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What Do We Care About in Bandits with Noncompliance? BRACE: Bandits with Recommendations, Abstention, and Certified Effects

Nicolás Della Penna

TL;DR

BRACE is proposed, a parameter-free phase-doubling algorithm that performs IV inversion only after matrix certification and otherwise returns full-range but honest structural intervals and an orthogonal score whose conditional bias factorizes into compliance-model and outcome-model errors, clarifying what must be stabilized for anytime-valid semiparametric IV inference.

Abstract

Bandits with noncompliance separate the learner's recommendation from the treatment actually delivered, so the learning target itself must be chosen. A platform may care about recommendation welfare in the current mediated workflow, treatment learning for a future direct-control regime, or anytime-valid uncertainty for one of those targets. These objectives need not agree. We formalize this objective-choice problem, identify the direct-control regime in which recommendation and treatment objectives collapse, and show by example that recommendation welfare can strictly exceed every learner-measurable treatment policy when downstream actors use private information. For finite-context square-IV problems we propose BRACE, a parameter-free phase-doubling algorithm that performs IV inversion only after matrix certification and otherwise returns full-range but honest structural intervals. BRACE delivers simultaneous policy-value validity, fixed-gap identification of the operationally optimal recommendation policy, and fixed-gap identification of the structurally optimal treatment policy under contextual homogeneity and invertibility. We complement the theory with a finite-context empirical benchmark spanning direct control, mediated present-versus-future tradeoffs, weak identification, homogeneity failure, and rectangular overidentification. The experiments show that safety appears as regret on easy problems, as abstention and wide valid intervals under weak identification, as a reason to prefer recommendation welfare under homogeneity failure, and as tighter structural uncertainty when extra instruments are available. For rich contexts, we also derive an orthogonal score whose conditional bias factorizes into compliance-model and outcome-model errors, clarifying what must be stabilized for anytime-valid semiparametric IV inference.

What Do We Care About in Bandits with Noncompliance? BRACE: Bandits with Recommendations, Abstention, and Certified Effects

TL;DR

BRACE is proposed, a parameter-free phase-doubling algorithm that performs IV inversion only after matrix certification and otherwise returns full-range but honest structural intervals and an orthogonal score whose conditional bias factorizes into compliance-model and outcome-model errors, clarifying what must be stabilized for anytime-valid semiparametric IV inference.

Abstract

Bandits with noncompliance separate the learner's recommendation from the treatment actually delivered, so the learning target itself must be chosen. A platform may care about recommendation welfare in the current mediated workflow, treatment learning for a future direct-control regime, or anytime-valid uncertainty for one of those targets. These objectives need not agree. We formalize this objective-choice problem, identify the direct-control regime in which recommendation and treatment objectives collapse, and show by example that recommendation welfare can strictly exceed every learner-measurable treatment policy when downstream actors use private information. For finite-context square-IV problems we propose BRACE, a parameter-free phase-doubling algorithm that performs IV inversion only after matrix certification and otherwise returns full-range but honest structural intervals. BRACE delivers simultaneous policy-value validity, fixed-gap identification of the operationally optimal recommendation policy, and fixed-gap identification of the structurally optimal treatment policy under contextual homogeneity and invertibility. We complement the theory with a finite-context empirical benchmark spanning direct control, mediated present-versus-future tradeoffs, weak identification, homogeneity failure, and rectangular overidentification. The experiments show that safety appears as regret on easy problems, as abstention and wide valid intervals under weak identification, as a reason to prefer recommendation welfare under homogeneity failure, and as tighter structural uncertainty when extra instruments are available. For rich contexts, we also derive an orthogonal score whose conditional bias factorizes into compliance-model and outcome-model errors, clarifying what must be stabilized for anytime-valid semiparametric IV inference.
Paper Structure (35 sections, 7 theorems, 51 equations, 8 figures, 1 table)

This paper contains 35 sections, 7 theorems, 51 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Pre-bandit roots of the assignment--receipt split.
  4. Bandits with observable noncompliance.
  5. IV bandits and causal regret.
  6. Strategic or induced compliance.
  7. Adaptive experimentation and anytime-valid inference.
  8. IV-based policy learning and confounded decision making.
  9. Recent adaptive-IV work under noncompliance.
  10. Terminology and objective choice.
  11. The Three Objectives and the Operational Advantage
  12. Finite-Context Model and Identification
  13. The BRACE Algorithm
  14. Matrix certification.
  15. Local intervals.
  16. ...and 20 more sections

Key Result

Proposition 3.1

Suppose $\mathcal{Z}=\mathcal{X}$ and $C(z)=z$ almost surely for every $z\in\mathcal{X}$. Then for every policy $\pi:\mathcal{W}\to\mathcal{X}$,

Figures (8)

  • Figure 1: Core empirical comparisons. Left: in the direct-control, private-signal, workflow-redesign, and homogeneity-failure cases, RECERT matches a strong recommendation baseline while Actual-UCB breaks once recommendation and treatment incentives diverge. Right: in representative TRT settings, the relevant comparison is not just error rate but the split between correct deployment, abstention, and wrong deployment. BRACE abstains in the hard cases, unsafe baselines can fail in different ways, and strong-IV settings still allow correct structural deployment.
  • Figure 2: Selective structural-uncertainty comparison. Weak square-IV settings produce wide honest intervals, while rectangular overidentification and the rescued-IV design materially tighten them. The appendix shows the full inference panel; this figure isolates the design comparison that matters most for the paper's main claim about certification and extra instruments.
  • Figure 3: Recommendation-track summary plots. Left: mean estimated REC value. Right: mean operational regret. Together they show that many methods agree on the final REC answer in easy environments, while differing sharply in how much operational welfare they spend during learning.
  • Figure 4: Treatment-track summary plots. Left: mean estimated TRT value. Right: abstention rate. The combination makes the main BRACE tradeoff visible: structural caution can preserve validity but often appears as delayed or missing deployment.
  • Figure 5: Structural risk and uncertainty validity. Left: wrong-non-abstain rate for TRT algorithms. Right: coverage of the best-policy TRT value in the inference track. The left panel highlights where unsafe methods make incorrect structural decisions; the right panel shows that the interval procedures remain conservative in the benchmark.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Proposition 3.1: REC and TRT coincide under direct control
  • proof
  • Proposition 3.2: Strict operational advantage under private discretion
  • proof
  • Remark 3.3: Why this does not contradict structural learning
  • Remark 3.4: Choosing between REC and TRT
  • Lemma 4.3: Contextual IV identification
  • proof
  • Remark 5.1: Structural output versus operational deployment
  • Lemma 6.1: Simultaneous concentration
  • ...and 9 more