PAC-Bayesian PROWL for Robust ITR Learning

• The paper introduces PROWL—a framework that robustly optimizes individualized treatment rules by leveraging a one-sided uncertainty certificate and finite-sample PAC-Bayesian guarantees.
• It reformulates policy learning as a cost-sensitive classification problem using a split-free reduction and a certified hinge surrogate to navigate noisy proxy rewards.
• Empirical results demonstrate that PROWL effectively minimizes target and robust regret, outperforming standard methods in scenarios with significant reward uncertainty.

PAC-Bayesian Reward-Certified Outcome Weighted Learning (PROWL) is a framework for learning individualized treatment rules (ITRs) when the observed outcome is only a noisy or optimistic proxy for the true clinical utility of interest. It combines a one-sided uncertainty certificate that lower-bounds true utility from the observed reward, an exact reduction to a split-free cost-sensitive classification problem, and a PAC-Bayesian/generalized Bayes analysis that yields finite-sample lower bounds on the true value of randomized ITRs together with a fully automated, bounds-based learning-rate calibration procedure (Ishikawa et al., 2 Apr 2026).

1. Problem setting and motivation

PROWL is formulated in the standard single-stage ITR setting. Each subject has covariates $X \in \mathcal X \subset \mathbb R^p$, receives a binary treatment $A\in\{-1,1\}$, and has potential rewards $R^1, R^{-1}$ for a proxy reward and $(R^\ast)^1,(R^\ast)^{-1}$ for the true target reward. An ITR is a measurable function

$d:\mathcal X\to\{-1,1\},$

with counterfactual rewards

$$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$

and values

$$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$

Under consistency, ignorability, and overlap, one observes i.i.d. data

$Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$

with treatment propensity $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$, and the policy values admit the inverse-propensity representations

$$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$

Outcome weighted learning (OWL) exploits the representation for $A\in\{-1,1\}$0 by recasting policy learning as weighted classification with labels $A\in\{-1,1\}$1 and weights $A\in\{-1,1\}$2; PROWL starts from the same OWL perspective but targets $A\in\{-1,1\}$3 rather than the proxy value $A\in\{-1,1\}$4 (Ishikawa et al., 2 Apr 2026).

The central motivation is reward uncertainty. In the PROWL formulation, the observed reward $A\in\{-1,1\}$5 may be a surrogate endpoint, may be noisy, or may be an optimistic aggregation of multiple outcomes, whereas the object of interest is an unobserved target reward $A\in\{-1,1\}$6. If one directly optimizes $A\in\{-1,1\}$7 using $A\in\{-1,1\}$8, policies can be selected on the basis of inflated apparent performance. The framework is designed to replace nominal optimization of proxy reward with optimization of a rigorous lower bound on the true value.

2. Conservative rewards and the exact certified reduction

The defining structural assumption is a one-sided uncertainty certificate: there exists a measurable nonnegative function

$A\in\{-1,1\}$9

such that

$R^1, R^{-1}$0

This induces the certified reward

$R^1, R^{-1}$1

By construction,

$R^1, R^{-1}$2

and therefore the certified value

$R^1, R^{-1}$3

satisfies

$R^1, R^{-1}$4

for every measurable policy $R^1, R^{-1}$5. This lower bound is strictly policy-dependent. If additionally $R^1, R^{-1}$6 almost surely, then $R^1, R^{-1}$7 and

$R^1, R^{-1}$8

The second term depends on $R^1, R^{-1}$9 unless $(R^\ast)^1,(R^\ast)^{-1}$0 is treatment-free, so the certificate can change the ranking of policies (Ishikawa et al., 2 Apr 2026).

PROWL then introduces a nuisance-augmented representation. For $(R^\ast)^1,(R^\ast)^{-1}$1, where $(R^\ast)^1,(R^\ast)^{-1}$2 parameterizes the policy score and $(R^\ast)^1,(R^\ast)^{-1}$3 parameterizes nuisance functions, define

$(R^\ast)^1,(R^\ast)^{-1}$4

$(R^\ast)^1,(R^\ast)^{-1}$5

and the certified advantage objects

$(R^\ast)^1,(R^\ast)^{-1}$6

For fixed $(R^\ast)^1,(R^\ast)^{-1}$7, the weighted $(R^\ast)^1,(R^\ast)^{-1}$8–$(R^\ast)^1,(R^\ast)^{-1}$9 risk is

$d:\mathcal X\to\{-1,1\},$0

Theorem 1 in the PROWL paper states an exact certified reduction: for every measurable ITR $d:\mathcal X\to\{-1,1\},$1 and every $d:\mathcal X\to\{-1,1\},$2,

$d:\mathcal X\to\{-1,1\},$3

so maximizing the certified value is exactly equivalent to minimizing a weighted $d:\mathcal X\to\{-1,1\},$4–$d:\mathcal X\to\{-1,1\},$5 classification risk. The Bayes-optimal certified rule chooses the treatment with the larger certified conditional mean $d:\mathcal X\to\{-1,1\},$6. A crucial feature is that the identity $d:\mathcal X\to\{-1,1\},$7 holds for every $d:\mathcal X\to\{-1,1\},$8 without sample splitting. The paper describes this as a split-free formulation: nuisance estimation and policy learning can be carried out jointly on the full sample without losing finite-sample control (Ishikawa et al., 2 Apr 2026).

