Papers

A popular selling point of diffusion large language models (dLLMs) is their capacity for parallelism: the ability to generate sequences of text far more efficiently than autoregressive models, which require one forward pass per token. Yet among the many competing paradigms for dLLMs, from masked to uniform to Gaussian diffusion, principled understanding of how these different proposals compare in parallelism remains limited. In this work, we initiate a fine-grained comparison of the capacity for parallelism among these three leading approaches and prove the following: - Uniform and Gaussian diffusion can sample in a number of forward passes which scales with the dual total correlation of the underlying distribution, a measure of intrinsic complexity which can be much smaller than the context length. Previously, it was only known how to achieve this using masked diffusion. - For a certain family of random empirical measures, we show that $\widetilde{\Theta}(\sqrt{d})$ forward passes are necessary and sufficient to sample using uniform or Gaussian diffusion, yet there exist approximate score oracles for which $\widetilde{\Omega}(d)$ forward passes are needed for masked diffusion. This establishes the first provable separation in parallelism between the three prevailing dLLM paradigms. Contrary to popular intuition that masked diffusions are harder to parallelize because they must commit to token values, the latter separation instead comes from the fact that the critical windows in masked diffusion sampling are asymptotically narrower than those in uniform and Gaussian diffusion sampling.

We study simultaneous inference for maxima of canonical order-two $U$-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos and establish a general approximation result that permits indefinite kernels. The general anti-concentration bound is too crude for high-dimensional inference, and we obtain sharper bounds under additional spectral structure. We also identify a phase transition from a non-Gaussian signed-chaos maximum to its covariance-matched Gaussian counterpart driven by the effective rank. For feasible inference, we propose a Gaussian multiplier bootstrap that avoid estimating eigensystems, and establish its validity. Two applications and extensive numerical simulations further illustrate the scope and practical performance of the proposed framework.

We prove a sharp concentration inequality for the spectral norm of sparse random tensors with independent Bernoulli entries. Let $T$ be an order-$k$ tensor of dimension $n\times\cdots\times n$ with independent Bernoulli$(p)$ entries, where $k$ is fixed. For any $c,r>0$, we show that $\|T-\mathbb E T\|\le C_{k,r,c}\sqrt{np}$ with probability at least $1-n^{-r}$ whenever $np\ge c\log n$. We extend this bound to inhomogeneous Bernoulli sampling with deterministic entrywise weights. This removes the logarithmic factor in the work of Zhou and Zhu (2021). The proof follows the Kahn–Szemerédi light–heavy decomposition with a refined estimate on the heavy tuple part. We also obtain a log-free second eigenvalue bound for the random hypergraph model of Friedman and Wigderson (1995).

We construct a symmetric operad $\mathfrak{C}$ on the class of all multivariate copulas, where operadic composition is given by Sklar substitution. We prove that the absolutely continuous subclass $\mathfrak{C}^{ac}$—which coincides with the $L^1$ class of copula densities—forms a suboperad; under composition, the density of the composite copula is given by the explicit Sklar substitution density formula $g(v)=\phi\big(\Psi_1(v^{(1)}),\dots,\Psi_n(v^{(n)})\big)\prod_{k=1}^{n}\psi_{k}(v^{(k)})$. Furthermore, we show that copulas with finite copula entropy—identified with the $L\log L$ class of copula densities—are closed under substitution and hence constitute a suboperad $\mathfrak{C}^{L\log L}$. On this suboperad, copula entropy is strictly additive: $H(\gamma(\Phi;\Psi_{1},\ldots,\Psi_{n}))=H(\Phi)+\sum_{k=1}^{n}H(\Psi_{k})$.

The classical Whittle formula provides a closed-form expression for the asymptotic Fisher information matrix (FIM) rate of multidimensional, real-valued, zero-mean, purely nondeterministic stationary Gaussian processes (GPs), expressed in terms of their parameterized spectra within a maximum-likelihood framework. However, the Gaussian assumption underlying this result restricts its applicability to many real-world signals exhibiting heavy-tailed or non-Gaussian behavior. In this paper, we extend Whittle's result to multidimensional, real-valued stationary compound Gaussian processes (CGPs) with arbitrary mean, under a unified framework encompassing fully known, parameterized, and completely unknown density generators. Building upon the Slepian-Bangs formula for $n$ consecutive observations and extending it to the semiparametric setting, we leverage the asymptotic properties of block Toeplitz matrices to obtain a closed-form spectral-domain expression for the asymptotic FIM rate. The resulting expression, common to all density generator families, generalizes the standard zero-mean Gaussian formula by incorporating a distribution-dependent contribution associated with the nonzero mean, and an additional covariance term that is invariant to the choice of non-Gaussian distribution. This extension enables efficient and theoretically grounded performance analysis for heavy-tailed signal processing applications.

