Neural network mixed-effects models (NMMs) have gained traction by combining the strong representation and predictive power of artificial neural networks with the capacity of mixed-effects modeling to capture complex correlation structures. However, existing estimation approaches rely heavily on manual derivations of objective functions and gradients, which inherently forces simplifying approximations and severely constrains the complexity and accuracy of NMMs. In this work, we introduce a general framework for implementing NMMs using Template Model Builder (TMB). By leveraging automatic differentiation and Laplace approximation, TMB requires users to specify only the negative joint log-likelihood and any regularization terms. The framework automatically integrates out random effects and evaluates the marginal objective function alongside its exact gradients, eliminating the need for manual derivations or ad hoc approximations. We demonstrate the efficiency, flexibility, and statistical performance of TMB-based NMMs across two numerical examples, including an application to monotonic NMMs. Reproducible code is provided to facilitate broader adoption.
Nan Zheng, Hoi Yiu Cheung, Vibhu Sharma et al.· 0 citations
Scientific discovery often requires reasoning over competing hypotheses that are consistent with experimental observations. For mixed-variable and combinatorial hypothesis spaces, however, constructing probabilistic representations remains challenging because both the active model components and their associated parameters are unknown. In this work, we present a framework for learning continuous latent representations of admissible partial differential equations (PDEs) by embedding a scientific inductive bias directly into the training distribution. Progressively richer structural principles (e.g., sparsity, logical dependencies, common PDE families, and physical admissibility) are used to generate a structured distribution of hypotheses from which a gated variational autoencoder learns a continuous latent manifold. Experimental results show that the resulting 11-dimensional representation accurately reconstructs a broad collection of representative PDEs, while exhibiting smooth geometric transitions both within and across equation families. Through an ablation study we further demonstrate that introducing scientific principles reduces both structural misclassifications of equation forms and parameter estimation errors when reconstructing a representative benchmark set of admissible partial differential equations. These results show that embedding a scientific inductive bias in the training distribution enables the learning of compact and geometrically meaningful hypothesis manifolds, providing a principled foundation for future inference over competing governing equations.
James Crowley, Faez Ahmed, Anton van Beek· 0 citations
Informative label missingness can change the usual efficiency ordering between completely and partially labelled classifiers because the pattern of missing labels may itself carry information about the classification model. We develop a general likelihood-based theory for this phenomenon in parametric multiclass classification. An efficient-information decomposition separates information lost through unavailable class memberships from information contributed by the missing-label mechanism. We then derive a quadratic expansion of plug-in excess risk over the active pairwise faces of the multiclass Bayes boundary, showing that classification efficiency depends on how information gains and losses align with directions that perturb the decision boundary. This yields a classification-weighted generalized-eigenvalue criterion under which informative partial classification may have smaller asymptotic classification risk without globally dominating complete classification in Fisher information. Near missing completely at random, with the marginal missing-label proportion fixed, redistribution of missing labels changes lost class-label information at first order, whereas efficient information from the missingness pattern appears only at second order. Three-class quadratic discriminant calculations, finite-sample experiments, and a semi-synthetic multiclass application illustrate the resulting regime-dependent behaviour.
Fariborz Setoudehtazang, Geoffrey J. McLachlan· 0 citations
We estimate the conditional population-risk curve of a realized smooth nonconvex gradient flow from the training sample. Flow approximate leave-one-out (Flow-ALO) propagates a deletion response and evaluates omitted observations at approximate deleted paths. The risk-curve error decomposes into response approximation, exact-LOO fluctuation, and deletion-to-full risk transfer. On each fixed finite horizon, bounded centered training-loss gradients, a one-sided Hessian lower bound, locally Lipschitz Hessians, and a strict tube-closure condition yield an explicit $(n-1)^{-2}$ bound for the deletion-response error. Bounded evaluation-loss gradients transfer the deletion-response bound to the score without requiring the Hessian to be invertible. Direct first-order jackknife cancellation and exact-LOO concentration control deletion-to-full risk transfer and fluctuation, respectively, completing recovery of the conditional population-risk curve. For bounded smooth two-layer mean-field networks training both layers, the score-error bound is uniform in width.
Mingzhi Song· 0 citations
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We propose a new design of fair classifiers for multi-class classification problems in the presence of vector-valued sensitive attributes. In that scenario each sensitive attribute has multiple values and forms several groups relevant to the fairness consideration. Naturally those groups are overlapping and one should also analyze the interaction of factors. Additionally, the decision makers aided by the classification should not violate individual rights at the expense of satisfying fairness metrics at the group level. We propose an approach using the theory and methods of coherent measures of risk aiming at resolving the fairness challenges. Further, we propose a specialized numerical method for solving the resulting optimization problem. The method scales well with the increase of the number of observations. Additionally, we note that the obtained classifier is robust with respect to corrupted data or to situation when data is scarce. We demonstrate the advantages of the proposed framework in comparison to the support-vector machine framework and other methods handling fairness.