3. PAC-Bayesian lower bounds and generalized Bayes structure

The PAC-Bayesian layer is imposed on the joint parameter space $d:\mathcal X\to\{-1,1\},$9 with prior $$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$0 independent of the policy-learning sample. For $$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$1, the empirical certified value is

$$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$2

A posterior $$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$3 defines a randomized ITR by drawing $$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$4 and deploying $$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$5. The target and empirical quantities under $$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$6 are

$$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$7

To place the problem in bounded-loss PAC-Bayes form, the paper defines

$$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$8

and shows that $$R^d \coloneq R^{d(X)},\qquad (R^\ast)^d \coloneq (R^\ast)^{d(X)},$$9. Hence the value loss

$$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$0

lies in $$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$1. Its expectation and empirical average satisfy

$$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$2

The main PAC-Bayes guarantee is a nonasymptotic lower bound on the unobserved target value. For any $$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$3 and any $$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$4, with probability at least $$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$5,

$$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$6

simultaneously for all posteriors $$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$7. This is the reward certificate: a finite-sample lower bound on the true value of the randomized ITR in terms of empirical certified value and a KL complexity penalty (Ishikawa et al., 2 Apr 2026).

For fixed $$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$8, maximizing the right-hand side over $$V(d)\coloneq E[R^d],\qquad V^\ast(d)\coloneq E[(R^\ast)^d].$$9 yields the unique generalized Bayes posterior

$Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$0

equivalently,

$Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$1

Thus the generalized Bayes update is not merely compatible with the PAC-Bayes bound; it is exactly the posterior that maximizes the certified lower bound. This matches the standard PAC-Bayesian variational structure in which a Gibbs posterior optimizes an empirical criterion regularized by $Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$2 (Guedj, 2019).

4. Automated temperature calibration and the certified hinge surrogate

A central practical issue in generalized Bayesian inference is the learning rate or inverse temperature. PROWL addresses this with a fully automated, bounds-based calibration procedure. Define

$Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$3

$Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$4

and for any $Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$5,

$Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$6

With probability at least $Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$7, the paper proves that

$Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$8

for all $Z_i=(X_i,A_i,R_i),\quad i=1,\dots,n,$9 and all $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$0, and that for fixed $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$1 the map $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$2 is uniquely maximized by the generalized Bayes posterior $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$3. Over a finite grid $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$4, one selects

$\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$5

so temperature selection is itself certified by the same lower-confidence-bound construction (Ishikawa et al., 2 Apr 2026).

Because the exact value loss embeds a non-convex $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$6–$\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$7 structure, PROWL also introduces a certified hinge surrogate. For $\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$8,

$\pi(a\mid x)=P(A=a\mid X=x)\ge\varepsilon>0$9

The associated practical Gibbs posterior is

$$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$0

If the prior has density $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$1, the MAP estimator solves a regularized weighted hinge-risk minimization problem that is computationally similar to SVM or cost-sensitive margin-based methods.

The certified hinge loss is Fisher-consistent for maximizing the certified value. Proposition 5 states that the hinge-risk minimizer induces the same Bayes-optimal rule as the exact certified objective, and that for any measurable score $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$2,

$$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$3

Thus excess hinge risk upper-bounds certified value regret, providing a surrogate-calibration bridge between tractable optimization and the target certified-value problem (Ishikawa et al., 2 Apr 2026).

5. Implementation structure and empirical results

At the algorithmic level, PROWL takes as inputs the observed data $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$4, known or estimated propensities, a reward certificate $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$5, policy and nuisance parameterizations, a prior $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$6, a confidence level $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$7, and grids for $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$8 and $$V(d)=E\Biggl[\frac{R\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr],\qquad V^\ast(d)=E\Biggl[\frac{R^\ast\,1\{A=d(X)\}}{\pi(A\mid X)}\Biggr].$$9. It then constructs conservative rewards

$A\in\{-1,1\}$00

forms augmented scores $A\in\{-1,1\}$01, pseudo-labels $A\in\{-1,1\}$02, and weights $A\in\{-1,1\}$03, builds either the value-based or hinge-based posterior family, and selects $A\in\{-1,1\}$04 by maximizing the certified lower confidence bound. Final deployment may use a MAP rule, a mean rule, or Gibbs deployment; the formal guarantees are stated for the randomized posterior policy (Ishikawa et al., 2 Apr 2026).