Time series of random objects, such as covariance matrices, probability distributions, and functional data, call for forecasting methods that do not rely on standard arithmetic operations. We introduce geodesic exponential smoothing, a generalization of exponential smoothing to time series in Hadamard spaces: the forecast level moves a fixed fraction of the way along the geodesic toward each new observation. The smoothing parameter is estimated by minimizing the average squared distance between observations and their forecasts. We further introduce an innovations mechanism under which each observation has conditional Fréchet mean equal to the current level, providing the metric-space analog of the innovations state-space model. In contrast to autoregressive models for object-valued time series, the framework involves a single scalar parameter, assumes no stationarity, and updates online in constant time per observation. Under this mechanism, we establish sample-path properties of the generative process via the quasilinearization available in Hadamard spaces, and prove almost-sure consistency of the smoothing-parameter estimator. Three real-data applications, spanning covariance-matrix, distributional, and functional time series, assess the forecasting performance of the method against structurally heavier alternatives.

This paper develops a design-based asymptotic theory for two-stage sampling over a continuous spatial domain. The target parameters are integral totals, or smooth functions of such totals, defined over a fixed bounded territory partitioned into an increasingly fine collection of primary sampling units. Within this fixed-area framework, we derive the order of the variance components of the Horvitz-Thompson estimator when secondary sampling units are points selected from continuous sub-regions. We establish design consistency of the estimator, as well as consistency of variance estimators under explicit regularity conditions on inclusion probabilities, sampling densities, and the study variable. The results are extended to plug-in estimators of smooth scaleinvariant functions of totals. For high-entropy first-stage designs, we further show that a Hájek-type variance estimator based only on first-order inclusion probabilities is consistent, and we prove asymptotic normality of both total and plug-in estimators. A simulation study inspired by forest inventory applications illustrates the finite-sample performance of the proposed estimators and variance estimators.

The Brunner-Munzel (BM) test is a nonparametric test for two independent samples that evaluates whether observations from one group tend to be greater than observations from another group, or vice versa. The BM test has a broader scope of application than the Mann-Whitney $U$ test because it does not assume equal variances between the two groups. However, the meaning of the BM test statistic is difficult to understand intuitively, which may be one of the factors hindering the widespread use of the BM test. To alleviate this problem, in this paper, I introduce an alternative interpretation of the BM test statistic from the viewpoint of local linear approximation. It is shown that the variance estimator for the sample stochastic superiority used in the BM test can be derived using local linear approximation, in which the influence of each observation on the sample stochastic superiority is assumed to be additive. This simple interpretation will help practitioners decide to use the BM test without hesitation.

Phase-type (PH) distributions are versatile semi-parametric models for lifetime duration and can be used in survival and reliability analysis. In this paper we put forward methods and software for using PH distributions in a competing-risks model. The resulting multi-absorbing phase-type (MAPH) distribution records both the time until absorption and its cause. After formulating basic properties of this family, we develop an EM algorithm for parameter inference from exact event-time/cause observations and independently right-censored observations. For a censored subject, the E-step conditions on survival up to the censoring time and imputes both the latent transient path and its eventual cause of absorption; closed-form cause-resolved sufficient-statistic expectations are obtained from matrix exponentials. We illustrate applications and numerical properties and describe two accompanying Julia packages, in which both the exact-event and the censored E-step are implemented. Applied to intensive-care length-of-stay data in full, censored records included, the method attains a higher likelihood than the phase-type competing-risks fit previously published for those data.

We determine the exact attainable region of Spearman's rho and Gini's gamma over all bivariate copulas, resolving the remaining pairwise exact-region problem among Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta and Spearman's footrule. We give an explicit parametrization of the rho-maximal boundary for prescribed Gini's gamma and construct copulas attaining every boundary point. The boundary consists of an elementary arc up to a single junction and, beyond it, countably many algebraic pieces accumulating at comonotonicity. This description also gives the largest possible value of $|\rho-\gamma|$ and the sharp thresholds beyond which rho and gamma must have the same sign. Our proof separates the signs and magnitudes of centred ranks, reducing the optimization of linear combinations of $\rho$ and $\gamma$ to an optimal transport problem for two uniform magnitudes. The extremizers combine a central antidiagonal block with rescaled auxiliary transport optimizers, and global optimality follows from a Kantorovich dual potential obtained by gluing the corresponding dual pieces.