Token prediction is a central pre-training objective for modern language models. Despite its empirical success, why token prediction learns broadly useful representations remains incompletely understood. We develop a statistical framework connecting token prediction with representation geometry, encoder approximation, and downstream performance. Under a softmax prediction head, we show that accurate token prediction organizes token embeddings according to similarities between the distributions of contexts in which different token types appear, as measured by Hellinger distance, with explicit errors governed by prediction accuracy and token frequency. Meanwhile, the contextual representation provides a low-dimensional coordinate for the conditional distribution of the target token relative to these embeddings. We further introduce a self-consistency principle showing that repeated applications of a shared representation block can progressively refine the contextual representation without introducing additional block parameters. Among representations with the same prediction accuracy, this recurrent construction favors those that can be stably reconstructed from their contexts. Finally, we establish downstream guarantees for token generation, token community recovery, and classification by a linear probe, showing how prediction accuracy and recovered geometry translate into performance beyond the pre-training objective. Together, these results explain how the simple objective of predicting tokens can recover semantic geometry and produce broadly useful representations. A controlled simulation illustrates the theoretical mechanisms.
Missing data, measurement error, and population heterogeneity are pervasive challenges in analyzing data arising from modern observational studies and machine learning applications. Although these problems frequently coexist and interact, they are often treated separately in existing works. We propose a unified probabilistic framework that jointly addresses these issues utilizing deep latent variable representation. The proposed method integrates a novel hierarchical tree-routed variational autoencoder with pattern-aware latent representations and calibration-based denoising. The framework accommodates missing data mechanisms, including MCAR, MAR, and MNAR, while simultaneously learning subgroup-specific and globally shared latent structure. The introduced reconvergent routing mechanism enables selective parameters to be shared across related subpopulations, which offers flexibility as well as improved statistical efficiency. Simulation studies demonstrate substantial improvements over existing deep generative imputation approaches under complex heterogeneous missingness and measurement-error settings. The proposed framework provides a principled approach for learning from noisy and incomplete data in modern healthcare and other high-dimensional applications.
Yasin Khadem Charvadeh, Grace Y. Yi, Mithat G\"onen et al.· 0 citations
Longitudinal cohort studies produce repeated data that enable the assessment of time-varying association patterns between exposures and health outcomes. Classical linear mixed-effects models (LMMs) can accommodate a large variety of association patterns while accounting for the irregularly spaced, partially observed measurement. But they require the analyst to pre-specify the functional form linking the exposure history to the outcome. We propose the Neural ODE-LMM, which embeds a Neural Ordinary Differential Equation (Neural ODE) within the linear mixed-effects framework: a learned vector field encodes covariate trajectories into a continuous-time latent state that drives both the fixed- and random-effect design, while preserving the standard LMM observation model. This retains classical likelihood-based inference while learning complex, potentially cumulative, covariate effects flexibly. All parameters are estimated by maximising a penalised marginal likelihood. To quantify covariate effects, we introduce contrasts of counterfactual predictions that compare the expected outcome under alternative covariate trajectories with variance estimated via the delta method. In simulations, the model recovers both instantaneous and cumulative-burden effects without prior specification of the functional form. Applied to the Trois-Cit\'{e}s (3C) cohort, a population-based study of 7{,}324 participants, the method reveals trajectory-dependent associations of BMI and fasting glucose with cognitive decline.
Deployed decisions are often optimized once and retained because updates impose operational, regulatory, or switching costs. As operating conditions change, when should such decisions be re-optimized? We study this question for stochastic optimization when the objective's functional form is known but the decision maker's trade-offs are encoded by an unknown preference parameter. Standard distribution-shift tests are poorly aligned with this goal: they can flag detectable yet decision-irrelevant changes without determining whether the incumbent decision has become materially suboptimal. We propose \texttt{RADAR} (Regret-based Assessment of Decision Adequacy and Risk), a decision-focused framework that uses inverse optimization to infer latent preferences and tests the deployed decision's optimality gap under the current distribution. By targeting regret, \texttt{RADAR} ignores decision-irrelevant shifts while detecting changes that warrant re-optimization. We develop two-sample and sequential changepoint procedures and establish asymptotic guarantees for Type-I error and power. Across synthetic optimization problems, a semi-synthetic capacity allocation task, and police-zone planning, \texttt{RADAR} more reliably distinguishes harmful from harmless shifts than decision-agnostic alternatives.
