Market microstructure has long classified trading activity by its informational role: an informed trader demands liquidity by trading in the direction of private information, while a market maker supplies liquidity by absorbing that order flow and earning the spread in compensation Kyle (1985); Glosten and Milgrom (1985). This classification is typically recovered from the data the classifier requires: signed order flow, quote revisions, or the sequential-trade structure of the market. The classification is harder to apply to an algorithmic strategy whose internal logic is unobservable. However, the signals or optimization problems generating the decisions of a typical quantitative fund are not visible, even though the trades and reported positions may be available. This paper shows that the liquidity role of such a strategy (consumer or provider) can be recovered from realized portfolio costs and trade decisions alone, without observing quotes, order flow, or any other microstructure-specific signal.

A central question in the theory of machine learning concerns the identification of classes of data distributions for which one can provide computationally efficient learning algorithms with provable statistical learning guarantees. Indeed, in the context of probably approximately correct (PAC) learning, there has been much interest in exploring intermediate PAC learning models that, unlike the realizable PAC learning setting, allow for some stochasticity in the labels, and unlike the fully agnostic PAC learning setting, also admit computationally efficient learning algorithms with finite sample complexity bounds. Some examples of such models include random classification noise (RCN), probabilistic concepts, Massart noise, and generalized linear models (GLMs); in general, most of this work has focused on binary classification problems. In this paper, we study what we call realizablestatistic models (RSMs), wherein we allow stochastic labels but assume that some vector-valued statistic of the conditional label distribution comes from some known function class. RSMs are a flexible class of models that interpolate between the realizable and fully agnostic settings, and that also recover several previously studied models as special cases.

Constraint-based causal discovery relies on repeated conditional independence tests, but fast nonparametric tests often sacrifice calibration, especially when variables depend on the conditioning set through nonlinear relationships. We introduce BLITZ (Broad-to-Local Independence Testing via residualiZation), a nonparametric conditional independence test designed to run well under a second while maintaining the accuracy needed for the thousands of queries performed by constraint-based causal discovery algorithms. BLITZ first removes broad smooth dependence on the conditioning set using low-order polynomial regression, then applies a small nonlinear feature map and residualizes those features with shallow tree regressions. The resulting statistic tests residual cross-covariance, with a moment-matched chi-square approximation to the null distribution. We show theoretically that the two-stage design reduces the effective complexity faced by the tree residualizers, allowing shallow trees to control residual conditional-mean bias while avoiding excessive overfitting. In simulations, BLITZ provides better null calibration than fast kernel, random-feature, and regression-based competitors while remaining among the fastest methods tested. In causal discovery experiments on synthetic graphs and flow-cytometry data, BLITZ yields more reliable endpoint orientations among retained adjacencies and competitive structural recovery. These results suggest that broad-to-local residualization is a practical route to calibrated, scalable nonparametric conditional independence testing for causal discovery.

We consider the concentration properties of functions of infinitely exchangeable random variables. By conditioning on the de Finetti directing measure, we show that the deviation of any function with bounded-difference constants $c_1, \dots, c_n$ decomposes into a conditional sampling fluctuation and a latent mixture fluctuation. When this latent mixture is $σ_{\mathrm{mix}}^2$-subgaussian, we establish a concentration inequality with an effective variance proxy of $\frac{1}{4}\sum_i c_i^2 + σ_{\mathrm{mix}}^2$. Crucially, we demonstrate that for zero-sum linear contrasts, such as the difference between a subsample mean and a full population mean, the latent mixture term cancels exactly. This cancellation yields a tight, mixture-free Hoeffding-type bound that provides a direct de Finetti mechanism for the infinite-extendibility limit of recent finite-exchangeable concentration results. We apply this framework to quantify uncertainty in composite AI benchmarks, such as MMLU, where question items naturally exhibit exchangeable dependence across domains. Our results provide both a domain-stratified hierarchical model for bounding the uncertainty of accuracy scores, and a distribution-free, cost-saving statistical guarantee for accurately estimating full benchmark scores from random subsets.

A central question in the theory of machine learning concerns the identification of classes of data distributions for which one can provide computationally efficient learning algorithms with provable statistical learning guarantees. Indeed, in the context of probably approximately correct (PAC) learning, there has been much interest in exploring intermediate PAC learning models that, unlike the realizable PAC learning setting, allow for some stochasticity in the labels, and unlike the fully agnostic PAC learning setting, also admit computationally efficient learning algorithms with finite sample complexity bounds. Some examples of such models include random classification noise (RCN), probabilistic concepts, Massart noise, and generalized linear models (GLMs); in general, most of this work has focused on binary classification problems. In this paper, we study what we call realizable-statistic models (RSMs), wherein we allow stochastic labels but assume that some vector-valued statistic of the conditional label distribution comes from some known function class. RSMs are a flexible class of models that interpolate between the realizable and fully agnostic settings, and that also recover several previously studied models as special cases.

Modern data marketplaces and data sharing consortia increasingly rely on incentive mechanisms to encourage agents to contribute data. However, schemes that reward agents based on the quantity of submitted data are vulnerable to manipulation, as agents may submit fabricated or low-quality data to inflate their rewards. Prior work has proposed comparing each agent's data against others' to promote honesty: when others contribute genuine data, the best way to minimize discrepancy is to do the same. Yet prior implementations of this idea rely on very strong assumptions about the data distribution (e.g.

