Papers

Techniques for concept extraction, such as sparse autoencoders and transcoders, aim to extract high-level symbolic concepts from low-level nonsymbolic representations. When these extracted concepts are used for downstream tasks such as model steering and unlearning, it is essential to understand their guarantees, or lack thereof. In this work, we present a unified theoretical framework for unsupervised concept extraction, in which we frame the task of concept extraction as identifying a generative model. We present a general meta-theorem for identifiability, which reduces the problem of establishing identifiability guarantees to the problem of characterizing the intersection of two sets. As we demonstrate on a range of widely-used approaches, this meta-theorem substantially simplifies the task of proving such guarantees, thus paving the way for the development of new, principled approaches for concept extraction.

This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system of equations, including the slope coefficients, to be unknown. For this reason, we describe the linear system as having unknown (as opposed to known) coefficients. This hypothesis testing problem arises naturally when constructing confidence sets for possibly partially identified parameters in the analysis of nonparametric instrumental variables models, treatment effect models, and random coefficient models, among other settings. To rule out certain instances in which the testing problem is impossible, in the sense that the power of any test will be bounded by its size, we begin our analysis by characterizing the closure of the null hypothesis with respect to the total variation distance. We then use this characterization to develop novel testing procedures based on sample-splitting. We establish the validity of our testing procedures under weak and interpretable conditions on the linear system. An important feature of these conditions is that they permit the dimensionality of the problem to grow rapidly with the sample size. A further attractive property of our tests is that they do not require simulation to compute suitable critical values. We illustrate the practical relevance of our theoretical results in a simulation study.

Hyperbolic space is increasingly used for hierarchical, tree-like, and network-structured data, but likelihood-based density modeling on hyperbolic space remains relatively limited. This paper develops finite mixture modeling with isotropic Riemannian Gaussian distributions on hyperbolic space under the hyperboloid model. We derive the density, radial normalizing constant, and a finite-sum representation involving the complementary error function. We then formulate weighted maximum likelihood estimation, which is the fundamental subproblem in mixture fitting: the location estimator is the weighted Fréchet mean, while the inverse-scale estimator is obtained from a one-dimensional strictly convex profile problem. For finite mixtures, we derive exact EM and generalized EM algorithms. The generalized version replaces exact barycenter solves with truncated hyperbolic majorization-minimization updates. We establish existence and uniqueness of the weighted single-component estimator, singularity of the unrestricted mixture likelihood, existence of a constrained mixture estimator, and monotonicity properties of the EM-type algorithms. Simulations show accurate weighted estimation, reliable mixture recovery, effective model selection, and substantial computational savings from generalized EM. Real network examples based on hyperbolic embeddings illustrate the method as an exploratory likelihood-based clustering tool for non-Euclidean data.

We provide a systematic recipe for translating ReLU approximation results to softmax attention mechanism. This recipe covers many common approximation targets. Importantly, it yields target-specific, economic resource bounds beyond universal approximation statements. We showcase the recipe on multiplication, reciprocal computation, and min/max primitives. These results provide new analytical tools for analyzing softmax transformer models.

The increased use of differential privacy (DP) has allowed the sharing of large amounts of data while reducing the risk of disclosure of sensitive information at the individual level. However, the noise introduced by DP methods makes performing statistical inference more challenging. While various methods have been proposed to address different inferential tasks, they often require strong parametric assumptions and/or do not scale well with sample sizes (e.g. U.S. Census products). In response to these limitations, we propose an approximate Bayesian method to analyze privatized data products, which uses a two-step approach of imputing the confidential data and then sampling from the non-private posterior, and which is inspired by the method of Guha and Reiter (2025). We prove that this approximate sampler is asymptotically valid under mild assumptions. While this approach is motivated by Bayesian theory, we show through simulations that it provides conservative frequentist properties as well. We demonstrate the utility of our method by applying it in simulated settings as well as for an analysis on the drivers of homeownership via the 2022 American Community Survey.

