Generalized linear models (GLMs) are fundamental tools for statistical modeling, with maximum likelihood estimation (MLE) serving as the classical method for parameter inference. While MLE performs well in canonical GLMs, it can become computationally inefficient near the true parameter value. In more general settings with non-canonical or fully general link functions, the resulting optimization landscape is often non-convex, non-smooth, and numerically unstable. To address these challenges, we investigate an alternative estimator based on solving the variational inequality (VI) formulation of the GLM likelihood equations, originally proposed by Juditsky and Nemirovski as an alternative for solving nonlinear least-squares problems. Unlike their focus on algorithmic convergence in monotone settings, we analyze the VI approach from a statistical perspective, comparing it systematically with the MLE. We also extend the theory of VI estimators to a broader class of link functions, including non-monotone cases satisfying a strong Minty condition, and show that it admits weaker smoothness requirements than MLE, enabling faster, more stable, and less locally trapped optimization. Theoretically, we establish both non-asymptotic estimation error bounds and asymptotic normality for the VI estimator, and further provide convergence guarantees for fixed-point and stochastic approximation algorithms. Numerical experiments show that the VI framework preserves the statistical efficiency of MLE while substantially extending its applicability to more challenging GLM settings.

Many rare diseases offer limited established treatment options, leading patients to switch therapies when new medications emerge. To analyze the impact of such treatment switches within the low sample size limitations of rare disease trials, it is important to use all available data sources. This, however, is complicated when usage of measurement instruments change during the observation period, for example when instruments are adapted to specific age ranges. The resulting disjoint longitudinal data trajectories, complicate the application of traditional modeling approaches like mixed-effects regression. We tackle this by mapping observations of each instrument to a aligned low-dimensional temporal trajectory, enabling longitudinal modeling across instruments. Specifically, we employ a set of variational autoencoder architectures to embed item values into a shared latent space for each time point. Temporal disease dynamics and treatment switch effects are then captured through a mixed-effects regression model applied to latent representations. To enable statistical inference, we present a novel statistical testing approach that accounts for the joint parameter estimation of mixed-effects regression and variational autoencoders. The methodology is applied to quantify the impact of treatment switches for patients with spinal muscular atrophy. Here, our approach aligns motor performance items from different measurement instruments for mixed-effects regression and maps estimated effects back to the observed item level to quantify the treatment switch effect. Our approach allows for model selection as well as for assessing effects of treatment switching. The results highlight the potential of modeling in joint latent representations for addressing small data challenges.

The capability of a novel Kullback-Leibler divergence method is examined herein within the Kalman filter framework to select the input-parameter-state estimation execution with the most plausible results. This identification suffers from the uncertainty related to obtaining different results from different initial parameter set guesses, and the examined approach uses the information gained from the data in going from the prior to the posterior distribution to address the issue. Firstly, the Kalman filter is performed for a number of different initial parameter sets providing the system input-parameter-state estimation. Secondly, the resulting posterior distributions are compared simultaneously to the initial prior distributions using the Kullback-Leibler divergence. Finally, the identification with the least Kullback-Leibler divergence is selected as the one with the most plausible results. Importantly, the method is shown to select the better performed identification in linear, nonlinear, and limited information applications, providing a powerful tool for system monitoring.

Diverse learning algorithms, optimization methods, and natural selection share a common mathematical structure, despite their apparent differences. Here I show that a simple notational partitioning of change by the Price equation reveals a universal force-metric-bias (FMB) law: $Δ\mathbfθ = \mathbf{M}\,\mathbf{f} + \mathbf{b} + \mathbfξ$. The force $\mathbf{f}$ drives improvement in parameters, $Δ\mathbfθ$, in proportion to the slope of performance with respect to the parameters. The metric $\mathbf{M}$ rescales movement by inverse curvature. The bias $\mathbf{b}$ adds momentum or changes in the frame of reference. The noise $\mathbfξ$ enables exploration. This framework unifies natural selection, Bayesian updating, Newton's method, stochastic gradient descent, stochastic Langevin dynamics, Adam optimization, and most other algorithms as special cases of the same underlying process. The Price equation also reveals why Fisher information, Kullback-Leibler divergence, and d'Alembert's principle arise naturally in learning dynamics. By exposing this common structure, the FMB law provides a principled foundation for understanding, comparing, and designing learning algorithms across disciplines.

Transformers have revolutionized nucleotide sequence analysis, yet capturing long-range dependencies remains challenging. Recent studies show that autoregressive transformers often exhibit Markovian behavior by relying on fixed-length context windows for next-token prediction. However, standard self-attention mechanisms are computationally inefficient for long sequences due to their quadratic complexity and do not explicitly enforce global transition consistency. We introduce CARMANIA (Context-Aware Regularization with Markovian Integration for Attention-Based Nucleotide Analysis), a self-supervised pretraining framework that augments next-token (NT) prediction with a transition-matrix (TM) loss. The TM loss aligns predicted token transitions with empirically derived n-gram statistics from each input sequence, encouraging the model to capture higher-order dependencies beyond local context. This integration enables CARMANIA to learn organism-specific sequence structures that reflect both evolutionary constraints and functional organization. We evaluate CARMANIA across diverse genomic tasks, including regulatory element prediction, functional gene classification, taxonomic inference, antimicrobial resistance detection, and biosynthetic gene cluster classification. CARMANIA outperforms the previous best long-context model by at least 7 percent, matches state-of-the-art on shorter sequences (exceeding prior results on 20 out of 40 tasks while running approximately 2.5 times faster), and shows particularly strong improvements on enhancer and housekeeping gene classification tasks, including up to a 34 percent absolute gain in Matthews correlation coefficient (MCC) for enhancer prediction. The TM loss boosts accuracy in 33 of 40 tasks, especially where local motifs or regulatory patterns drive prediction.

