Data imbalance across sensitive groups and class labels is common in classification problems and can contribute to substantial fairness disparities. We study how such imbalance affects Equal Opportunity and how synthetic data augmentation can mitigate the resulting unfairness. We first identify two mechanisms, prevalence imbalance and group-size imbalance, through which changes in the training distribution shift the learned decision boundary and induce disparities in group true positive rates. Motivated by this analysis, we study exact full-balance augmentation, which equalizes all group-label cells. We establish conditions under which full balancing strictly improves Equal Opportunity together with a prediction-error comparison under the oracle balanced distribution, and show that the Equal Opportunity gap of the synthetic-data classifier converges to that of the oracle balanced classifier as the sample size grows and the generator error vanishes. The finite-sample convergence rate agrees, up to logarithmic factors and generator-approximation error, with the information-theoretic lower-bound rate under the corresponding regularity regimes. Since exact balancing need not provide the best fairness-accuracy trade-off in practice, we further propose Accuracy-Constrained Adaptive Synthetic Augmentation (ACASA), which learns both where and how much to augment by minimizing estimated fairness disparity subject to a constraint on predictive accuracy. We provide a finite-sample guarantee comparing the selected augmentation allocation with the population-optimal accuracy-feasible allocation. Simulation studies and real data application support the theoretical findings and demonstrate that synthetic augmentation can substantially reduce fairness disparities while achieving favorable empirical fairness-accuracy trade-offs.
In this work, we study last-iterate convergence bounds for stochastic gradient descent (SGD) on convex and $\beta$-smooth loss functions and improve the bound from $\cO((\log T)^3/T)$ to $\cO((\log T)^3/(T\log\log T))$ in the interpolation regime. The improvement comes from applying a horizon-dependent multiscale estimate of gradients to the comparison of trajectories. The bound applies to both sampling with replacement and sampling without replacement within one pass.
Real-world machine learning systems often operate on imperfect measurements: sensors are noisy, annotations are produced by non-experts, and scientific observations are often indirect. In these settings, classic distribution comparison tools such as maximum mean discrepancy (MMD) can be systematically biased, matching artifacts of the measurement process rather than the underlying mechanisms of interest. We study a practical regime in which noisy observations are abundant, but only a small audited subset contains ground-truth measurements. We propose measurement-error-corrected MMD (MEC-MMD), an audit-based estimator that debiases kernel evaluations without requiring a noise model. MEC-MMD is exactly unbiased for the oracle MMD under simple random auditing, admits an explicit variance decomposition, and is consistent for the population MMD, achieving the standard $O_p(N^{-1/2})$ rate when the audit size scales with the dataset. We demonstrate the utility of distributional matching with MEC-MMD loss for both parameter estimation and unsupervised applications, showing that MEC-MMD consistently outperforms MMD and alternative methods. These results provide a principled foundation for robust distribution comparison, simulation-based inference, and learning in noisy real-world machine learning pipelines.
Fairness interventions often targets disparities associated with a single sensitive attribute in binary classification tasks. However, when multiple sensitive attributes are associated with a data subject, improving fairness with respect to one attribute does not necessarily improve it across all attributes: it may exacerbate disparities associated with other sensitive attributes, including those that are unavailable to the practitioner. We refer to these unintended effects as cross-attribute externalities of fairness. We study how fairness with respect to one sensitive attribute varies when an initially unfair classification or regression model is corrected with respect to another attribute. We characterize these externalities using Wasserstein barycentric disparities and develop a distributional framework that captures changes in both marginal and joint fairness measures. This highlights a limitation of conventional fairness evaluation, in which the success of an intervention is assessed mainly for the attribute it targets. In a jointly Gaussian model, we derive explicit trajectories for marginal and joint disparities and identify conditions under which an intervention eliminates disparities with respect to its target attribute while increasing disparities associated with another attribute. We also derive sharp partial-identification bounds on the resulting externalities when a sensitive attribute is unavailable and extend our results to categorical sensitive attributes. We illustrate these externalities using classification and regression real-data: in the classification experiments, interventions increase unfairness with respect to the hidden attribute in the majority of scenarios across more than half of the datasets considered; in the regression experiments on the ACSIncome dataset, correcting for race reduces sex disparities, whereas subsequently correcting for age increases them.