Industrial recommenders give new content initial views through budgeted exploration, then use early performance to decide further delivery. On many short-video platforms, exploration is the primary way new videos reach viewers. Viewer-side tests measure consumption; the published budget objectives we review omit creator response. We analyze four experiments on a major short-video platform. An eight-month creator ablation finds production exploration raises videos posted per creator by 8.55% and creators posting at least once by 7.10% relative to a minimal floor. A budget-matched reallocation raises creator participation with no detectable short-run viewer-side change. A year-long viewer ablation finds 1.74% more video views but 2.13% less view time. A delivered view creates immediate feed value, can trigger organic take-up, and can induce creator supply. Take-up and supply replenish a shared corpus, creating two measurement limits. Viewer-side A/B tests cancel the corpus effect when both arms consume the same corpus. Giving each arm its own corpus avoids cancellation, but turnover still controls the horizon. If the corpus turns over at rate w per posting cycle, a t-cycle experiment expresses at most wt of the eventual corpus effect. More users reduce noise but do not speed turnover. Before the corpus path visibly bends, data cannot distinguish a modest fast effect from an arbitrarily large slow one, so a valid confidence interval may lack a finite upper endpoint. As predicted, the three-week co-diverted experiment cannot determine the sign of the eventual corpus effect. Within the window, it identifies the direct feed effect, and an exploratory cohort analysis detects organic lift after exploration ends. The experiments establish a positive creator response, measure the gross corpus flow visible within three weeks, and show the design and duration needed to identify total value.
Yuanyuan Shen, Yiren Yan, Wenjie Li et al.· 0 citations
Kernel density estimation (KDE) is one of the most fundamental statistical estimators of density functions. Its direct implementation on a dataset of $N$ points incurs an $\mathcal{O}(N^{2})$ computational cost, which is prohibitive for large-scale datasets. Kernel approximation techniques can be applied to bring the computational cost down to $\mathcal{O}(N)$. The random Fourier features (RFF) technique, based on sampling from the spectral density of the kernel function, has become popular to speed up kernel estimators for machine learning applications. Unfortunately, it is restricted to positive definite kernels, while the majority of kernel functions popular in KDE, such as the parabolic kernel, do not satisfy this property. To overcome this limitation, this article introduces the signed random Fourier features (SRFF) technique. It is a generalization of RFF compatible with indefinite kernels whose inverse Fourier transform is absolutely integrable. The motivation for introducing this method is to speed up KDE in the case of multivariate compact kernels, which are generally not positive definite. We detail how to implement SRFF for both product kernels and isotropic kernels. For the class of Kuttner-Golubov kernels $K(\boldsymbol{x}_{i},\boldsymbol{x}_{j})=(1-\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert ^{\alpha})^{\beta}\mathbf{1}_{\{\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert \leq1\}}$ where $\boldsymbol{x}_{i}\in\mathbb{R}^{d}$, $\boldsymbol{x}_{j}\in\mathbb{R}^{d}$, $\alpha>0$, $\beta>0$, which includes the triangular, parabolic, biweight, triweight, and other kernel functions of interest for KDE as particular examples, we provide an explicit acceptance-rejection algorithm to sample from its signed spectral density. Our numerical tests on a dataset of one million points confirm the computational efficiency and accuracy of SRFF for large-scale KDE.
Xie Wang, Nicolas Langren\'e, Wen Chen· 0 citations
We study the empirical Sinkhorn estimator of the entropic optimal transport potentials under the uniform loss. Since the potentials are only unique up to additive constants, we measure the error using the quotient supremum norm, defined as $d_\infty([u],[v]) = \inf_{a\in\mathbb{R}}\|u-v-a\|_\infty$. For a fixed regularization parameter $\varepsilon>0$, we establish a non-asymptotic statistical rate of $n^{-1/2}$. This is achieved by combining the Birkhoff-Hopf contraction theorem with entropy bounds on normalized kernel sections. However, the constant in this bound grows exponentially with $1/\epsilon$. To improve this, we isolate geometric conditions under which the empirical estimator maintains the $n^{-1/2}$ rate but features polynomial dependence on $1/\varepsilon$. The key requirement is a polynomial residual-stability estimate for the population Sinkhorn map. We provide sufficient criteria for this, including a polynomial contraction property and a local inverse estimate. Furthermore, we introduce two rigorously verifiable model classes an $\varepsilon$-weak residual-interaction class obtained after separable centering and another based on connected tight-edge graphs for fixed discrete costs where the polynomial rate is guaranteed without relying on abstract resolvent assumptions. Finally, we establish matching minimax lower bounds demonstrating that the $\varepsilon n^{-1/2}$ rate cannot be uniformly improved in the bounded-interaction regime.