Data-adaptive two-sample testing assesses if two samples come from the same distribution, using a discrepancy learned from the data (e.g., via kernel-based feature representations). Such methods typically rely on data splitting to decouple learning from testing and control type I error. However, this paradigm is ill-suited to few-shot settings with severe sample-size imbalance: abundant reference samples are available, while only a handful of query samples arrive. In this paper, we show how this imbalance can be leveraged constructively. Using abundant reference data, we learn reference-dependent representations that summarize salient structure of the reference distribution and provide informative signals for detecting departures. We incorporate a collection of representation families that capture both global and local structure, and adaptively weight them using only reference samples via an uncertainty-guided principle. Theoretically, we establish permutation-based type I error control and show consistency of the aggregated test: as the sample sizes grow, the test power converges to one whenever the representation set contains at least one consistent representation. Empirically, our aggregation achieves strong performance across a range of benchmarks while retaining type I error control.

The Hilbert-Schmidt Independence Criterion (HSIC) and its joint-independence extension $d\mathrm{HSIC}$ are degenerate $V$-statistics whose data-dependent weighted-$χ^2$ null limits force a permutation calibration that multiplies the per-test cost by the number of permutations, in practice two orders of magnitude. Adapting the recent martingale MMD construction for two-sample testing to the (joint) independence problem, we introduce two studentised statistics whose null distributions are standard normal regardless of the data law, so that a single normal-quantile lookup replaces the permutation step entirely. The first, $m\mathrm{HSIC}$, is a self-normalised lower-triangular sum of the Hadamard product of two empirically centred Gram matrices. Under independence and bounded-fourth-moment kernels it converges to a standard normal. It is consistent against every fixed alternative, and runs at quadratic cost in the sample size without any sample split, matching the biased HSIC $V$-statistic. Our second statistic, $md\mathrm{HSIC}$, achieves finite-sample consistency with a single half-sample split: the centring is estimated on one half and the lower-triangular self-normalised martingale is run on the other, shrinking the conditional-mean residual to a quantity that is exponentially small in $d$, so the statistic is asymptotically standard normal at every fixed number of jointly tested variables, with a per-test cost that grows only linearly in $d$. On synthetic data with per-variable input dimension from $1$ to $500$ and between $2$ and $10$ jointly tested variables, both statistics match the empirical type-I error rate and test power of permutation-calibrated baselines while running $25$ to $60\times$ faster.

A local specialist LLM, fine-tuned with reinforcement learning from verifiable rewards (RLVR) on operator-local data, is installed in a regulated organization with per-deployment error budget $α$. The operator needs a safety certificate for this deployment's stream at every round: no pooling across deployments, no waiting for a long-run average. Existing wrappers cannot deliver this on adaptive, online-updated streams: offline conformal-risk methods require exchangeability; online-conformal methods bound only long-run averages; non-exchangeable extensions are marginally valid; and the closest anytime wrapper, A-RCPS, controls marginal rather than selective risk. Using a (test statistic, validity guarantee, deployment rule) framework, we identify one empty cell forced by deployment requirements: e-process per threshold, selective risk, anytime-pathwise validity, max-certified-threshold rule. Conformal Selective Acting (CSA) fills it as a per-round wrapper maintaining a Ville-type e-process per threshold on a Bonferroni grid, evaluated against the RLVR filtration. Under predictable updates and isotonic-calibrated monotone risk we prove (i) an anytime-pathwise selective-risk bound $R_T^{\mathrm{act}}\leα+O(N_T^{-1/2})$, (ii) rate-optimal certification matching $Θ(\barη^{-2}\log(1/δ))$, and (iii) a horizon-independent release-rate gap. Across eight specialist benchmarks ($480$ streams), sixteen adversarial distribution-shift cells ($160$ streams), and five live Expert-Iteration RLVR cells with online LoRA over four base models in three architecture families ($10{,}300$ rounds), CSA is the only method among ten compared that satisfies pathwise validity and non-refusing deployment on every cell. We do not propose a new LLM, training algorithm, or policy class; CSA is the deployment-side complement, orthogonal to the model, for operators who cannot use a frontier API.

The validity of statistical inference depends critically on how data are collected. When data gathered through active data collection (ADC) are reused for a post-hoc inferential task, conventional inference can fail because the sampling is adaptively biased toward regions favored by the collection strategy. This issue is especially pronounced in black-box optimization, where sequential model-based optimization (SMBO) methods such as the tree-structured Parzen estimator (TPE) and Gaussian process upper confidence bound (GP-UCB) preferentially concentrate evaluations in promising regions. We study statistical inference on actively collected data when the inferential target is constructed in a data-dependent manner after data collection. To enable valid inference in this setting, we propose post-ADC inference, a framework that accounts for the biases arising from both the active data collection process and the subsequent data-driven target construction. Our method builds on selective inference and provides valid $p$-values and confidence intervals that correct for both sources of bias. The framework applies to a broad class of ADC processes by imposing only assumptions on the observation noise, without requiring any assumptions on the underlying black-box function or the surrogate model used by the SMBO algorithm. Empirical results also show that post-ADC inference provides valid inference for data collected by GP-UCB and TPE.