As large language models (LLMs) increasingly make consequential decisions and solve problems alongside humans, we must understand how models arrive at their conclusions. Understanding how an LLM behaves requires observing it across multiple inputs, yet free-text explanations such as chain-of-thought (CoT) are generated per instance and cannot be easily verified on new inputs. We propose {code explanations to address the mismatch in how LLM explanations are generated versus how they are evaluated: a program explains LLM behavior across a dataset of inputs, and its execution enables automatic verification. We define Consistency metric that measures the fraction of inputs on which the program reproduces the LLM's outputs, and use it to evaluate and select candidate programs. Across three open-source code-capable LLMs and eight tasks, we find that LLMs can produce highly consistent code explanations in several settings. Code explanations are also 6$\times$ to 100$\times$ shorter than CoT, and better align with model behavior in 9 out of 24 settings. However, consistency is significantly lower on tasks with more complex linguistic components, where generated programs reproduce model outputs no more frequently than an uninformative explanation baseline, and these programs often rely on longer hard-coded patterns. In experiments with Claude Opus 5.5, providing the model with consistency feedback and code execution improves explanation consistency across tasks, but consistency remains significantly lower on more complex tasks. Our findings establish executable code as a novel, verifiable, and compact approach to explaining LLM behavior and highlight verification feedback as a promising direction for improving explanation consistency. The code for this paper is at https://github.com/rajesh-lab/code-as-explanations.
Thompson sampling is natural randomized strategy for optimizing unknown functions in a black-box manner. In continuous global optimization, Thompson sampling has largely been restricted to Gaussian process priors, as these are essentially the only class for which computing the posterior distribution has been tractable in practice. In this work, we introduce \emph{prior-fitted diffusion transformers}, which are diffusion models that perform Thompson sampling in-context at inference time, after being pre-trained using the prior-fitted network paradigm. To apply these models, we develop a set of practical non-Gaussian priors for general black-box functions---including those that have characteristics that cannot be achieved under Gaussianity, such as priors over unimodal functions. We also quantify the tradeoff between model size and the number of pre-training iterations, and show that these models follow scaling laws akin to those observed in transformers defined over other modalities. Across a set of global optimization benchmarks, we show that prior-fitted diffusion transformers with non-Gaussian priors can achieve improved performance compared to traditional pipelines built on Gaussian processes.
Understanding the persistence of learned behaviors in AI safety motivates an investigation of the dynamical counterpart to the equilibrium geometry studied in singular learning theory. We analyze this persistence through transitions between singular wells. We establish an Eyring-Kramers formula for a low temperature Langevin diffusion between two compact connected zero components of a real-analytic potential. The wells may be singular and nonisolated, and the unique minimizing gate is a nondegenerate index-one saddle, with gates understood as inclusion-minimal sets meeting every minimizing path. A boundary-gate construction shows that uniqueness of this inclusion-minimal gate is equivalent to every minimizing path passing through the saddle and one such path meeting the communication level only there. Compact semianalytic topology provides a separating function away from a fixed saddle neighborhood; analytic Morse coordinates give matching capacity bounds by first sending the temperature to zero and then shrinking that neighborhood. An anchored energy estimate and a small-ball potential-theoretic identity give uniformity for deterministic starting points. The leading constant is expressed in terms of the sublevel-volume coefficient, a Gamma factor, and the saddle Hessian.