While the optimal sample complexity of binary classification in terms of the VC dimension is well-established, determining the optimal sample complexity of multiclass classification has remained open. The appropriate complexity parameter for multiclass classification is the DS dimension, and despite significant efforts, a gap of $\sqrt{\text{DS}}$ has persisted between the upper and lower bounds on sample complexity. Recent work by Hanneke et al. (2026) shows a novel algebraic characterization of multiclass hypothesis classes in terms of their DS dimension. Building up on this, we show that the maximum hypergraph density of any multiclass hypothesis class is upper-bounded by its DS dimension. This proves a longstanding conjecture of Daniely and Shalev-Shwartz (2014). As a consequence, we determine the optimal dependence of the sample complexity on the DS dimension for multiclass as well as list learning.

We study learning with Chain-of-Thought (CoT) supervision from multiple thinkers, all of whom provide correct but possibly systematically different solutions, e.g., step-by-step solutions to math problems written by different thinkers, or step-by-step execution traces of different programs solving the same problem. We consider classes that are computationally easy to learn using CoT supervision from a single thinker, but hard to learn with only end-result supervision, i.e., without CoT (Joshi et al. 2025). We establish that, under cryptographic assumptions, learning can be hard from CoT supervision provided by two or a few different thinkers, in passive data-collection settings. On the other hand, we provide a generic computationally efficient active learning algorithm that learns with a small amount of CoT data per thinker that is completely independent of the target accuracy $\varepsilon$, a moderate number of thinkers that scales as $\log \frac{1}{\varepsilon}\log \log \frac{1}{\varepsilon}$, and sufficient passive end-result data that scales as $\frac{1}{\varepsilon}\cdot poly\log\frac{1}{\varepsilon}$.

A parametric theory of statistical inference is developed for the moderate deviation probability zone. The new approach to the proofs is based on the Taylor series expansion of the logarithm of the likelihood ratio based on the Hellinger distance. The Large Deviation Principle in the moderate deviation probability zone is proven for Bayesian estimators and maximum likelihood estimators. A uniform approximation of the logarithm of the likelihood ratio and Theorem on concentration of the posterior Bayesian measure are also established for the zone of moderate deviation probabilities.

We consider debiased inference on finite-dimensional functionals of infinite-dimensional least-squares solutions to inverse problems as a way to avoid having to assume exact solutions exist. Such assumptions are substantive and not innocuous, and their failure may imperil inference when we impose them on the statistical model. Our approach instead allows us to conduct inference on a quantity that is defined regardless of solutions existing and coincides with the usual estimands when they do. For the case of instrumental variables, this means we can motivate the analysis with structural models but these do not need to hold exactly for the semiparametric inferential procedure to remain valid.

Multi-armed bandits are widely used for sequential experimentation in clinical trials, recommendation systems, and online platforms. While regret minimization and valid inference from adaptively collected data have each been studied extensively, a basic question remains: when does adaptivity improve estimation precision relative to uniform designs, and how should inference be balanced against the online cost of experimentation? We first study arm-level mean estimation under mean-squared-error (MSE) objectives. We characterize when an adaptive Neyman allocation, which allocates samples according to arm variance, yields strict MSE improvements over uniform sampling. When there is variance heterogeneity across arms, these improvements arise at modest sample sizes, clarifying that adaptivity can be preferable for inference not only asymptotically, but also in many practical finite-sample settings. We then study a joint inference-regret objective that accounts for the cost of assigning units to inferior arms during experimentation. We propose the Static-Allocation Rate Policy (SARP) and Neyman-Adaptive Rate Policy (NARP), which interpolates between inference- and regret-oriented policies by adjusting exploration to the local structure of the instance. We show that SARP and NARP converge to the complete-information benchmark at the optimal rate as the sampling budget grows. Our proposed policies are practically attractive as it linearly interpolates between any standard regret-minimizing algorithm and inference-targeting adaptive policies. Yet we show it still enjoys the oracle-based asymptotic optimal rate. Simulations support the theory by demonstrating improved precision over uniform allocation while controlling performance loss across a range of instances.

We derive first-order (in the stepsize) bounds on the bias in Wasserstein distances of the invariant measure of stochastic gradient kinetic Langevin dynamics with minimal assumptions on the stochastic gradient noise. These bounds sharpen existing non-asymptotic guarantees for stochastic-gradient MCMC methods and provide a quantitative resolution of a previously open problem on invariant measure accuracy. The main technical ingredients are new Gaussian convolution inequalities controlling the Wasserstein-$p$ distance between a Gaussian convolved with a mean-zero perturbation and the Gaussian itself. We anticipate that these inequalities will be of independent interest beyond the present application. To complement the theory, we illustrate the benefits of stochastic gradient sampling methods on a one-dimensional model, a Bayesian logistic regression example and a Bayesian random-effect model including comparison with pseudo-marginal MCMC methods.

