reward, and with perfect rewards, it often achieves near-perfect accuracy. With imperfect rewards from reward models, however, BoN fails to reliably find the correct answer and its performance degrades drastically. We consider the distribution of BoN's outputs and highlight that, although the correct answer does not usually have a probability close to one under imperfect rewards, it is often the most likely outcome. This suggests that the mode of this distribution can be more reliably correct than a sample from it. Based on this idea, we propose Majority-of-the-Bests (MoB), a novel selection mechanism that estimates the output distribution of BoN via bootstrapping and selects its mode. Experimental results across five benchmarks, three different base LLMs, and two reward models demonstrate consistent improvements over BoN in 25 out of 30 setups. We also provide theoretical results for the consistency of the bootstrapping.

Adapting billion-parameter language models to a downstream task is still costly, even with parameter-efficient fine-tuning (PEFT). We re-cast task adaptation as output-distribution alignment: the objective is to steer the output distribution toward the task distribution directly during decoding rather than indirectly through weight updates. Building on this view, we introduce Steering Vector Decoding (SVDecode), a lightweight, PEFT-compatible, and theoretically grounded method. We start with a short warm-start fine-tune and extract a task-aware steering vector from the Kullback-Leibler (KL) divergence gradient between the output distribution of the warm-started and pre-trained models. This steering vector is then used to guide the decoding process to steer the model's output distribution towards the task distribution. We theoretically prove that SVDecode is first-order equivalent to the gradient step of full fine-tuning and derive a globally optimal solution for the strength of the steering vector. Across three tasks and nine benchmarks, SVDecode paired with four standard PEFT methods improves multiple-choice accuracy by up to 5 percentage points and open-ended truthfulness by 2 percentage points, with similar gains (1-2 percentage points) on commonsense datasets without adding trainable parameters beyond the PEFT adapter. SVDecode thus offers a lightweight, theoretically grounded path to stronger task adaptation for large language models.

It is common to reject undesired outputs of Large Language Models (LLMs); however, current methods to do so require an excessive amount of computation to re-sample after a rejection, or distort the distribution of outputs by constraining the output to highly improbable tokens. We present a method, Approximately Aligned Decoding (AprAD), to balance the distortion of the output distribution with computational efficiency, inspired by algorithms from the speculative decoding literature. AprAD allows for the generation of long sequences of text with difficult-to-satisfy constraints, while amplifying low probability outputs much less compared to existing methods. We show through a series of experiments that the task-specific performance of AprAD is comparable to methods that do not distort the output distribution, while being much more computationally efficient.

In this paper, we study the problem of sampling from a distribution under the constraint of differential privacy (DP). Prior works measure the utility of DP sampling with density ratio-based measures such as KL divergence. However, such formulations suffer from two key limitations: 1) they fail to capture the geometric structure of the support, and 2) they are not applicable when the supports of the distributions differ. To deal with these issues, we develop a novel framework for DP sampling with Wasserstein distance as the utility measure. In this formulation, we propose Wasserstein Projection Mechanism (WPM), a minimax optimal mechanism based on Wasserstein projection. Furthermore, we develop efficient algorithms for computing the proposed mechanisms approximately and provide convergence guarantees.

Measurement-based quantum computation (MBQC) is a framework for quantum information processing in which a computational task is carried out through one-qubit measurements on a highly entangled resource state. Due to the indeterminacy of the outcomes of a quantum measurement, the random outcomes of these operations, if not corrected, yield a variational quantum channel family. Traditionally, this randomness is corrected through classical processing in order to ensure deterministic unitary computations. Recently, variational measurement-based quantum computation (VMBQC) has been introduced to exploit this measurement-induced randomness to gain an advantage in generative modeling. A limitation of this approach is that the corresponding channel model has twice as many parameters compared to the unitary model, scaling as $N \times D$, where $N$ is the number of logical qubits (width) and $D$ is the depth of the VMBQC model. This can often make optimization more difficult and may lead to poorly trainable models. In this paper, we present a restricted VMBQC model that extends the unitary setting to a channel-based one using only a single additional trainable parameter. We show, both numerically and algebraically, that this minimal extension is sufficient to generate probability distributions that cannot be learned by the corresponding unitary model.