Recursive estimators on drifting streams must forget stale data fast enough to track the target, yet slowly enough to average out noise. In prior-anchored ridge regression, this trade-off admits a closed-form spectral decomposition in which each eigen-direction interpolates between the data solution and the prior with pull weight $s_i = \rho / (\mu_i + \rho)$. When the anchor is refreshed every $k$ steps, we show that the moving-frame offset converges to a finite-cadence equilibrium that lags the pure-leaky path by an explicit $\Delta_k(s)$ while reducing estimation variance by a factor $\Psi(s, r_k)$. Minimizing the cycle-averaged risk gives $\omega^* = (4\nu^2 \Sigma / c')^{1/3}$, $c' = \sigma^2 d_{\mathrm{eff}}(\rho)$, and the plug-in strength $\rho^* = \sigma^2 / (n\delta^2)$. A random-matrix deterministic equivalent extends both laws to finite dimension, renormalizing the anchor to $\rho_{\mathrm{eff}} = \rho (e^{\zeta q} - 1) / (\zeta q)$; it matches simulation within 5% at fixed operating points and within 13% at the method's own, and rank-deficient populations reduce exactly to their active subspace. The resulting method, SPAR, calibrates window and strength online from residuals. On a 27-configuration grid the calibrated window opens on drift without divergence, and the grid defaults transfer to an hourly electricity stream and to BCI IV 2b. The core algebra, refresh chain, and regret bounds are formalized in Lean 4.
The integration of mechanistic signaling networks into clinical research has become a cornerstone of precision oncology. By capturing the complex biochemical interactions within a cell, these networks provide a functional map of disease progression that goes beyond simple genomic mutations. Consequently, the development of specific signaling models of patients has significant relevance, allowing for a more granular representation of individual disease profiles. In this setting, there is a growing need for robust methodologies capable of comparing graph based pathway structures to identify similarities and differences across patient cohorts. Such comparisons are essential for the identification of clinically meaningful subgroups, and the discovery of novel biomarkers. Standard graph kernels, while mathematically rigorous, often remain too generic for this purpose. They frequently fail to capture features specific to gene signaling networks, resulting in suboptimal clustering performance. In this paper, we present a domain specific graph kernel designed for mechanistic signaling networks. Unlike generic approaches, our method explicitly incorporates biological constraints and gene interaction patterns into the kernel formulation, enabling a more accurate characterization of both topological structure and functional behavior. We assess its performance on breast cancer datasets, focusing on its ability to support disease subtyping. Our results suggest that our approach can effectively contribute to patient stratification.
Causal mediation analysis decomposes a treatment effect into indirect pathways through mediators and direct pathways through other processes. Tabular foundation models perform classification and regression through in-context learning, providing estimates of conditional means and propensity scores used in causal mediation analysis. Constructing confidence intervals from these estimates requires accounting for uncertainty introduced by fitting and combining several nuisance functions. We introduce MediPFN, an amortized inference framework that adapts a pretrained tabular foundation model to nuisance prediction and combines its predictions through a cross-fitted influence-function estimator. A separate head learns quantiles of the estimator's standardized error from simulated tables with known indirect effects. Supervised fine-tuning, symmetrization, and independent calibration then produce a confidence interval. We establish coverage bounds for the resulting intervals under standard assumptions. Across experiments, MediPFN attains coverage closer to the nominal level than several comparators. It also reduces the computational cost of uncertainty quantification, running 59.1 times faster than bootstrap in matched comparisons.
Bayesian optimization is widely used in active learning loops for materials science. Gaussian processes serve as the de-facto surrogate model family, but require task-dependent kernel choices and can lack the modeling capacity for complex relationships. Tabular foundation models (TFMs) have recently emerged as the state-of-the-art for predictive tasks on structured data, without requiring bespoke kernels or hyperparameter tuning. We evaluate two TFMs, TabPFN and TabICL, as out-of-the-box surrogate models on two standard materials science benchmarks, comparing against a large field of baselines encompassing Gaussian processes, random forests, and CatBoost. Our findings indicate that TFMs are highly competitive surrogate models across the two benchmarks, without task-specific training for active learning or GP kernel selection. Our results motivate developing TFMs further as surrogates for active learning in materials discovery.
This paper examines robustness and causal interpretation of panel estimates using double/debiased machine learning. The prototype treatment setup is the 2018 US banking regulation. Under common treatment timing, we characterize identification and nuisance restrictions and derive exact bank-level representations of specified "within" and "first-difference" scores. Simulations assess panel model transformation-induced specification bias and inference under weaker overlap. Empirically, estimates vary with asset-size classification and prior growth. Prior-growth adjustment reduces overlap-weighted estimates across four Machine Learners (ML); doubly robust estimates depend on the fitted propensity and concentrated weights. Both bank groups received relief, so we estimate differential exposure effects. The findings highlight specification issues and the relation between functional-form robustness and causal identification.
