Differentiable optimization layers are traditionally integrated in predict-then-optimize frameworks where a neural model estimates parameters that subsequently serve as fixed inputs to downstream decision-making optimization problems. In this work, we introduce the concept of a "fairness layer": a differentiable optimization layer appended to a model's output layer that guarantees a chosen notion of output parity is satisfied when integrated into a neural network. Additionally, we introduce an online primal-dual inference algorithm that provides provable aggregate fairness guarantees for streaming predictions with arbitrarily small batch sizes, where traditional per-batch constraints become overly restrictive. Numerical experiments demonstrate the effectiveness of the fairness layer and associated algorithm, and theoretical analysis characterizes the layer's differentiability and stability properties during model training and backpropagation. Our code for these experiments is publicly available on GitHub (https://github.com/dtroxell19/FairDL-ICML-2026.git) and our public Python package documentation can be found online: https://dtroxell19.github.io/fairness_training/.

Many practical data analysis tasks reduce to learning, from observed samples, how a collection of variables depend on each other. A widely used approach is to fit a Gaussian graphical model, which represents the dependence structure as a graph connecting the variables. In a number of important applications, such as financial returns, gene co-expression, and climate or network analysis, the dependencies tend to be positive: variables move together rather than offset each other. Encoding this positivity through the constraint of multivariate total positivity of order two (MTP2) yields an attractive estimator that produces accurate fits with no tuning required. The resulting graphs are, however, typically much denser than the underlying ground-truth model, which makes them hard to interpret and slow to use in any downstream task that operates on the graph. In this work, we propose a novel highly-scalable approach for learning Gaussian graphical models from data using spectral sparsification; we call it Spectral-MTP2. Spectral graph sparsification is a fundamental method which aims to preserve meaningful properties of a dense graph with a sparser subgraph. We theoretically and empirically investigate and validate our method, and show that learning Gaussian Graphical Models under MTP2 using spectral sparsification preserves MTP2 and approximates well the original model in terms of Kullback-Leibler divergence and Gaussian log-likelihood. In simulations and applications to equity returns and gene expression, we observe that Spectral-MTP2 retains most of the fit quality of the denser MTP2 baseline, while producing substantially sparser and more interpretable graphs.

Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the state space size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes dimension-free convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our theory relies only on a single standard rate-matrix regularity assumption and is compatible with time-inhomogeneous schedules. Four novel techniques drive our improvements: working in the space of observables via adjoint equations rather than directly with probability measures, a regularity analysis that yields bounds on any IPM, a coupling argument that removes $S$-dependence under uniform transitions, and a score-marginal cancellation technique that removes $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models.

We study the dynamic joint assortment selection and positioning problem, where the attraction of each product depends on both its intrinsic appeal and its display position under a Multinomial Logit (MNL) choice framework. Our study ranges from the multiplicative position effects model, in which each product's attraction is scaled by a position-specific factor, to a general position effects model assigning independent attraction parameters to every product--position pair to capture heterogeneous synergies. For both models, we design round-based learning algorithms that update decisions after every single feedback, and establish the first regret-optimal characterization. Besides, our round-based algorithms provide the prompt operations needed by modern platforms. For the multiplicative model, we develop a cross-position pairwise maximum likelihood estimator with a clipping mechanism, and prove that our algorithm P2MLE-UCB attains a regret of $\tilde{O}(\sqrt{NT})$, matching the lower bound and closing the $\sqrt{K}$ gap left by prior epoch-based analyses. For the general model, we establish a minimax lower bound and propose GP2-UCB with a matching upper bound. Moreover, we design an efficient subroutine for the per-round joint assortment and positioning optimization based on Dinkelbach's method and maximum-weight bipartite matching. Numerical experiments on synthetic data and the Expedia dataset show that our algorithms consistently outperform state-of-the-art benchmarks.

Spectral deconvolution is essential for extracting peak structures that encode material properties and chemical structures, but conventional automated methods often fail when spectra contain high-intensity noise or unknown background components. In practice, scientists rarely interpret spectra in isolation. Instead, they identify physically meaningful peaks by relating spectral structures to auxiliary information such as physical-property values, chemical structures, and trends across related measurements. Here, we propose a Bayesian framework that integrates spectral deconvolution with a model of expert scientific reasoning. In this work, expert scientific reasoning refers to the practice of evaluating candidate spectral structures by their consistency with independently measured physical-property values, rather than to manual expert intervention during inference. We formalize this reasoning as a physical-property regression layer, implemented using Gaussian process regression, and couple it with Bayesian spectral deconvolution. By averaging the physical-property likelihood over posterior predictive spectra inferred from Bayesian spectral deconvolution, the proposed method selects spectral models according to the consistency between inferred spectral structures and physical-property information. We validate the framework using synthetic spectra with high-intensity noise or unknown backgrounds and infrared spectra of poly(lactic acid). The method recovers physically meaningful peak structures that conventional Bayesian spectral deconvolution misses or misidentifies from spectra alone, including weak peaks in poly(lactic acid) IR spectra related to measured degradation rates. These results demonstrate that integrating expert scientific reasoning with Bayesian spectral deconvolution enables robust peak estimation under conditions where spectrum-only inference is unreliable.

