We study online individualized treatment selection when only the outcome under the assigned treatment is observed. Our objective is to preferentially treat individuals whose potential outcomes under treatment exceed a covariate-specific control reference, while limiting false discoveries among treated individuals. We propose Partial-Feedback Adaptive Individualized Treatment Selection (PAITS), which initializes learning from historical data and combines sequential estimation of local null risks, randomized exploration, and feedback-driven updates of the treatment selection threshold. Under regularity conditions, we establish asymptotic false discovery rate control and sublinear gap-weighted assignment regret relative to a population oracle. We further establish a minimax lower bound on assignment regret under prescribed uniform nuisance-estimation requirements and give conditions under which PAITS attains the corresponding constrained minimax rate up to logarithmic factors. These results quantify the tradeoff between accurate treatment assignment and nuisance learning under partial feedback. Simulation studies and a real-data analysis using the Jester recommendation dataset illustrate the finite-sample performance of PAITS.
Figures & tables
Figure 1: Empirical FDR over online time for Models M1–M3. The top row compares four control-reference estimators at χ=0.5 , and includes the known-nuisance setting. The bottom row varies χ , using weighted B-spline estimation for both nuisance functions. The horizontal line denotes the nominal FDR level α=0.1 .
Figure 2: Log–log plots of the mean cumulative gap-weighted assignment regret for Models M1–M3. The top row compares four control-reference estimators at χ=0.5 . The bottom row varies χ , using weighted B-spline estimation for both nuisance functions.
Figure 3: Sensitivity to the historical sample size for Models M1–M3. The historical sample sizes are nhist∈{1000,2000,3000,4000} . The top row reports the empirical FDR over online time, and the bottom row reports the log–log plots of the mean cumulative gap-weighted assignment regret.
Figure 4: Sensitivity to the step-size exponent ζ for Models M1–M3. The adaptive cutoff uses a polynomial step-size schedule ηt≍t−ζ . The top row reports the empirical FDR over online time, and the bottom row reports the log–log plots of the mean cumulative gap-weighted assignment regret.
Figure 5: Performance of PAITS on the Jester data over five randomized replay orders. Left: cumulative false discovery proportion over online time, where the thick solid line represents the pointwise average over the five orders and the dashed line denotes the target level α=0.10 . Right: log–log trajectories of cumulative gap-weighted assignment regret relative to the finite-population oracle, together with the fitted log–log slopes.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Supplementary log–log fits for the mean cumulative gap-weighted assignment regret in Models M1–M3, corresponding to Figure 2 . The top row compares four control-reference estimators at χ=0.5 , with weighted B-spline estimation of F1 throughout. The bottom row varies χ , using weighted B-spline estimation for both nuisance functions. The lines show the fitted linear trends over the displayed time interval, and the legends report their slopes.
Figure 7: Supplementary log–log fits for the historical-sample-size comparison in Figure 3 . The panels correspond to Models M1–M3, with nhist∈{1000,2000,3000,4000} . Both nuisance functions are estimated by weighted B-spline regression, with χ=0.5 in M1 and M2 and χ=0.4 in M3. The lines show fitted linear trends in the log mean cumulative gap-weighted assignment regret against logt over the displayed time interval. The legends report the fitted slopes.
Figure 8: Supplementary log–log fits for the step-size comparison in Figure 4 . The panels correspond to Models M1–M3, with ζ∈{0.55,0.60,0.65,0.70,0.75} . Both nuisance functions are estimated by weighted B-spline regression, with χ=0.5 in M1 and M2 and χ=0.4 in M3. The lines show fitted linear trends in the log mean cumulative gap-weighted assignment regret against logt over the displayed time interval. The legends report the fitted slopes.
When can we learn from biased samples? We study regression when outcomes are observed only after passing through selection filters that depend on both covariates and outcomes themselves, a ubiquitous challenge spanning clinical trials with patient dropout, labor markets with self-selection, and auctions with strategic entry. Ignoring such selection yields systematically biased conclusions with real-world consequences. This challenge has a long history in econometrics and statistics, starting with Heckman's seminal two-stage model and followed by numerous generalizations. While these works provide various sufficient conditions for identification, a complete characterization of when such regression is possible has remained elusive. In this work, we provide a characterization for when regression is possible in the presence of sample selection bias. Our results establish the minimal assumptions required on the functional forms of selection processes under which regression remains possible, which are particularly relevant in modern settings where selection mechanisms are increasingly complex and opaque. As a corollary of our characterization, we show that there are settings where the regression function can be identified even when the selection filter itself cannot. This observation already goes beyond the ``estimate selection filter, then debias regression'' paradigm that is followed by virtually all existing approaches. Under natural strengthenings of our identification conditions, we also establish finite-sample estimation guarantees with explicit convergence rates and provide oracle-efficient algorithms. This yields the first general-purpose estimation method for this broad class of selection problems. Finally, we explore the implications of our results for several well-studied econometric settings with complex selection mechanisms such as auctions with entry costs and labor markets.
Randomized controlled trials typically assume that prognostic covariates are known and available at no cost. In practice, obtaining high-dimensional pretreatment data is costly, forcing a trade-off between covariate-adaptive precision and a measurement budget. We introduce Dynamic Adaptive Rerandomization via Thompson Sampling (DARTS), which treats covariate acquisition as a sequential optimization problem embedded within a design-based causal inference task. A budgeted combinatorial Thompson sampler learns which covariates are most prognostic across successive batches; selected covariates then drive rerandomization and regression adjustment to reduce batch-level average treatment effect variance. Our primary theoretical contribution is a decoupling result: adaptive covariate selection based on past batches preserves batch-level randomization validity, and the cumulative inverse-variance weighted estimator achieves at least nominal asymptotic coverage. We further derive a Bayes risk bound for the acquisition layer that matches the minimax lower bound up to logarithmic factors. Empirically, DARTS systematically concentrates the budget on informative features, significantly closing the efficiency gap to oracle designs while maintaining strict inferential validity.
Kateryna Husar, Alexander Volfovsky
Department of Statistical Science Duke University Durham, NC 27705
We address the problem of conformal selection, where an agent must select a minimal subset of options to ensure that at least one success'' is identified with a pre-specified target probability $φ$. While traditional online conformal prediction focuses on maintaining validity for the observed sequence, minimizing the resource cost (efficiency) of such selections, especially under limited feedback, remains a significant challenge. In this work, we consider settings with the most limited bandit'' feedback, and demonstrate that the simple Adaptive Conformal Inference (ACI) update rule, when applied to the appropriate control parameter or dual variable, is both adversarially valid, ensuring the success target is met on average for any input sequence (and hence under distribution shifts), and stochastically efficient, achieving sublinear efficiency regret for i.i.d. inputs against an appropriate stochastic benchmark. We show such guarantees under canonical models capturing bandit and semi-bandit feedback to the agent via a unifying algorithmic technique, and analytic framework involving Lyapunov functions. Our approach handles more complex settings than prior work, while requiring significantly less feedback, and our results provide a new theoretical bridge between efficient online learning with limited feedback and distribution-free uncertainty quantification.