Conditional generation of time-series paths is a central problem in probabilistic modeling, where the goal is to simulate future trajectories that are both realistic and responsive to external context. It has direct applications in finance and economics. Neural stochastic differential equations provide a natural framework for this task by combining continuous-time stochastic dynamics with learnable drift and diffusion functions. However, directly modeling the target state can be unstable in conditional settings, especially when the target and conditioning variables have strong level dependence, limited training data, or different dimensions. We propose a residual-anchor Neural SDE architecture that decomposes the target into a learned conditional anchor and a stochastic residual. The anchor captures the context-dependent baseline, while the Neural SDE (Stochastic Differential Equation) models the residual dynamics around this baseline. This separation reduces the burden on the stochastic dynamics and generalizes spread-based representations to more general conditioning settings. We evaluate the approach on real-world financial time series, including conditional interest-rate scenarios and a comparison with a time-series foundation model. The results show that the proposed model can generate plausible conditional paths, maintain competitive point accuracy, and provide calibrated uncertainty estimates.
Learning independence is fundamental to statistical inference and machine learning, yet learning independence between entire time series by a test remains challenging when only a single observation record is available. Existing guarantees often rely on structural or mixing assumptions, while stationarity and ergodicity alone do not permit consistent testing. In this work, we identify a class-level boundary governed by shared nontrivial Koopman eigenvalues, which represent common persistent oscillatory frequencies of the two processes. When no such eigenvalues are shared, we extend shift-based rank calibration with finite-window dependence witnesses and a data-adaptive threshold to construct a deterministic, weakly consistent test with finite-sample false-positive control, without quantitative mixing assumptions. Conversely, we show that no possibly randomized test can achieve vanishing false-positive probabilities under every independent jointly totally ergodic law while consistently detecting every dependent jointly totally ergodic law with a shared nontrivial eigenvalue. This impossibility already holds when the shared point spectrum is exactly the cyclic group generated by any single prescribed irrational eigenfrequency, identifying a spectral obstruction to universal learning of full-process independence.
Model selection and hyperparameter tuning pose a central challenge in off-policy evaluation (OPE) because estimation errors in fitted $Q$-functions and occupancy ratios cannot be directly evaluated using held-out transition data. Existing Bellman-residual-based validation methods often require auxiliary regressions, while many minimax-based methods require optimization over separate critic classes. We propose pairwise Bellman validation (PBV), a family of penalized Bellman tournament methods that reduces validation to computing held-out sample averages. PBV constructs critics from pairwise candidate differences, on the log scale for occupancy ratios, and selects the candidate with the smallest worst-case penalized score. This construction approximates oracle critics given by $Q$-function errors and log ratios of candidate to true occupancy ratios, whose Bellman and adjoint Bellman moments control estimation error. Critic penalties account for approximation error under candidate-class misspecification. We establish finite-sample oracle inequalities relative to the best available candidate, even under candidate-class misspecification. For $Q$-function selection, unweighted PBV avoids importance weighting but requires a coercivity condition on a quadratic form induced by the Bellman operator. Importance-weighted PBV (IW-PBV) uses marginalized importance weighting to exploit Bellman contraction under the target-policy occupancy measure, avoiding this condition. For occupancy-ratio selection, adjoint PBV (A-PBV) exploits KL contraction of the adjoint Bellman operator to obtain oracle guarantees under the offline data law. On Hopper and DeepOPE, unweighted PBV is competitive with LSTD-Tournament, often achieving higher accuracy on Hopper, while importance weighting can further improve performance under controlled overlap.
Natural direct and indirect effects are defined through cross-world interventional counterfactuals. We propose an alternative construction based on backtracking counterfactuals. Rather than intervening on the treatment or mediator and change the structural equations, we use a backtracking conditional distribution to construct an alternative exogenous state satisfying a desired treatment-mediator constraint and then evaluate the unchanged structural equations on that state. This yields the backtracking direct and indirect effects, and also the backtracking total effect. Likewise, we adapt the population intervention effects.