Bayesian inference in hidden Markov models (HMMs) can be challenging due to the presence of multimodality in the likelihood function, and consequently in the joint posterior distribution, even after correcting for label switching. The parallel tempering (PT) algorithm, a state-space augmentation method, is a widely used approach for dealing with multimodal distributions. Nevertheless, standard implementation of the PT algorithm may not always be sufficient to effectively explore the high-dimensional, complex multimodal posterior distributions that arise in HMMs. In this work, we demonstrate common pitfalls when implementing the PT algorithm for HMMs, approaches to remedy them, and introduce new non-informative prior distributions that facilitate effective posterior distribution exploration. We analyse time series of blue whale dive data with two 3-state HMMs in a Bayesian framework, one of which includes a categorical covariate in the transition probability matrix to account for the effect of sound stimuli on the whale's behavior. We demonstrate how effective implementation of the modified PT algorithm for Bayesian inference leads to effective exploration of the resultant multimodal posterior distribution and how that affects inference for the underlying movement patterns of the blue whales.

Machine-learning interatomic potentials (MLIPs) have enabled molecular dynamics at near ab initio accuracy, yet remain limited to energies and forces by construction, leaving electronic observables such as dipole moments and polarizabilities inaccessible. We introduce DenSNet, a density-first approach to machine-learned electronic structure that learns the Hohenberg–Kohn map from nuclear configurations to the ground-state electron density. Our approach employs an SE(3)-equivariant neural network to predict density coefficients of a flexible atom-centered Gaussian basis, combined with a $\Delta$-learning strategy that uses superposed atomic densities as a prior to accelerate training. A second equivariant network then maps the predicted density to the total energy, providing a unified framework for molecular dynamics and electronic structure. We validate DenSNet on ethanol, ethanethiol, and resorcinol, where infrared spectra from machine-learned trajectories show excellent agreement with experimental gas-phase measurements. To test scalability, we train on polythiophene oligomers with 1–6 monomers and extrapolate to chains of up to 12 monomers, generating stable long-time trajectories whose infrared spectra agree with reference density functional theory calculations. Here, we show that reinstating the electron density as the central learned quantity opens a practical route to transferable prediction of spectroscopic and electronic observables in large-scale molecular simulations.

We consider online learning problems under a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem.

In many areas of medicine, security, and life sciences, we want to allocate limited resources to different sources in order to detect extreme values. In this paper, we study an efficient way to allocate these resources sequentially under limited feedback. While sequential design of experiments is well studied in bandit theory, the most commonly optimized property is the regret with respect to the maximum mean reward. However, in other problems such as network intrusion detection, we are interested in detecting the most extreme value output by the sources. Therefore, in our work we study extreme regret which measures the efficiency of an algorithm compared to the oracle policy selecting the source with the heaviest tail. We propose the ExtremeHunter algorithm, provide its analysis, and evaluate it empirically on synthetic and real-world experiments.

We study the problem of global maximization of a function f given a finite number of evaluations perturbed by noise. We consider a very weak assumption on the function, namely that it is locally smooth (in some precise sense) with respect to some semi-metric, around one of its global maxima. Compared to previous works on bandits in general spaces (Kleinberg et al., 2008; Bubeck et al., 2011a) our algorithm does not require the knowledge of this semi-metric. Our algorithm, StoSOO, follows an optimistic strategy to iteratively construct upper confidence bounds over the hierarchical partitions of the function domain to decide which point to sample next. A finite-time analysis of StoSOO shows that it performs almost as well as the best specifically-tuned algorithms even though the local smoothness of the function is not known.

