Classical ReLU-based Input Convex Neural Networks (ICNNs) are equivalent to the optimal value functions of Linear Programming (LP). This intrinsic structural equivalence restricts their representational capacity to piecewise-linear polyhedral functions. To overcome this representational bottleneck, we propose the SOC-ICNN, an architecture that generalizes the underlying optimization class from LP to Second-Order Cone Programming (SOCP). By explicitly injecting positive semi-definite curvature and Euclidean norm-based conic primitives, our formulation introduces native smooth curvature into the representation while preserving a rigorous optimization-theoretic interpretation. We formally prove that SOC-ICNNs strictly expand the representational space of ReLU-ICNNs without increasing the asymptotic order of forward-pass complexity. Extensive experiments demonstrate that SOC-ICNN substantially improves function approximation, while delivering competitive downstream decision quality. The code is available at https://anonymous.4open.science/r/SOC-ICNN-4B18/.

Robust uncertainty quantification is increasingly important in modern data analysis and is often formalized under Huber's model, which allows an $\varepsilon$-fraction of arbitrary corruptions. In many experimental sciences, however, the measurement protocol is well controlled, and contamination is more plausibly introduced upstream. Motivated by this noise-oblivious nature of adversaries, we study confidence intervals for the null location parameter $θ$ in Efron's Gaussian two-groups model, where an unknown fraction $\varepsilon$ of observations have arbitrarily shifted means, but all samples share the same law of additive Gaussian measurement noise with variance $σ^2$. We characterize the minimax-optimal length among confidence intervals with a prescribed coverage level uniformly over the unknown contamination proportion and all noise-oblivious adversaries. Although prior work has shown that the minimax point estimation rate of theta does not deteriorate when $\varepsilon$ becomes unknown, our results reveal that, with a given $σ^2$, the minimax-optimal length of confidence intervals that are adaptive to unknown $\varepsilon$ is of order $σ(n^{-1/4}+\varepsilon^{1/2}/\max\{1, \log(en \varepsilon^2)\}^{1/2})$, which is polynomially worse than the optimal length when $\varepsilon$ is known. When the variance $σ^2$ is also unknown, we show a further degradation: no adaptive confidence interval can be shorter than $Ω(σn^{-1/8})$. Algorithmically, we introduce a Fourier-based certification procedure built on Carathéodory's positive-semidefiniteness constraints. By scanning candidate points and accepting those whose residual characteristic function is certifiably consistent with a Gaussian location mixture, our algorithm attains the minimax lower bound in the known-variance setting and is computable in polynomial time.

Evolutionary methods have long been useful for analysis and explanation in genetics, biology, ecology, and related fields. In this work, we extend these methods to neural networks, specifically large language models (LLMs), to better analyze and explain relationships among models. We show how relating weights to genotypes and output text to phenotypes can improve our understanding of model lineage, important datasets, the roles of different model layers, and visualization of model relationships. We demonstrate this in a controlled experiment, where our estimated evolutionary trees reliably recover the topology of the ground-truth training tree. We further identify the most important weight layers according to weight differences and show through phenotypic experiments that one training dataset appears to contribute more useful information than the others. Finally, we generate an unsupervised evolutionary tree of black-box foundation models. Throughout, we provide visualizations that support a clearer understanding of evolutionary relationships among LLMs.

We propose a novel method of understanding disease transformation from a healthy baseline with biomarker-level explainability. By modeling the biomarker covariance matrices of healthy controls and disease states, the perturbation can be individually characterized to accomplish mechanistic explanations of disease trajectories, both at a molecular level and for individual patients. Given a cohort of n patients each measured on p biomarkers, we define the biomarker "Hamiltonian" H = X^T X / n \in R^{p \times p}, where X \in R^{n \times p} is the covariant biomarker matrix. The eigenvectors of H define a set of normal modes of biomarker coordination, and the eigenvalues quantify the energy carried by each mode. In the healthy state, the reference Hamiltonian H_0 governs this structure where disease perturbs H_0 by an additive operator ΔH, thus shifting eigenvalues and rotating eigenvectors in proportion to the severity of pathological disruption. We formalize this framework, derive the spectral change given a disease perturbation, and demonstrate that the projection of a newly diagnosed patient's cumulative biomarker covariance structure onto disease-discriminant eigenmodes constitutes an optimal prognostic statistic for greater precision in disease prognosis. This work serves as a veritable white paper with application across a panoply of disease frameworks from cancer to neurodegenerative disorders.

Deep learning models for drug--target interaction (DTI) prediction often achieve strong benchmark performance without necessarily relying on mechanistically meaningful molecular features, a limitation that standard accuracy-based evaluation cannot detect. We introduce ISAAC (Intervention-based Structural Auditing Approach for Causal Reasoning), a post-hoc framework that evaluates prior-relative structural sensitivity by probing frozen models through matched mechanistic and spurious input-level interventions, independently of predictive accuracy. Applied to three sequence-based DTI architectures on the Davis benchmark, ISAAC reveals approximately 25\% relative differences in reasoning scores across models with comparable AUROC (within around 3\%), stable across training and intervention seeds and two distinct perturbation operators. These discrepancies, undetectable under conventional accuracy metrics, motivate the use of post-hoc structural auditing as a complement to standard performance evaluation in scientific machine learning for molecular modeling.

