The achievement of the 2030 Sustainable Development Goals (SDGs) is dependent upon strategic resource distribution. We propose a causal discovery framework using Panel Vector Autoregression, along with both country-specific fixed effects and PCMCI+ conditional independence testing on 168 countries (2000-2025) to develop the first complete causal architecture of SDG dependencies. Utilizing 8 strategically chosen SDGs, we identify a distributed causal network (i.e., no single 'hub' SDG), with 10 statistically significant Granger-causal relationships identified as 11 unique direct effects. Education to Inequality is identified as the most statistically significant direct relationship (r = -0.599; p < 0.05), while effect magnitude significantly varies depending on income levels (e.g., high-income: r = -0.65; lower-middle-income: r = -0.06; non-significant). We also reject the idea that there exists a single 'keystone' SDG. Additionally, we offer a proposed tiered priority framework for the SDGs namely, identifying upstream drivers (Education, Growth), enabling goals (Institutions, Energy), and downstream outcomes (Poverty, Health). Therefore, we conclude that effective SDG acceleration can be accomplished through coordinated multi-dimensional intervention(s), and that single-goal sequential strategies are insufficient.

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.

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.

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.

We introduce ECSEL, an explainable classification method that learns formal expressions in the form of signomial equations, motivated by the observation that many symbolic regression benchmarks admit compact signomial structure. ECSEL directly constructs a structural, closed-form expression that serves as both a classifier and an explanation. On standard symbolic regression benchmarks, our method recovers a larger fraction of target equations than competing state-of-the-art approaches while requiring substantially less computation. Leveraging this efficiency, ECSEL achieves classification accuracy competitive with established machine learning models without sacrificing interpretability. Further, we show that ECSEL satisfies some desirable properties regarding global feature behavior, decision-boundary analysis, and local feature attributions. Experiments on benchmark datasets and two real-world case studies i.e., e-commerce and fraud detection, demonstrate that the learned equations expose dataset biases, support counterfactual reasoning, and yield actionable insights.

We study partial-feedback online learning, where each instance admits a set of correct labels, but the learner only observes one correct label per round; any prediction within the correct set is counted as correct. This model captures settings such as language generation, where multiple responses may be valid but data provide only a single reference. We give a near-complete characterization of minimax regret for both deterministic and randomized learners in the set-realizable regime, i.e., in the regime where sublinear regret is generally attainable. For deterministic learners, we introduce the Partial-Feedback Littlestone dimension (PFLdim) and show it precisely governs learnability and minimax regret; technically, PFLdim cannot be defined via the standard version space, requiring a new collection version space viewpoint and an auxiliary dimension used only in the proof. We further develop the Partial-Feedback Measure Shattering dimension (PMSdim) to obtain tight bounds for randomized learners. We identify broad conditions ensuring inseparability between deterministic and randomized learnability (e.g., finite Helly number or nested-inclusion label structure), and extend the argument to set-valued online learning, resolving an open question of Raman et al. [2024b]. Finally, we show a sharp separation from weaker realistic and agnostic variants: outside set realizability, the problem can become information-theoretically intractable, with linear regret possible even for $|H|=2$. This highlights the need for fundamentally new, noise-sensitive complexity measures to meaningfully characterize learnability beyond set realizability.

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.

We present the convergence rates of synchronous and asynchronous Q-learning for average-reward Markov decision processes, where the absence of contraction poses a fundamental challenge. Existing non-asymptotic results overcome this challenge by either imposing strong assumptions to enforce seminorm contraction or relying on discounted or episodic Markov decision processes as successive approximations, which either require unknown parameters or result in suboptimal sample complexity. In this work, under a reachability assumption, we establish optimal $\widetilde{O}(\varepsilon^{-2})$ sample complexity guarantees (up to logarithmic factors) for a simple variant of synchronous and asynchronous Q-learning that samples from the lazified dynamics, where the system remains in the current state with some fixed probability. At the core of our analysis is the construction of an instance-dependent seminorm and showing that, after a lazy transformation of the Markov decision process, the Bellman operator becomes one-step contractive under this seminorm.

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.