We study the mean function of longitudinal functional data, where each subject contributes a small number of complete profiles over a general domain, observed at random visit times. The mean is projected onto an orthonormal basis in the time direction, and each coefficient function is estimated by a weighted average of the profiles. We consider deterministic and random weights, including closed-form, data-driven weights that dispense with the design density entirely. For each weighting scheme we derive non-asymptotic bounds on the integrated quadratic risk and an explicit optimal truncation level. All schemes share the same convergence rate, which we show to be minimax optimal over the corresponding coefficient-decay class, only the leading constants differ. We also provide a goodness-of-fit test for the mean function, and derive non-asymptotic bounds for the Gaussian approximation of its distribution under both the null and alternative hypotheses. Our estimation and testing procedures are easy to implement, fast, and perform well in applications.

Several institutions test one family of hypotheses under family-wise error control but cannot pool their data, so each site releases only a quantized report of its p-value for each hypothesis, such as whether the p-value is significant. The report's resolution is the number of values it can take. We quantify the power lost to such reports relative to the most powerful centralized procedure given all p-values (the oracle), with reports and combining rule chosen optimally. A report from which the p-value can be recovered loses no power, so the loss is due to compression, not decentralization. Under the stated regularity and power-objective conditions, every finite resolution loses power, and the optimal loss decays as the inverse square of the resolution: equal-width intervals attain this order, and no partition into as many cells improves it. In two-site Beta examples, optimized one-bit reports are nearly lossless for a single hypothesis but lose several times as much power for two. At level 0.05 with two Beta(1,2) hypotheses, optimally combined three-bit equal-width reports retain 99.2% of oracle power. We give the power-optimal combining rule for given reports, and a rule using only their exactly known null distribution, valid under arbitrary dependence across hypotheses.

Human pairwise comparisons provide a reference for evaluating large language models (LLMs), but collecting sufficient judgments for each new release is costly and time-consuming. LLM judges offer a scalable alternative, although their comparisons may differ systematically from human preferences and across judges. We study the ranking of a new model that has received LLM-judge comparisons but no human comparisons. We propose ANCHOR (ANchored Comparisons for Human-reference inference with Orthogonal Riesz correction), which uses historical human and LLM comparisons to learn judge-specific sensitivities to human score differences and feature-dependent judge biases. These estimates are then used to infer the new model's human-reference score from its judge comparisons. The framework allows the feature distribution to change between historical and new-model comparisons. For inference, we construct a Neyman-orthogonal estimator through a joint Riesz correction that removes the first-order effects of estimating the historical human scores, judge sensitivities, and bias functions. We establish identification, convergence rates, and asymptotic normality with consistently estimable variance, and show that ANCHOR attains the semiparametric efficiency bound. Simulations demonstrate gains in score estimation and ranking accuracy. On Chatbot Arena, ANCHOR achieves the lowest score RMSE and insertion MAE among competing methods, with narrower score intervals on average.

This paper establishes a unified non-linear regularization framework for high-dimensional $M$-estimation, encompassing both linear models and Cox's proportional hazards models. Rather than relying on traditional additive non-convex penalties, the proposed paradigm embeds sparsity directly into the transformation for the physical parameter $\theta=\phi(\beta)$ using a smooth ($C^2$) component-wise "ReParametrization map for Sparsity (RePS)" $\phi$, and the penalty term is $\lambda \Vert \beta \Vert_1$ rather than $\lambda \Vert \theta \Vert_1$. This structural formulation dynamically adapts to local parameter scales, suppressing high-dimensional noise while simultaneously recovering unbiased oracle asymptotic normality under the large-sample limit. Through the Primal-Dual Witness method, we establish a unified oracle equivalence for both model classes under a liberated micro-penalty scaling regime $\lambda \ll n^{-1/2}$, accommodating model-specific structures such as the shift-invariance in survival analysis. Extensive Monte Carlo simulations demonstrate that the proposed framework consistently achieves superior false-positive control and high 95% confidence interval coverage in high-dimensional linear models, while also delivering performance comparable or superior in all aspects to state-of-the-art methods like SCAD and MCP in proportional hazards models.