Artificial intelligence-generated content (AIGC) has emerged as a transformative paradigm for automating the creation of diverse and customized content, giving rise to rapidly growing computational workloads in cloud data centers. It is imperative for AIGC service providers (ASPs) to strategically schedule AIGC workloads to reduce data center energy costs while guaranteeing high-quality content generation. However, the distinctive characteristics of AIGC services pose critical challenges, including model heterogeneity across ASPs, implicit service quality evaluation, and complex inference process control. To tackle these challenges, we propose a joint energy management and coordinated AIGC workload scheduling framework, which introduces an explicit mathematical characterization of service quality to promote both job transfer among ASPs and fine-grained inference process configuration. Moreover, various energy resources within data centers are jointly considered to enhance power usage flexibility. Subsequently, a system utility maximization problem is formulated to balance AIGC service revenue with operational penalties and costs. Nevertheless, the strong coupling among job scheduling decisions induces severe reward sparsity, which limits the effectiveness of existing deep reinforcement learning (DRL) algorithms. To address this issue, we develop a diffusion model-aided reward shaping approach to synthesize complementary reward signals through a multi-step denoising process. This approach is seamlessly integrated with DRL to enable efficient learning of scheduling policies under sparse environmental feedback. Experiments based on real-world models and datasets demonstrate that our scheme effectively accommodates electricity price fluctuations and AIGC model heterogeneity, while achieving superior learning convergence and system utility compared with benchmark methods.

This manuscript contains preprint of a chapter under consideration for inclusion in the forthcoming third edition of {\em Cover and Thomas's Elements of Information Theory}, posted with permission from Wiley. The table of contents EIT-3 ToC of the new edition can be found at: https://docs.google.com/document/d/1L-m4oQEJw1PJhoxBeMwrrBD8S_HmvzMEkPbYvS24980/edit?usp=sharing . For feedback, please contact abbas@ee.stanford.edu Learning and information theory intersect in both model training and the characterization of fundamental performance limits. This manuscript provides a concise and accessible treatment of the first intersection, requiring only basic background in information theory and statistics at the senior undergraduate or first-year graduate level. End-of-chapter exercises make the material well suited for classroom use as well as self-study. The chapter focuses on the role of divergence measures in model training, with examples ranging from linear and logistic regression to autoregressive models, variational autoencoders, diffusion models, generative adversarial networks, and score-based models. It introduces the evidence lower bound (ELBO), $f$\!-divergences, and the Fisher divergence. In particular, the treatment of the generative diffusion model provides a more systematic and explicit derivation than is typical in the literature.

We introduce a general framework that extends Bayesian inference by allowing the researcher to explicitly encode confidence in each source of uncertainty within the model. This mechanism provides a new handle for model design and regularisation control. Building on this framework, we develop a general approach for inducing sparsity in statistical models and illustrate its use in linear and logistic regression, as well as in Bayesian neural networks.

Conformal prediction provides distribution-free predictive intervals with finite-sample marginal coverage. However, achieving conditional validity and interval efficiency (in terms of short interval length) remains challenging, particularly in complex settings with heteroskedasticity, skewed responses, or estimation errors. We propose a conformal-style calibration method for responses obtained by the probability integral transform (PIT) of the conditional cumulative distribution function (CDF) estimated via neural networks to construct a finite-sample-adjusted percentile interval with the shortest length determined by the estimated conditional CDF. Calibrating in PIT space is effective because PIT values are asymptotically feature-independent when the CDF estimator is accurate, which mitigates feature-dependent miscoverage and improves conditional calibration. On the other hand, our percentile calibration adapts to the empirical PIT distribution, which is robust against a possibly imperfect estimation of the conditional CDF. We prove the finite-sample marginal coverage property of the proposed method and show its asymptotic conditional coverage under mild consistency conditions. Experiments on diverse synthetic and real-world benchmarks demonstrate better conditional calibration and substantially shorter intervals than existing methods.

Here we introduce Partially Observed Structural Causal Models (POSCMs) that formalize causal systems where latent contexts co-determine both the interaction structure and downstream mechanisms on observed variables. POSCMs provide an extension of structural causal models (SCMs), as a self-contained causal modeling framework for endogenous graphs, allowing for an intervention hierarchy spanning node- and edge-level context and endogenous variable interventions. To enable surgical edge interventions, we adopt a Kolmogorov-Arnold-Sprecher edge-functional decomposition, an existence theorem for representing each node mechanism as a sum of univariate functions of its parents, yielding an explicit parametrization of dyadic functional contributions. We provide an identifiability theory that clarifies which intervention families would suffice to disentangle structure formation from mechanisms. We empirically validate these predictions in a biophysically detailed virtual human retina simulator, constructing intervention protocols that (i) reproduce the non-identifiability predicted when context is latent and no context-level interventions are available, (ii) exhibit structure-mechanism confounding under latent edges when only node interventions are observed, and (iii) recover synaptic input-output relationships via targeted node interventions, consistent with our positive kernel identifiability result. Our work generalizes SCMs in a way that allows it to work in a world closer to the one we live in.