Large-scale hypothesis testing is central to modern science, where controlling the False Discovery Rate (FDR) has become the standard approach to managing false positives across many simultaneous tests. Hypotheses rarely exist in isolation; they often exhibit structure through proximity, connectivity, or hierarchy. This structure represents both a challenge and an opportunity: while classical methods treat these dependencies as obstacles requiring conservative correction, leveraging them can substantially increase discovery power. Here, we reframe structured FDR control as a regularized learning problem. By optimizing within a suitable Reproducing Kernel Hilbert Space (RKHS), we introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone. This formulation enables smooth solutions in place of the piecewise-constant fits of prior methods, principled likelihood-based hyperparameter selection rather than heuristic tuning, and inference at unobserved locations which in turn supports sample-efficient experimental design. Building on this estimator, we provide two decision rules which we prove to control the FDR. We validate our method on two sources: spatial locations derived from high-dimensional real-world datasets, and a differential gene expression task utilizing protein-protein interaction graphs.

Transformers have become the dominant architecture in modern machine learning, yet the theoretical understanding of their training dynamics remains limited. This paper develops a rigorous mathematical framework for analyzing gradient-based training of transformers in the mean-field regime, where both the depth (number of layers) and width (number of attention heads) tend to infinity. While ResNet training can be understood as controlling a neural ODE, transformer training corresponds to controlling a neural PDE, due to the coupling of multiple token distributions through the attention mechanism. Our mean-field model features two types of measure representations: token distributions evolving through layers and attention parameters at each layer. We establish well-posedness of the forward pass through infinitely deep transformers, characterizing token evolution via flow maps that satisfy ODEs in function spaces. Using adjoint sensitivity analysis, we derive an explicit formula for the conditional Wasserstein gradient of the training risk, involving adjoint variables governed by backward ODEs. We prove the existence and uniqueness of gradient flow curves in the conditional Wasserstein metric space, establishing a rigorous foundation for gradient-based transformer training. A key technical contribution is providing necessary and sufficient conditions for injectivity of the Neural Tangent Kernel (NTK) for attention mechanisms: we show that NTK injectivity is equivalent to linear independence of log-sum-exp functions modulo affine functions, a condition satisfied by diverse token distributions, including discrete distributions, uniform distributions, and Gaussian mixtures. Under this NTK injectivity assumption, we prove that gradient flow converges to global minima when the initial loss is sufficiently small, eliminating spurious local minima from the optimization landscape.

Panel data, in which multiple units are repeatedly observed over time, arise throughout science and engineering. Quantifying predictive uncertainty in such settings is challenging because conformal prediction, while distribution-free and model-agnostic, classically relies on exchangeability assumptions that fail under temporal dependence and unit heterogeneity. We propose a simple online conformal framework for non-exchangeable panel data. The method exploits a key feature of online panel prediction: when a forecast is required for one unit, contemporaneous outcomes from related units may already be observed and can serve as a calibration panel. At each round, prediction sets are formed using currently observed calibration units together with two adaptive quantities: history-based similarity weights that emphasize calibration units resembling the target, and an adaptive miscoverage level that is updated whenever target feedback is revealed. This two-state design yields a stepwise coverage bound and a long-run coverage guarantee. Empirically, across synthetic and real panel data sets, the method improves coverage on the worst-covered target units through adaptive interval-width allocation rather than uniform inflation. The two states are complementary: similarity weights protect coverage when target feedback is sparse, while the adaptive level further improves coverage as feedback accumulates.

This article proposes an inferential framework for comparing predictor importance in classification problems with categorical response variables. The approach is based on the categorical Gini correlation (CGC) proposed by Dang et al. (2020), a measure of dependence between numerical predictors and categorical outcomes. Predictor importance is evaluated by testing differences in CGCs across competing predictor groups. The proposed methodology accommodates predictors of arbitrary and unequal dimensions and allows for dependence between predictor groups. Asymptotic normality of the test statistic is established under both the null and alternative hypotheses, and the resulting test is shown to be consistent. In addition to deriving the asymptotic distribution, a nonparametric bootstrap procedure is developed as an alternative approach to inference. Simulation studies, along with applications to breast cancer and human activity recognition datasets, demonstrate the effectiveness of the proposed framework.

Modern generative models have emerged as a powerful Diffusion-based generative models increasingly paradigm for learning complex, high-dimensional data distributions. In particular, diffusion models (Ho et al., 2020; rely on inference-time guidance, adding a drift Sohl-Dickstein et al., 2015; Song and Ermon, 2019; Song term or reweighting mixture of experts, to imet al., 2020) and flow-based methods (Zhang et al., 2018a; prove sample quality on task-specific objectives. However, most existing techniques reLipman et al., 2022; Albergo and Vanden-Eijnden, 2022; Liu quire repeated score or gradient evaluations, inet al., 2022) provide a principled and scalable framework for generative modeling, achieving state-of-the-art performance troducing bias, high computational overhead, or across diverse applications, including video generation (Ho both. We introduce URGE, approximation-free et al., 2022), protein design (Gruver et al., 2023), and largeResampling via Girsanov Estimation, a derivativefree inference-time scaling algorithm that perscale text generation (Li et al., 2022; Nie et al., 2025). A forms pathwise importance reweighting via a Girunifying perspective underlying these approaches is their formulation in terms of stochastic differential equations sanov change of measure.