Suppose we observe the first $n$ points of a sequence of random variables having length $n+1$, and wish to estimate a functional of the unobserved final point and the empirical measure of the $n$ observed training points. Such next-token functionals include the probability that the next token is novel (also known as the surprise probability), the tail probability of the minimum distance between the next token and training points, and the test error of a classifier trained on the observed points. All of these quantities are classically estimated by the leave-one-out method, which is inconsistent under temporal dependence. We propose a leave-a-window-out estimator, which deletes a window of length $\tau$ after each index before forming the empirical measure and reduces to leave-one-out at $\tau = 1$. Under natural assumptions, we show that the error of our estimator decays at a parametric rate for any stationary $\beta$-mixing process that also admits a Marton coupling. Our results thus cover several natural functionals on a large class of stochastic processes. We complement these upper bounds with a sharp minimax lower bound for estimating the surprise probability on mixing Markov chains. Simulations on Markov chains, moving-average processes, and autoregressive processes show that our estimator succeeds in many scenarios where leave-one-out and add-constant baselines fail.

Despite the widespread use and success of generative AI techniques today, theoretical guarantees on learning a distribution supported in $d$ dimensions from $n$ samples degrade as $O(n^{-1/\Theta(d)})$, though shown to be minimax optimal. We hypothesize that present bounds are too pessimistic because smoothness assumptions are not enough to capture the structure of distributions that often appear in real applications. Consequently, we introduce the class of sparse priors and define the "Sparse Dimension" as a measure of sparsity of a prior over the space of all distributions. We show that distribution learning under a $k$-sparse prior achieves a Bayesian risk lower bound of $\Omega(\sqrt{k/n})$ under common distance metrics, and show a matching (up to logarithmic terms asymptotically in $n,k$) upper bound for the TV distance under mild additional assumptions. We show the statistical equivalence of distribution learning and learning to sample in the Bayesian setting so that our results apply to learning to sample as well. While $k$ can still depend on the dimension $d$, or a notion of intrinsic dimension, our results show that learning under an appropriate prior overcomes the curse of dimensionality with respect to the dependence on $n$.

While kernel-based learning methods for scalar-valued functions have been extensively studied, comparatively little work considers nonlinear vector-valued operators. Such operators with vector-valued outputs, e.g., state vectors and trajectories, commonly arise in system identification problems. In this paper, we study the problem of learning a vector-valued operator under the framework of vector-valued kernel ridge regression. We assume that the outputs take values in an arbitrary Hilbert space and are corrupted by additive noise with bounded norm. Under the assumption that the unknown operator belongs to a vector-valued reproducing kernel Hilbert space, we derive a pointwise deterministic bound on the discrepancy between the kernel ridge regression estimator and the ground truth in the corresponding Hilbert space norm. Numerical simulations under two different scenarios are presented to validate the theoretical results.

In evidence-based policymaking, typically one experimental sample is observed, then a learned policy recommendation is implemented at scale. Policies learned from the experimental data can perform well in expected welfare, yet random sampling in the experiment can produce recommendations with poor welfare outcomes. In this paper, we ask: how should policy learning algorithms balance expected welfare against sampling risk? Our main contribution is to show that algorithmic stability plays a central role in characterizing and navigating the tradeoff. Intuitively, if a policy learning algorithm's recommendation remains stable when one experimental unit is replaced, then that algorithm has limited sampling risk. We propose a method for policy learning called policy-vote bagging, which learns treatment decisions on many subsamples then averages their votes into treatment probabilities. Relative to using one subsample, averaging across subsamples preserves expected welfare and improves expected utility for a risk-averse researcher. We derive sharp bounds linking estimation accuracy, subsample size, and welfare variation, including an exact guarantee under CARA utility.

We propose a two-view weighted-concordance operator (WCO) framework for offline change-point detection in weakly dependent text networks. Weighted word-co-occurrence graphs yield first-order direct-co-occurrence and second-order shared-context views, projected into paired Euclidean observations using maps learned from an independent pilot corpus. A WCO scan detects changes in cross-view dependence. A dependent multiplier bootstrap tests for a change, while a segment-wise block bootstrap gives a descriptive stability interval for its location. We establish null weak convergence, bootstrap validity, asymptotic size control, consistent localization under a single identifiable change, and perturbation bounds for token-level errors. CUSUM and Gaussian-kernel MMD scans provide complementary benchmarks. In the 1915-1921 New Youth corpus, the method detects a change in November 1919, with a descriptive 95 percent stability interval from April 1919 to June 1920, consistent with the linguistic transition surrounding China's May Fourth and New Culture movements.

