We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For this subclass we prove a sample-level almost-sure convergence theorem and an explicit almost-sure rate of convergence, under coupled rate conditions on the penalty parameter, the random-feature count, the sample size, and the optimization tolerance. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

When treatment effects are naturally expressed as ratios -- as in medicine, pricing, and marketing -- the ratio-based CATE $τ(x) = E[Y|W=1,X=x] / E[Y|W=0,X=x]$ is the appropriate estimand. Yet existing estimators either impose a log-linear parametric structure or apply generic regression without robustness guarantees for this functional. We introduce the Q-Learner, which decomposes $τ(x)$ into a product of two odds ratios, reducing ratio-CATE estimation for binary outcomes to two propensity classification tasks. We further derive doubly robust augmentations for both S/T- and Q-style ratio learners and characterize their distinct robustness properties. In benchmarks on seven RCT datasets, the Q-Learner is the most consistently competitive method in low-conversion regimes, where its propensity-only construction sidesteps the imbalanced regression that hurts outcome-based estimators. On four observational datasets, where propensity must be estimated and confounding cannot be ruled out, the DR learners introduced here decisively come out on top, making them practitioners' natural default for confounded observational data.

Causal discovery in time series is increasingly performed using nonlinear machine-learning models, yet the resulting causal relationships are almost always summarized by scalar edge scores. We argue that this practice obscures the true object learned by nonlinear autoregressive models: a state-dependent function whose effect varies across regimes, magnitudes, and contexts. We formalize function-valued causal influence for additive, contribution-decomposable architectures and show that scalar causal scores constitute a severe information bottleneck, conflating between-state variation with within-state residual noise. Using Neural Additive Vector Autoregression as a representative architecture, we introduce a practical framework based on Individual Conditional Expectation for estimating causal response functions directly from trained models. Through controlled synthetic experiments, we demonstrate that edges with indistinguishable scalar scores can exhibit qualitatively different functional behaviors, including monotonic, thresholded, saturating, and sign-changing effects. An applied case study on democratic development further shows that function-valued analysis reveals regime-specific and asymmetric causal structure systematically missed by score-centric approaches.

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.

The classic concept of "calibrated forecasts" and its more recent refinement, "calibeating," are defined with respect to the standard quadratic scoring rule. We extend these notions to the class of $\textit{proper}$ scoring rules (for which the best forecast is the true distribution) and define $\textit{proper-calibration}$ and $\textit{proper-calibeating}$ by requiring the errors to converge to zero uniformly over all bounded proper scoring rules. We first establish that calibration always implies proper-calibration, whereas calibeating need not imply proper-calibeating. Second, we show how to guarantee proper-calibeating and proper-multicalibeating. Finally, we demonstrate the equivalence between proper-calibration and universal no regret when best replying to forecasts in decision-making under uncertainty.

Logged advertising auctions make offline reserve-price evaluation attractive but risky. Replay tables can identify policies with large apparent yield gains, yet they can also hide weak threshold support, multiple-comparison effects, subgroup harm, and bidder-response uncertainty. Existing replay and off-policy evaluation methods estimate or rank policy values, but they do not directly answer the operational question of whether the available evidence is strong enough to justify validation. This paper develops a support-aware offline decision framework for reserve-policy selection. Rather than outputting a single point-estimate winner, the framework converts logged evidence into a conservative decision object consisting of certified policies, statistically dominated alternatives, and unresolved candidates requiring further validation. The main theoretical result gives a unified finite-catalog guarantee showing that, under simultaneous uncertainty control and conservative support gates, the framework preserves the best gate-passing policy while eliminating only policies with certified regret. Supporting results characterize support-localized replay generalization, establish information-theoretic threshold-resolution limits, and quantify when heterogeneous bidder response can overturn localized replay rankings. Experiments on iPinYou real-time-bidding logs show that the leading reserve rule achieves a 47.66% replay lift in season two, a 40.71% simultaneous lower-bound lift, and a 43.87% frozen out-of-time replay lift in season three. The framework reduces a 19-policy catalog to a two-policy validation shortlist while certifying non-harm across 44 advertiser, exchange, and region segments. The results support the central claim that offline reserve-policy evaluation should produce certified validation decisions rather than point-estimate rankings alone.

