We introduce the (Wishart) projection mechanism, a randomized map of the form $S \mapsto M f(S)$ with $M \sim W_d(1/r I_d, r)$ and study its differential privacy properties. For vector-valued queries $f$, we prove non-asymptotic DP guarantees without any additive noise, showing that Wishart randomness alone can suffice. For matrix-valued queries, however, we establish a sharp negative result: in the noise-free setting, the mechanism is not DP, and we demonstrate its vulnerability by implementing a near perfect membership inference attack (AUC $> 0.99$). We then analyze a noisy variant and prove privacy amplification due to randomness and low rank projection, in both large- and small-rank regimes, yielding stronger privacy guarantees than additive noise alone. Finally, we show that LoRA-style updates are an instance of the matrix-valued mechanism, implying that LoRA is not inherently private despite its built-in randomness, but that low-rank fine-tuning can be more private than full fine-tuning at the same noise level. Preliminary experiments suggest that tighter accounting enables lower noise and improved accuracy in practice.

The problem of optimising functions with intractable gradients frequently arise in machine learning and statistics, ranging from maximum marginal likelihood estimation procedures to fine-tuning of generative models. Stochastic approximation methods for this class of problems typically require inner sampling loops to obtain (biased) stochastic gradient estimates, which rapidly becomes computationally expensive. In this work, we develop sequential Monte Carlo (SMC) samplers for optimisation of functions with intractable gradients. Our approach replaces expensive inner sampling methods with efficient SMC approximations, which can result in significant computational gains. We establish convergence results for the basic recursions defined by our methodology which SMC samplers approximate. We demonstrate the effectiveness of our approach on the reward-tuning of energy-based models within various settings.

We develop a near-optimal testing procedure under the framework of Gaussian differential privacy for simple as well as one- and two-sided tests under monotone likelihood ratio conditions. Our mechanism is based on a private mean estimator with data-driven clamping bounds, whose population risk matches the private minimax rate up to logarithmic factors. Using this estimator, we construct private test statistics that achieve the same asymptotic relative efficiency as the non-private, most powerful tests while maintaining conservative type I error control. In addition to our theoretical results, our numerical experiments show that our private tests outperform competing DP methods and offer comparable power to the non-private most powerful tests, even at moderately small sample sizes and privacy loss budgets.

The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard approaches typically require high-fidelity training data. In many practical settings, the data are indirectly observed through noisy and nonlinear measurement. The challenge lies not only in approximating the coefficients of the SDEs, but in simultaneously inferring the posterior updates given the observations. In this work, we present a neural path estimation approach to solve stochastic dynamical systems based on variational inference. We first derive a stochastic control problem that solve filtering posterior path measure corresponding to a pathwise Zakai equation. We then construct a generative model that maps the prior path measure to posterior measure through the controlled diffusion and the associated Randon-Nykodym derivative. Through an amortization of sample paths of the observation process, the control is learned by an embedding of the noisy observation paths. Thus, we learn the unknown prior SDE and the control can recover the conditional path measure given the observation sample paths and we learn an associated SDE which induces the same path measure. In the end, we perform experiments on nonlinear dynamical systems, demonstrating the model's ability to learn multimodal, chaotic, or high dimensional systems.

Evaluating large language models (LLMs) on open-ended tasks without ground-truth labels is increasingly done via the LLM-as-a-judge paradigm. A critical but under-modeled issue is that judge LLMs differ substantially in reliability; treating all judges equally can yield biased leaderboards and misleading uncertainty estimates. More data can make evaluation more confidently wrong under misspecified aggregation. We propose a judge-aware ranking framework that extends the Bradley-Terry-Luce model by introducing judge-specific discrimination parameters, jointly estimating latent model quality and judge reliability from pairwise comparisons without reference labels. We establish identifiability up to natural normalizations and prove consistency and asymptotic normality of the maximum likelihood estimator, enabling confidence intervals for score differences and rank comparisons. Across multiple public benchmarks and a newly collected dataset, our method improves agreement with human preferences, achieves higher data efficiency than unweighted baselines, and produces calibrated uncertainty quantification for LLM rankings.