The test error of a model plotted against its number of parameters $d$ falls, peaks when the model can just fit the training data, and falls again, exhibiting the double descent phenomenon. We explain the phenomenon with statistical mechanics. The training trajectory of a stochastic gradient-based method is a particle wandering over the energy landscape of the training loss at an induced temperature $T$, and a run that has equilibrated visits every parameter vector of a given training loss equally often, the fundamental postulate of statistical mechanics, with probability given by the Boltzmann distribution. Because training starts at an initial point and has only finite time to diffuse, it carries an effective weight decay, which makes every parameter a quadratic degree of freedom. The equipartition theorem then distributes the energy among the $d$ degrees of freedom in shares of $T/2$, so at a fixed training loss adding parameters lowers the temperature and drives the Boltzmann distribution toward the stationary path. Finally, adding parameters can only lower the $L^2$ norm of the stationary path, so a solution sampled at fixed loss is less likely to be large with increasing $d$, effectively increasing weight regularization.

We consider the problem of noisy gradient-free minimization of the k-th order partial derivative of a $\beta$-Hölder function supported on a d-dimensional cube. We show that T ^{($\beta$+d+k)/(2$\beta$+d)} log(T )^{(\beta-k)/(2\beta+d)} is a non-asymptotic minimax rate of the T step cumulative regret for all $\beta$ \ge 0. In the special case k = 0, our results cover the problem of noisy gradient-free minimization of $\beta$-Hölder functions, closing the existing gap between the known upper and lower bounds. We show that a minimizer of a suitably chosen local polynomial estimator is rate-optimal. The minimax optimal upper bound is achieved under the passive design, that is, when the query points are i.i.d. Thus, there is no advantage in considering sequential designs when it is only known that f is a $\beta$-Hölder function with no additional property. We propose an algorithm feasible in polynomial time that constructs a proxy of the minimizer of the local polynomial estimator. The procedure requires computing the estimator on auxiliary random points. The resulting polynomial time algorithm matches the lower bound.

Randomization or design-based inference is becoming an increasingly popular tool for analysing data from randomized experiments: It does not require modelling assumptions on the distribution of outcomes or covariates, and hypothesis testing and estimation are respectively valid and unbiased in finite samples. Yet, confidence intervals for the sample average treatment effect (SATE) are still constructed via finite-population central limit theorems and their coverage is only asymptotic. In this work, we explore an alternative approach: We use concentration inequalities to construct confidence intervals for the SATE with non-asymptotic guarantees. We develop this approach for the most common experimental designs (Bernoulli trials and completely randomized experiments) and provide Hoeffding and Bernstein-type confidence intervals. Moreover, we extend these results to matched-pair, cluster and stratified randomized experiments. Our key technical contributions are a novel Bernstein-type concentration inequality for i.i.d. data points as well as a concentration result for Neyman's variance estimator.

The practical use of statistical depth for fuzzy data requires regularity, inferential stability, and computational feasibility. This paper studies these aspects for the first time for the fuzzy depth, in particular for the fuzzy Tukey depth. In particular, we investigate the continuity of this depth with respect to each of its arguments. As a function of the elements of the underlying space, we prove its upper semicontinuity under the main metrics used in fuzzy spaces. As a function of the distribution with respect to which the depth is computed, we establish the almost sure uniform consistency of its empirical version for continuous fuzzy random variables. These results ensure closed depth regions and support the use of sample depth values to approximate their population counterparts. We also derive consistency of empirical deepest-point estimators and study a finite-grid approximation of the depth, supported by theoretical and simulation results.