Uncertainty is ubiquitous in real-world data, and the assumptions underlying classical linear regression models are often violated in practice. Inspired by the theory of sublinear expectation, we consider a linear regression model where the random intercept term has mean uncertainty and the error term has variance uncertainty. We develop a novel two-step approach, named Expectation-Maximization with Moving Block (EMMB), to estimate the model parameters. The proposed method requires no prior knowledge of group structures or change points. Theoretical properties of the estimators are established under mild regularity conditions. Simulation studies and a real-data application to PM2.5 concentration modeling in Beijing demonstrate the superiority of the proposed method: it captures substantial intercept heterogeneity overlooked by ordinary least squares and yields more accurate and interpretable estimates.

Digital travel platforms often operate multiple marketing journeys simultaneously, resulting in overlapping user exposures that bias the standard A/B lift estimation. Because traditional lift experiments assume treatment isolation, the observed lifts reflect only marginal effects and may substantially underestimate the total incremental impact of each journey. This work introduces a Hierarchical Causal Lift Model that decomposes pure and global effects under journey overlap. Each journey is modeled as a multiplicative causal factor, and the interaction terms capture potential synergies or cannibalizations. The model is estimated through a Monte Carlo framework that incorporates uncertainty in overlap proportions, observed lifts, and single-journey effects. Regularized non-linear least squares are complemented with Monte Carlo simulation to quantify parameter uncertainty and assess the robustness of the solution. Applied to an active user base of approximately three million users, the model reveals positive but modest synergies between journeys and shows that pure lifts are significantly larger than those observed experimentally. The predicted global lift closely matches the experimentally measured value, demonstrating the ability of the model to recover incremental effects in an interpretable manner.

Information theory is a powerful framework to capture aspects of dynamical systems with multiple degrees of freedom. Mathematically, the dynamics can be represented as a continuous curve $\mathcal{C}$ on a suitable hyperplane in flat space and the Fisher information provides the norm of an infinitesimal displacement along this curve. In many applications, however, we do not have direct access to $\mathcal{C}$. Instead, we have to reconstruct the latter from a time-series of measurements (obtained as samples of size $n$), which are represented by an ordered set of points $\widehat{\mathcal{C}}$ on the same hyperplane. In this work, we calculate the bias of the Fisher information for large $n$, which provides a quantitative estimation for how accurately the dynamics of a system can be reconstructed from a given set of sampled data. Based on this result, we show that a clustering of the degrees of freedom reduces the bias and thus improves the accuracy with which the new system can be described with the same data. Inspired by a recent proposal for such a clustering, we provide a quantitive assessment of the loss of information, which allows to estimate how much information about the dynamics of a system can reliably be extracted based on a given set of data. We illustrate our findings in the case of a simple compartmental model. Although the latter is inspired by epidemiology, the results of this work are applicable to very general dynamical models with multiple degrees of freedom.

Measures of association in contingency tables, such as odds ratios and their generalizations, are often studied under different sampling schemes that either fix or leave random the margins of the table. While classical results show that certain odds ratios are unaffected by constraining the margins, it is less clear when this invariance holds more generally. This paper studies posterior inference for a broad class of multiplicative contrasts of multinomial cell probabilities, which we refer to as generalized odds ratios, and addresses exactly when fixing a margin alters inference about them. We consider Bayesian inference under multinomial sampling and under models in which partition sums of the table are fixed in advance, and assume that the marginal and conditional parameters are independent a priori. Under additional mild assumptions, we show that the posterior distribution of a generalized odds ratio is invariant to fixing a margin if and only if the coefficients defining the contrast sum to zero within the margin.

A novel data-driven methodology is presented for the joint selection of prior parameters for both fixed and random effects in Linear Mixed Models (LMMs). This approach facilitates the estimation of complex random-effects structures, as well as potentially high-dimensional data. Although Bayesian frameworks require the specification of informative prior parameters, such values are often unavailable a priori - especially for random-effect covariances. The proposed method automates this selection through an Empirical Bayes framework, which maximizes the marginal likelihood using an efficient Laplace approximation. Numerical simulations demonstrate that this methodology significantly enhances parameter estimation accuracy and predictive performance. Finally, an application to a real-world air pollution and health dataset illustrates how the method enables the use of more sophisticated and statistically appropriate models to improve predictive outcomes.