A natural assumption is that if each model is individually calibrated, the aggregate prediction will also be well calibrated. We show that this assumption fails in multi-agent settings: individually calibrated predictors can become collectively miscalibrated when their predictions interact strategically--where "strategically" refers to the game-theoretic sense of Brier-optimal local response, not deliberate gaming or collusion, and arises naturally whenever agents are independently trained on overlapping data. This phenomenon affects multiple independent agents in federated healthcare, multi-vendor intrusion detection, and crowdsourced forecasting, where agents optimize their own objectives. Specifically, we prove that under Brier-score-based aggregation with positively correlated beliefs each agent's individually optimal report systematically underestimates the positive-class probability, yielding a Price of Anarchy strictly greater than one whenever Cov(bi,bj) > 0. At our canonical setting (n=5 agents, pairwise correlation ρ=0.5, base rate µ=0.3, threshold τ=0.3) the empirically measured PoA in false-negative rate is 7.25 (mean aggregate bias 0.375). In contrast, VCG-based aggregation, which rewards each agent's marginal contribution to aggregate accuracy, achieves dominant-strategy incentive compatibility and the lowest empirical PoA among all mechanisms studied (PoA 1.0). On three real-world datasets (NSL-KDD, UNSW-NB15, Credit Card Fraud) with featurepartitioned agents, VCG provides the strongest robustness guarantees among the aggregation methods we evaluate, while maintaining comparable accuracy. In data-sparse regimes (n 500), VCG consistently outperforms stacking and majority voting; under adversarial agents, VCG maintains substantially lower false-negative rates than robust aggregation baselines. Adaptive weight updates further reduce false negatives by 20-22% under distribution shift, with O( T) online regret guarantees. These results establish that how probabilistic predictions are aggregated matters as much as how well individual models are calibrated.

Calibrated probability outputs of trained classifiers are increasingly used as inputs to downstream regression estimands such as effects, prevalences, or disparities for a latent group observed only on a small labelled subset. A standard practice is to threshold the calibrated score at a confidence cutoff and treat the hard label as the truth. Building on a recent identification result for the underlying moment equation, we develop a calibration-aware diagnostic apparatus for pseudo-labelling pipelines. We derive a closed-form expression for the attenuation bias that confidence thresholding induces in the downstream regression coefficient, and show that the bias can be predicted, before any inference is run, from the residual score variance $V^{*}=\mathbb{E}[\operatorname{Var}(p\mid X)]$ on the unlabelled set after partialling out the downstream controls $X$. We further obtain a sharp sensitivity bound under bounded calibration drift, and identify the boundary $V^{*}=0$, which holds iff $p$ is a deterministic function of $X$; this motivates a structural separation between classifier features $W$ and downstream controls $X\subsetneq W$. Five controlled simulations and a UCI Adult illustration trace the predictions. The contribution is operational: a $(V^{*}, κ)$ decision rule that practitioners can compute from any classifier output to decide whether confidence thresholding is safe.

Probabilistic partial least squares (PPLS) is a central likelihood-based model for two-view learning when one needs both interpretable latent factors and calibrated uncertainty. Building on the identifiable parameterization of Bouhaddani et al.\ (2018), existing fitting pipelines still face two practical bottlenecks: noise--signal coupling under joint EM/ECM updates and nontrivial handling of orthogonality constraints. Following the fixed-noise scalar-likelihood line of Hu et al.\ (2025), we develop an end-to-end framework that combines noise pre-estimation, constrained likelihood optimization, and prediction calibration in one pipeline. Relative to Hu et al.\ (2025), we replace full-spectrum noise averaging with noise-subspace estimation and replace interior-point penalty handling with exact Stiefel-manifold optimization. The noise-subspace estimator attains a signal-strength-independent leading finite-sample rate and matches a minimax lower bound, while the full-spectrum estimator is shown to be inconsistent under the same model. We further extend the framework to sub-Gaussian settings via optional Gaussianization and provide closed-form standard errors through a block-structured Fisher analysis. Across synthetic high-noise settings and two multi-omics benchmarks (TCGA-BRCA and PBMC CITE-seq), the method achieves near-nominal coverage without post-hoc recalibration, reaches Ridge-level point accuracy on TCGA-BRCA at rank $r=3$, matches or exceeds PO2PLS on cross-view prediction while providing native calibrated uncertainty, and improves stability of parameter recovery.

The standard constraint-based paradigm for causal discovery with incomplete data -- impute first, test second -- is frequently miscalibrated: any consistent conditional independence (CI) test rejects a true null with probability approaching 1 when imputation error induces spurious conditional dependence. We introduce PAIR-CI, a nonparametric CI test that restores calibration by integrating multiple imputation directly into the inferential procedure via a paired permutation design. PAIR-CI compares cross-validated models that include and exclude the candidate variable while receiving the same imputed conditioning set, forcing imputation error to cancel in their loss difference rather than contaminate the test statistic. A provably consistent variance estimator jointly accounts for uncertainty arising from cross-validation and multiple imputation -- to our knowledge, the first formal unification of these two inferential frameworks. In simulations, existing imputation-based CI tests exhibit false positive rates of 28--45% when data are missing not at random (MNAR), whereas PAIR-CI averages below the nominal 5% level across data-generating processes and missingness mechanisms. These gains are largest in nonlinear settings and grow with causal graph size: when integrated into the PC algorithm, PAIR-CI reduces structural Hamming distance by 8% on 10-variable nonlinear graphs, 15% on 30-variable equivalents, and up to 44% on the 56-variable HAILFINDER network, with stable performance in all settings.