We investigate the minimal width of deep neural networks that can approximate all continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ in the topology of uniform convergence on compact subsets. We show that whenever the real projective space $\mathbb{RP}^{n-1}$ admits a smooth immersion into $\mathbb{R}^n$, the minimal width is at least $n+\min\{n,m\}$, assuming that the activation function is continuous and monotone. As a consequence, the minimal width of ReLU networks is $n+m$ whenever $n$ is equal to 2, 3, 4, 7 or 8 and $1\le m \le n$, extending results of Hanin-Sellke and Hwang.
We develop a bias-aware digital-twin calibration and control framework for multiscale bioprocess models within a biological systems-of-systems (Bio-SoS) paradigm. The digital twin is represented by a stochastic differential equation (SDE) model and calibrated from sparse observations using quasi-likelihood estimation and adjoint sensitivity analysis. SDE generator-based moment expansions characterize truncation-induced parameter bias, while forward-backward adjoints quantify how calibration uncertainty propagates to value functions and policy performance. The resulting parameter-error distribution supports both policy-directed data acquisition and uncertainty-aware policy optimization through a second-order Gaussian-averaged objective. We characterize the asymptotic behavior of the resulting exploration criterion and derive a physical-system performance under the optimized policy. To implement these ideas, we develop an Actor-Simulator algorithm that jointly updates model parameters, selects informative experiments, and optimizes control policies. Numerical studies demonstrate improved calibration accuracy, sample efficiency, and control performance relative to state-of-the-art baselines.
Large language models (LLMs) are increasingly used as classifiers, yet they operate as opaque systems whose decisions are difficult to interpret, which complicates their use in regulated domains such as credit scoring or medical diagnosis. We propose an evolutionary framework that iteratively discovers natural language feature definitions (rubrics) for interpretable classification. An LLM generates candidate binary features, evaluates each sample against them, and the resulting vectors can be used to train a transparent classifier such as logistic regression. The feature set evolves over multiple iterations guided by classification errors, per-class activation rates, and feature ablation scores. We evaluate across three benchmarks, including a credit risk dataset representative of regulated domains, comparing single-shot LLM rubrics, evolved rubrics, and direct zero-shot LLM classification. Evolved features improve over single-shot rubrics by +2.9 pp on average and outperform zero-shot LLM classification on two of three tasks, while providing fully auditable decision logic. On the credit risk task, the zero-shot LLM performs at chance (50.7%) with a strong bias toward a single class, whereas evolved features achieve balanced, interpretable predictions. Crucially, this failure is invisible in aggregate accuracy and surfaces only under per-class auditing. Analysis reveals that evolution is most effective when label boundaries cannot be inferred from category names alone or when the LLM lacks reliable domain-specific reasoning.
Pharmaceutical process characterization is central to Quality by Design because it defines how variations in process parameters affect the ability to meet product quality specifications, thereby supporting proven acceptable ranges and robust manufacturing. In practice, however, characterization still relies largely on factorial design of experiments (DOE) approaches, which are inefficient for resolving multivariate pass/fail boundaries in higher-dimensional spaces. While Bayesian optimization has transformed process optimization, adaptive methods for multi-objective process characterization remain lacking. Here, we introduce JAREX (Joint Acceptable Region EXploration), a Bayesian active-learning acquisition function for multi-objective process characterization. JAREX formulates characterization as a joint boundary-learning problem and adaptively selects experiments to recover the joint pass region defined by simultaneous satisfaction of threshold criteria across multiple objectives. JAREX combines an optimistic joint-feasibility mask with a multi-objective extension of randomized straddle, focusing sampling on the joint edge of failure. Our benchmark study suggests that JAREX provides more accurate and sample-efficient recovery of the joint pass region than factorial DOE, space-filling designs, and greedy objective-wise strategies over the full experimental budget range. For batched experimentation, it reduces the number of iterative process characterization experiments by more than half while preserving high accuracy for the boundary-identification task. Implemented in the open-source obsidian package, JAREX provides a modular framework for adaptive, data-efficient multi-objective algorithmic process characterization, supporting sample-efficient range finding in high-dimensional spaces.