The empirical study includes two simulated single-stage scenarios. Scenario 1 is benign, with a linear decision boundary, policy-invariant reward uncertainty, balanced randomization, and modest uncertainty. Scenario 2 is intended to be challenging and realistic, with correlated covariates, nonlinear outcome surfaces, a beneficial treated subgroup confined to a moderate-risk region, treatment-specific reward optimism concentrated in vulnerable clinical regions, and nonuniform logging propensities. The comparisons include standard OWL, RWL, Q-learning, Policy Tree, PROWL with certificate $A\in\{-1,1\}$05, and PROWL $A\in\{-1,1\}$06. The reported metrics are target regret

$A\in\{-1,1\}$07

and robust regret

$A\in\{-1,1\}$08

In Scenario 1, all strong methods, including Q-learning, RWL, and PROWL, achieve similar low regrets, and PROWL does not pay a noticeable robustness cost. In Scenario 2, as the uncertainty scale $A\in\{-1,1\}$09 increases, standard methods’ regrets increase, especially PROWL $A\in\{-1,1\}$10, whereas PROWL maintains low regret across all $A\in\{-1,1\}$11, with the most pronounced improvement at higher uncertainty. Additional diagnostics show that certificates are valid, clipping rates are essentially zero, PROWL reduces the proxy–target gap more than competing methods in hard settings, and the split-free model substantially outperforms a sample-split PROWL, especially in Scenario 2 (Ishikawa et al., 2 Apr 2026).

The real-data application concerns the AKI alert trial (ELAIA-1), where the objective is to learn a selective alerting policy robust to uncertainty in how multiple hard outcomes are aggregated into a utility score. The proxy reward is constructed from a nominal weight vector $A\in\{-1,1\}$12, the certified reward is the minimum over a preference-uncertainty set $A\in\{-1,1\}$13, and evaluation is performed over repeated hospital- and treatment-stratified $A\in\{-1,1\}$14 splits using an AIPW estimator. The strongest rule overall is Never alert, which achieves the highest certified and composite-free values. Among learned policies, PROWL achieves the highest certified value and the highest composite-free value, the lowest mortality among adaptive methods, and an alert rate around $A\in\{-1,1\}$15. It slightly improves over PROWL $A\in\{-1,1\}$16 on all metrics and reduces alert rate, while OWL variants are reported to be much more aggressive without better clinical performance. Alert-allocation analysis further shows that PROWL is more conservative in non-teaching hospitals, especially in the highest-risk quintile (Ishikawa et al., 2 Apr 2026).

6. Assumptions, limitations, and surrounding PAC-Bayesian context

The formal PROWL guarantees rely on standard single-stage causal assumptions, the one-sided certificate $A\in\{-1,1\}$17 almost surely, bounded rewards and certificates in $A\in\{-1,1\}$18, and a prior independent of the policy-learning sample. If the certificate $A\in\{-1,1\}$19 is itself learned, the paper requires it to be learned on an independent calibration sample. Within those conditions, the PAC-Bayes guarantees are finite-sample and nonasymptotic (Ishikawa et al., 2 Apr 2026).

The limitations stated in the paper are equally structural. Valid but loose certificates can make $A\in\{-1,1\}$20 overly pessimistic and flatten the learning objective. The method depends strongly on certificate quality; purely policy-invariant penalties do not change policy ranking. PAC-Bayesian hinge-based optimization with temperature tuning is computationally more complex than a single OWL, RWL, or Q-learning fit. The guarantees are for randomized Gibbs policies, and the gap between those guarantees and deterministic deployments such as MAP or mean rules is not yet fully quantified. Proposed extensions include better derandomization bounds, data-dependent certificates learned from the same sample, multiple or continuous treatments, alternative uncertainty structures arising from measurement error or partial identification, and higher-order corrections to reduce conservatism (Ishikawa et al., 2 Apr 2026).

Within the broader PAC-Bayesian literature, PROWL occupies a specific bounded-loss, i.i.d., decision-theoretic niche. The general prior/posterior/KL architecture comes directly from the PAC-Bayes framework for generalized Bayes learning (Guedj, 2019). For dependent or sequential data, martingale PAC-Bayesian inequalities extend the same style of reward certification beyond the i.i.d. regime (Seldin et al., 2011). For unbounded losses or heavy-tailed importance weights, Efron-Stein PAC-Bayesian inequalities provide variance-adaptive, semi-empirical concentration and truncation-free off-policy guarantees (Kuzborskij et al., 2019). Related work on self-certified neural networks emphasizes the use of all available data to learn and certify a predictor through PAC-Bayes bounds (Perez-Ortiz et al., 2021). In offline contextual bandits, importance-weighted learning with implicit exploration yields PAC-Bayesian regret guarantees without uniform coverage assumptions (Gabbianelli et al., 2023). Differentially private weighted ERM, including an application to OWL, supplies a privacy-preserving weighted-learning backend but does not by itself provide PAC-Bayesian reward certificates (Giddens et al., 2023). These adjacent developments do not define PROWL, but they clarify the methodological environment in which reward-certified outcome-weighted learning is situated.