The success of teams in robotics, nature, and society often depends on the division of labor among diverse specialists; however, a principled explanation for when such diversity surpasses a homogeneous team is still missing. Focusing on multi-agent task allocation problems, we study this question from the perspective of reward design: what kinds of objectives are best suited for heterogeneous teams? We first consider an instantaneous, non-spatial setting where the global reward is built by two generalized aggregation operators: an inner operator that maps the $N$ agents' effort allocations on individual tasks to a task score, and an outer operator that merges the $M$ task scores into the global team reward. We prove that the curvature of these operators determines whether heterogeneity can increase reward, and that for broad reward families this collapses to a simple convexity test. Next, we ask what incentivizes heterogeneity to emerge when embodied, time-extended agents must learn an effort allocation policy. To study heterogeneity in such settings, we use multi-agent reinforcement learning (MARL) as our computational paradigm, and introduce Heterogeneity Gain Parameter Search (HetGPS), a gradient-based algorithm that optimizes the parameter space of underspecified MARL environments to find scenarios where heterogeneity is advantageous. Across different environments, we show that HetGPS rediscovers the reward regimes predicted by our theory to maximize the advantage of heterogeneity, both validating HetGPS and connecting our theoretical insights to reward design in MARL. Together, these results help us understand when behavioral diversity delivers a measurable benefit.

The distribution of data changes over time; models operating in dynamic environments need retraining. But knowing when to retrain, without access to labels, is an open challenge since some, but not all shifts degrade model performance. This paper formalizes and addresses the problem of post-deployment deterioration (PDD) monitoring. We propose D3M, a practical and efficient monitoring algorithm based on the disagreement of predictive models, achieving low false positive rates under non-deteriorating shifts and provides sample complexity bounds for high true positive rates under deteriorating shifts. Empirical results on both standard benchmark and a real-world large-scale internal medicine dataset demonstrate the effectiveness of the framework and highlight its viability as an alert mechanism for high-stakes machine learning pipelines.

Training a team of agents from scratch in multi-agent reinforcement learning (MARL) is highly inefficient, much like asking beginners to play a symphony together without first practicing solo. Existing methods, such as offline or transferable MARL, can ease this burden, but they still rely on costly multi-agent data, which often becomes the bottleneck. In contrast, solo experiences are far easier to obtain in many important scenarios, e.g., collaborative coding, household cooperation, and search-and-rescue. To unlock their potential, we propose Solo-to-Collaborative RL (SoCo), a framework that transfers solo knowledge into cooperative learning. SoCo first pretrains a shared solo policy from solo demonstrations, then adapts it for cooperation during multi-agent training through a policy fusion mechanism that combines an MoE-like gating selector and an action editor. Experiments across diverse cooperative tasks show that SoCo significantly boosts the training efficiency and performance of backbone algorithms. These results demonstrate that solo demonstrations provide a scalable and effective complement to multi-agent data, making cooperative learning more practical and broadly applicable.

Idempotent Generative Networks (IGNs) are deep generative models that also function as local data manifold projectors, mapping arbitrary inputs back onto the manifold. They are trained to act as identity operators on the data and as idempotent operators off the data manifold. However, IGNs suffer from mode collapse, mode dropping, and training instability due to their objectives, which contain adversarial components and can cause the model to cover the data manifold only partially -- an issue shared with generative adversarial networks. We introduce Non-Adversarial Idempotent Generative Networks (NAIGNs) to address these issues. Our loss function combines reconstruction with the non-adversarial generative objective of Implicit Maximum Likelihood Estimation (IMLE). This improves on IGN's ability to restore corrupted data and generate new samples that closely match the data distribution. We moreover demonstrate that NAIGNs implicitly learn the distance field to the data manifold, as well as an energy-based model.

Large language models (LLMs) are often queried multiple times at test time, with predictions aggregated by majority vote. While effective, this self-consistency strategy (arXiv:2203.11171) requires a fixed number of calls and can fail when the correct answer is rare. We introduce Confidence-Guided Early Stopping (CGES), a Bayesian framework that forms posteriors over candidate answers using scalar confidence signals derived from token probabilities or reward models. CGES adaptively halts sampling once the posterior mass of a candidate exceeds a threshold. We provide theoretical guarantees for both perfectly calibrated confidences and realistic noisy confidence signals. Across five reasoning benchmarks, CGES reduces the average number of model calls by about 69 percent (for example, from 16.0 to 4.9) while matching the accuracy of self-consistency within 0.06 percentage points.