We study statistical variants of the classical largest average submatrix problem. For small submatrices (where the dimension is less than linearly proportional to the original matrix), the problem is typically well understood and believed to exhibit the statistical-computational gap (SCG). However, analytical treatment of both the information-theoretic and algorithmic aspects of the linear regime remains challenging, and no mathematically rigorous results have yet arrived anywhere close to proving or disproving SCG existence in this setting. Focusing on the linear regime, we make strong progress in several key directions: 1) We develop a generic Random Duality Theory (RDT) framework to characterize largest average submatrix values. 2) Using the plain RDT variant, we obtain closed-form upper bounds as explicit functions of dimensionality proportionality parameters. 3) We demonstrate that a lifted RDT variant strictly improves upon the plain RDT within a certain linear range of submatrix dimensions. 4) For small submatrices where the dimensional proportionality constants approach zero, we prove that our results match both the replica results from [34] (obtained via one-step replica symmetry breaking) and the sublinear results from [18,45].

We study when the logarithm $\psi$ of a positive finite exponential sum on $\mathbb{R}^d$ satisfies $\det\nabla^2\psi=C\exp(\langle b,\theta\rangle-\lambda\psi)$. This is the Kähler–Einstein equation for metrics induced by exponential maps into projective space and, for natural exponential families, the condition that the Jeffreys prior be a Diaconis–Ylvisaker conjugate prior. First, we classify the bivariate Laurent polynomials with unimodular support satisfying the generalized Einstein condition of Di Scala and Sombra: up to units and monomial changes of coordinates, they are powers of an affine trinomial or products of powers of two independent binomials. Second, we show in every dimension that a smooth compact toric manifold with a projectively induced Kähler–Einstein metric is a product of projective spaces with matched multiples of the Fubini–Study metrics, immersed by a complete Veronese–Segre system up to automorphisms. This proves the compact toric case of the homogeneity conjecture for such metrics and the fixed-point germ and univalent forms of a conjecture of Manno and Salis. Third, without lattice or rationality assumptions, the finite-support exponential families satisfying the equation are, up to affine changes of statistic, exactly the products of multinomial families with a common ratio of categories to trials; this settles the finite-support case of a question of Casalis.

We study inference on the drift of a critical Galton–Watson process with immigration, a count time series with a unit root. Climate change motivates such nonstationary models for weather-related disaster counts. The drift is the expected increase in the count per period. Ordinary least squares is inconsistent for the drift, so we study state-weighted least squares, giving greater weight to observations following small counts, where conditional variance is lower. Under strict recurrence, we apply null-recurrent regenerative limit theory to obtain a polynomial convergence rate and a standard normal studentized limit. Our main contribution is the recurrence boundary, where we establish a logarithmic convergence rate and a parameter-free non-Gaussian studentized limit. Estimating the optimal weights yields the same first-order limiting distribution as knowing them. Simulations show lower root mean squared error than time-weighted least squares with weights $1/t$, and improved state-weighted confidence-interval coverage when the unit root is imposed.

The problem of identifiability in linear parametric models (LPMs) whose nodes follow either an ordered logit model or a regular one-parameter exponential family is evaluated. The results go beyond classical structural equation models as well as results for nodes with observations from a homogeneous family of distributions. The main result establishes that the orientation of every edge joining an ordinal node to an exponential-family node is identifiable from the joint distribution alone at every parameter value, provided the ordinal node has at least three categories and the exponential-family node at least three points of support, with no restriction on the sufficient statistic. Converses show that both requirements are necessary: the three-category requirement is binding only for affine sufficient statistics, and the three-point requirement is binding under the canonical link. The guarantee extends to orienting every such mixed ordinal-exponential family edge of a given $d$-node undirected skeleton. Numerical experiments illustrate the theoretical results by successfully separating orientations within a Markov equivalence class, which are indistinguishable by conditional independence alone.

Exploiting the equality restrictions that nested Markov models encode requires their tangent-space geometry. For strictly positive finite-state models on arbitrary acyclic directed mixed graphs (ADMGs), we differentiate the recursive-head chart and prove that the range of its score map is the full tangent space. Intrinsic-set coordinate blocks form an algebraic direct sum, blocks of distinct districts are orthogonal, and the resulting Gram projection needs neither mb-shieldedness nor a district order. Exact counterexamples show that observational centering and kernel normalization alone do not certify tangency. For the node, complete-source edge, and compatible path-specific intervention targets considered here, boundary substitution yields a normalized configured active law and an order-free canonical-gradient formula, extended to finite mixtures by independent source redraws. A coherent one-step estimator with an exact remainder identity and a model-valid chart-flow targeted maximum likelihood estimator are efficient under stated local conditions. On a narrower source-isolated fixed-node subclass, a sequential estimator has an exact transition-factorized drift and up to $2^m$ nuisance-correctness regimes over $m$ active-district transitions; its all-correct influence function projects onto the canonical gradient, and an exact rational law exhibits a strict variance gap. These results separate order-free efficiency on arbitrary finite-state ADMGs from multiple robustness of a narrower construction.