Immune checkpoint inhibitor–based therapies often produce heterogeneous survival responses, including early risk, delayed treatment benefit, and durable long-term survival in a subset of patients. In these settings, conventional summary measures such as the hazard ratio may not adequately describe how treatment effects evolve over follow-up. We propose a milestone-based framework that separates long-term survival beyond a clinically meaningful time point from earlier outcomes and provides a practical way to characterize patient heterogeneity in treatment response. The framework summarizes treatment differences through milestone survival probabilities and, among patients who do not reach the milestone, characterizes short-term treatment ordering over time using a tau-based summary that helps identify hazard reversal. We illustrate the approach using reconstructed individual-level data from three landmark phase III trials: CheckMate~067, CheckMate~227, and CLEAR. Across these examples, the framework captures patterns that are difficult to summarize with conventional measures, including settings in which early disadvantage coexists with later durable benefit. It also helps clarify when treatment benefit begins to emerge and how short-term and long-term effects differ within the same trial. This approach provides a clinically interpretable and statistically principled way to evaluate heterogeneous and time-varying treatment effects in oncology trials with nonproportional hazards.

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou–Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the Nelson problem.

A drifting model is a one-step generator trained by moving each sample along a field of kernel-weighted attraction toward data samples and repulsion between model samples; training halts once this field vanishes. The soundness of this scheme rests on two questions: whether a zero-field equilibrium guarantees agreement with the data distribution, and how the error is controlled when the field is small. We answer both questions. We introduce the companion-elliptic kernel class, which contains the Laplace kernel, and prove that a vanishing field identifies arbitrary Borel probability measures within this class and its blockwise product extension. The scalar companion-elliptic kernels are exactly the Gaussian and Matérn families, and deconvolution, the recovery of a measure from its kernel-smoothed density, is possible precisely when the zero set of the kernel's Fourier transform has empty interior. At approximate equilibria, a mass-escape counterexample shows that field information on a bounded region cannot control the global error. We therefore prove an explicit stability inequality bounding the error by the field residual on the observation region plus the convolution mass outside it. In this inequality the Gaussian kernel requires only field values but amplifies high-frequency errors exponentially, whereas the Matérn kernel requires one additional order of derivative information in exchange for polynomial amplification. Since this gap widens exponentially with frequency, the balance favors the Matérn family and hence the Laplace kernel.

This paper uses a minimum divergence framework to introduce a new way of calculating model weights that can be used to average probabilistic predictions from statistical and machine learning models. The method is general and can be applied regardless of whether the models under consideration are fit to data using frequentist, Bayesian, or some other fitting method. The proposed method is motivated in two different ways and is shown empirically to perform better than or on a par with standard model averaging methods, including model stacking and model averaging that relies on Akaike-style negative exponentiated model weighting, especially when the sample size is small. Our theoretical analysis explains why the method has a small-sample advantage.

We propose a gate-based quantum algorithm for the prediction step of Bayesian state estimation based on the Fokker-Planck equation on a discretized position-velocity state space. The probability density is encoded in the amplitudes of a quantum state, enabling a compact representation of high-dimensional distributions. Exploiting the circulant structure of finite-difference operators, the evolution is realized in the spectral domain using quantum Fourier transforms and phase rotations. A key result is that the drift component can be implemented exactly in amplitude space, leading to an accurate reproduction of the classical transport dynamics. In contrast, the diffusion term does not admit a linear representation in amplitude space due to the nonlinear relation between probability density and wave function. To enable a quantum implementation, we introduce a unitary surrogate based on a Wick rotation, transforming diffusion into a dispersive phase evolution. This yields a fully unitary propagation that can be implemented efficiently on a gate-based quantum computer. The proposed method is evaluated numerically for different scenarios and shows strong agreement with the exact solution of the Fokker-Planck equation. The approach demonstrates the potential of quantum computing for Bayesian state estimation, as the representable state space grows exponentially with the number of qubits. This allows the efficient representation and propagation of probability densities that would otherwise require complex tensor decompositions on classical hardware, making the method a promising candidate for high-dimensional filtering problems.