We introduce a model-agnostic forward diffusion process for time-series forecasting that decomposes signals into spectral components, preserving structured temporal patterns such as seasonality more effectively than standard diffusion. Unlike prior work that modifies the network architecture or diffuses directly in the frequency domain, our proposed method alters only the diffusion process itself, making it compatible with existing diffusion backbones (e.g., DiffWave, TimeGrad, CSDI). By staging noise injection according to component energy, it maintains high signal-to-noise ratios for dominant frequencies throughout the diffusion trajectory, thereby improving the recoverability of long-term patterns. This strategy enables the model to maintain the signal structure for a longer period in the forward process, leading to improved forecast quality. Across standard forecasting benchmarks, we show that applying spectral decomposition strategies, such as the Fourier or Wavelet transform, consistently improves upon diffusion models using the baseline forward process, with negligible computational overhead. The code for this paper is available at https://anonymous.4open.science/r/D-FDP-4A29.

The performance of large language models (LLMs) on verifiable tasks is usually measured by pass@k, the probability of answering a question correctly at least once in k trials. At a fixed budget, a more suitable metric is coverage@cost, the average number of unique questions answered as a function of the total number of attempts. We connect the two metrics and show that the empirically-observed power-law behavior in pass@k leads to a sublinear growth of the coverage@cost (diminishing returns). To solve this problem, we propose Reset-and-Discard (ReD), a query method of LLMs that increases coverage@cost for any given budget, regardless of the pass@k form. Moreover, given a pass@k, we can quantitatively predict the savings in the total number of attempts using ReD. If pass@k is not available for the model, ReD can infer its power-law exponent. Experiments on three LLMs using HumanEval demonstrate that ReD substantially reduces the required attempts, tokens, and USD cost to reach a desired coverage, while also offering an efficient way to measure inference power-laws.

We introduce Statsformer, a principled framework for integrating large language model (LLM)-derived knowledge into supervised statistical learning. Existing approaches are limited in adaptability and scope: they either inject LLM guidance as an unvalidated heuristic, which is sensitive to LLM hallucination, or embed semantic information within a single fixed learner. Statsformer overcomes both limitations through a guardrailed ensemble architecture. We embed LLM-derived feature priors within an ensemble of linear and nonlinear learners, adaptively calibrating their influence via cross-validation. This design yields a flexible system with an oracle-style guarantee that it performs no worse than any convex combination of its in-library base learners, up to statistical error. Empirically, informative priors yield consistent performance improvements, while uninformative or misspecified LLM guidance is automatically downweighted, mitigating the impact of hallucinations across a diverse range of prediction tasks.

Partial Least Squares (PLS) learns shared structure from paired data via the top singular vectors of the empirical cross-covariance (PLS-SVD), but multimodal datasets often have missing entries in both views. We study PLS-SVD under independent entry-wise missing-completely-at-random masking in a proportional high-dimensional spiked model. After appropriate normalization, the masked cross-covariance behaves like a spiked rectangular random matrix whose effective signal strength is attenuated by $\sqrtρ$, where $ρ$ is the joint entry retention probability. As a result, PLS-SVD exhibits a sharp BBP-type phase transition: below a critical signal-to-noise threshold the leading singular vectors are asymptotically uninformative, while above it they achieve nontrivial alignment with the latent shared directions, with closed-form asymptotic overlap formulas. Simulations and semi-synthetic multimodal experiments corroborate the predicted phase diagram and recovery curves across aspect ratios, signal strengths, and missingness levels.

We introduce a flexible empirical Bayes approach for fitting Bayesian generalized linear models. Specifically, we adopt a novel mean-field variational inference (VI) method and the prior is estimated within the VI algorithm, making the method tuning-free. Unlike traditional VI methods that optimize the posterior density function, our approach directly optimizes the posterior mean and prior parameters. This formulation reduces the number of parameters to optimize and enables the use of scalable algorithms such as L-BFGS and stochastic gradient descent. Furthermore, our method automatically determines the optimal posterior based on the prior and likelihood, distinguishing it from existing VI methods that often assume a Gaussian variational. Our approach represents a unified framework applicable to a wide range of exponential family distributions, removing the need to develop unique VI methods for each combination of likelihood and prior distributions. We apply the framework to solve sparse logistic regression and demonstrate the superior predictive performance of our method in extensive numerical studies, by comparing it to prevalent sparse logistic regression approaches.