Let the exact homogeneous hard-margin support vector machine be trained on \(m\) independent observations from a Borel probability law on a real Hilbert space. We prove that, with score zero counted as an error, there is a universal numerical constant \(C\) such that \[ \Pp\left( \gamma_m>0,\quad \Risk(u_m)> \frac{C}{m} \left( K_m+\log\frac1\delta \right) \right) \le \delta . \] Here \(\gamma_m\) is the empirical homogeneous margin, \(u_m\) is the exact minimum-norm unit-margin separator, \(r_m\) is the largest training radius, and \(K_m:=r_m^2\norm{u_m}^2=r_m^2/\gamma_m^2\) on \(\{\gamma_m>0\}\). The proof is driven by a deterministic deletion problem. Given vectors \(x_1,\ldots,x_n\) in the unit ball, delete a set \(B\) of constraints and let \(u_B\) be the closest point to the origin that satisfies every retained unit-margin constraint. Suppose that \(\norm{u_B}^2\le k\) and that every deleted vector has nonpositive score under \(u_B\). We prove that a family of such deletion sets of cardinality \(q\) has size at most \(\exp(8k+2q)\). The conceptual step is an exact identity obtained from the KKT representation of \(u_B\). For a random deletion set, the identity converts the mean squared spread of the separators into a weighted sum of score deficits. It therefore forces a coordinate whose deletion status separates the two conditional means by a quantitatively large amount. Revealing that coordinate decreases the conditional separator variance enough to control the binary entropy of the split. An entropy induction gives the deletion count, and an exact factorial ghost-sample identity converts that count into the stated high-probability SVM bound.

We study computable error bounds and certified early stopping for regularized inverse problems, where a data-fidelity term is traded against a regularizer. The analysis relies on an exact duality-gap identity that splits the total gap of $F(\Phi\mu)+\lambda R(\mu)$ into a data-fidelity Fenchel–Young loss and a regularizer Fenchel–Young loss, $ \Delta(\mu,h)=L_F(\Phi\mu\parallel h)+\lambda L_R(\mu\parallel\eta),\qquad \eta=-\Phi^\star h/\lambda, $ valid for any primal point $\mu$ and any dual point $h$. The data-fidelity term $F$ is strictly convex, so wherever $F^\star$ is differentiable the loss $L_F(\Phi\mu\parallel h)$ is the Bregman divergence of~$F$ between the prediction $\Phi\mu$ and $\nabla F^\star(h)$, and it vanishes exactly at Mirror Alignment $h=\nabla F(\Phi\mu)$. Evaluated at a dual-feasible point $\tilde h$, the gap~$\Delta(\mu,\tilde h)$ is computable and oracle-free, meaning that it uses no knowledge of the solution, and it bounds the suboptimality of $\mu$. Under the source condition, the same Fenchel–Young losses give a priori bounds on the estimation and prediction errors. Their scale is the irreducible model and noise error $L_F(\Phi\mu^\star\parallel h^\star)$, which vanishes exactly when Mirror Alignment holds at the certificate. This gives an early-stopping rule: run the algorithm until the regularizer Fenchel–Young loss falls below a tolerance $\epsilon$. A constructive version of the Brøndsted–Rockafellar theorem then turns the current pair into an exact dual-feasible one, and this proxy lifts to an exact primal certificate. We build the proxy by a proximal step in the geometry of the fidelity, with Bregman kernel $F^\star$ and tilted by the prediction $\Phi\mu$: it recovers the Euclidean step of Carlier when $F$ is the squared error, and it reduces the duality gap by the regularizer Fenchel–Young loss, up to a second-order remainder that vanishes in the quadratic case. Our running example is the Generalized Beurling–Lasso (GBL), where $R$ is the total-variation norm on signed measures. It contains the classical Beurling–Lasso, obtained with the squared error, and also covers robust, logistic, entropic and inverse-optimal-transport losses. The same duality gap certifies deep-learning optimizers such as Lion-K and Muon, in their proximal form, as solvers of the regularized program. A companion paper by the same authors builds on these error bounds to establish exact support recovery for the GBL under a non-degenerate source condition.