Create an idea, prototype it, evaluate if users like it, then learn. It is the circle of business. If AI can operate in all parts of the circle, it will enable rapid iteration and learning speeds for businesses. Experiment platforms that deploy experiments to evaluate return on investment for businesses are abundant, but systems that help businesses learn personalization, mechanisms, and what to ideate next, are rare. Among technologies that do exist, they cannot be well orchestrated in a single software interface that can be safely and efficiently leveraged by an AI agent. These challenges make it difficult to teach an AI agent how to learn within a robust experimentation framework, and difficult for an AI agent to operate and iterate for the business. We offer a two part solution: one half that is rooted in mathematical reductions to contain complexity, and one half that is rooted in software design to optimize for orchestration, software safety, and multiplicity. Our solution, a software framework, moves beyond the simple treatment effect computed as a difference in means. To create a better understanding of a business and its customers, we enrich causal analysis with heterogeneous effects, policy algorithms, mediation analysis, and forecasts of effects. To have an AI complete the iteration cycle faster, we further enrich the analysis with variance reduction and anytime valid inference. The enrichments are made compatible across different types of experiments, and are presented in a single software interface that is usable in an AI agent. We evaluate the approach on various objectives in experiment analysis, and show that the framework improves code correctness, reduces lines of code, and is more performant than a baseline analysis constructed by a vanilla agent.

Effectively controlling the false discovery rate (FDR) in high-dimensional variable selection is a fundamental statistical problem that has garnered significant research interest. In this paper, we propose a novel, user-friendly, and computationally efficient method called Bi-Gaussian Mirrors (BGM), which offers a conceptually simple yet powerful approach for FDR control. Our method makes the first attempt to achieve FDR control in high-dimensional data with complex dependencies, while overcoming key limitations of existing approaches, such as prior knowledge of the joint distribution of data, significant power loss, the need for full symmetry in test statistics, and the theoretical restriction to linear regression models. Additionally, we present a self-guiding procedure designed to enhance the practicality and applicability of the BGM method. Theoretical guarantees for FDR control and asymptotic power are rigorously established under regularity conditions. Moreover, extensive numerical simulations and two real-data examples demonstrate that the BGM method outperforms existing approaches in terms of finite-sample performance, achieving a superior balance between FDR control and testing power.

Emergent intelligence have played a major role in the modern AI development. While existing studies primarily rely on empirical observations to characterize this phenomenon, a rigorous theoretical framework remains underexplored. This study attempts to develop a mathematical approach to formalize emergent intelligence from the perspective of limit theory. Specifically, we introduce a performance function E(N, P, K), dependent on data size N, model size P and training steps K, to quantify intelligence behavior. We posit that intelligence emerges as a transition from finite to effectively infinite knowledge, and thus recast emergent intelligence as existence of the limit $\lim_{N,P,K \to \infty} \mathcal{E}(N,P,K)$, with emergent abilities corresponding to the limiting behavior. This limit theory helps reveal that emergent intelligence originates from the existence of a parameter-limit architecture (referred to as the limit architecture), and that emergent intelligence rationally corresponds to the learning behavior of this limit system. By introducing tools from nonlinear Lipschitz operator theory, we prove that the necessary and sufficient conditions for existence of the limit architecture. Furthermore, we derive the scaling law of foundation models by leveraging tools of Lipschitz operator and covering number. Theoretical results show that: 1) emergent intelligence is governed by three key factors-training steps, data size and the model architecture, where the properties of basic blocks play a crucial role in constructing foundation models; 2) the critical condition Lip(T)=1 for emergent intelligence provides theoretical support for existing findings. 3) emergent intelligence is determined by an infinite-dimensional system, yet can be effectively realized in practice through a finite-dimensional architecture. Our empirical results corroborate these theoretical findings.

Cluster randomized trials are widely used when individual randomization is logistically infeasible or when correlations between observations cannot be ignored, especially in fields such as ophthalmology, infectious disease, vaccine research, and sociology. The desirability of outcome ranking (DOOR) framework evaluates patient-centric benefit-risk using an ordinal outcome and a Wilcoxon-Mann-Whitney statistic-based approach to compare outcome distributions between interventions. We propose a suite of new methods to extend DOOR to cluster trials based on properties of U-statistics and influence functions to estimate within-cluster and between-cluster treatment effects. These approaches can be applied in different scenarios, including mixtures of clusters with two treatment groups and clusters with only one group, and both small and large numbers of clusters. Simulations demonstrate that the proposed methods perform well under various scenarios regarding the number of clusters and cluster sizes. As an illustration, we apply the proposed methods to a cluster randomized crossover trial comparing delayed cord clamping and umbilical cord milking for newborns.