Metelkina and Pronzato (2017) formulated sequential treatment allocation with covariates as a compound design problem that trades off Fisher information about all treatment-response parameters against cumulative ethical regret, proved an equivalence theorem characterizing the optimal allocation, and gave a doubly-adaptive sequential rule whose compound criterion value converges almost surely. The asymptotic theory of the adaptive implementations is nonetheless incomplete, and the source states the open questions explicitly. When the compromise weight is allowed to grow with the sample size, that is, a Lagrange coefficient Cn→∞C_n\to\infty (equivalently αn↑1\alpha_n\uparrow1), it is open which increase rates of CnC_n preserve strong consistency of the running maximum-likelihood estimator θ^n\hat\theta^n, and whether any rate can attain the Lai-Robbins benchmark rate E{Rn}=O(log⁡n/n)\mathrm{E}\{R_n\}=O(\log n/n) of the covariate-free problem, or the best achievable expected-regret decay in the presence of covariates; at fixed compromise weight, the limiting covariance of the allocation proportions under the doubly-adaptive rule and the selection of a limiting design when the optimal design is non-unique remain uncharacterized.

Background

In a sequential clinical trial with KK competing treatments, subjects arrive one at a time with covariates XX (age, biomarkers, and so on) drawn independently from a measure μ\mu, and the probability of success of treatment kk for a subject with covariates xx is a known function ηk(x,θk)\eta_k(x,\theta_k) of unknown parameters, with logistic regression as the canonical example. The experimenter assigns each arriving subject to a treatment and faces two conflicting objectives: statistical information, meaning precise estimation of all parameters θk\theta_k including those of treatments that look weak early in the trial, and individual ethics, meaning assignment of each subject to the treatment that is best for that subject's covariates. Covariate-adjusted response-adaptive (CARA) designs assign the (n+1)(n+1)-st subject to treatment kk with a probability πk(Xn+1,θ^n)\pi_k(X_{n+1},\hat\theta^n) that depends on the subject's covariates and on the current maximum-likelihood (ML) estimate θ^n\hat\theta^n built from the previous nn responses.

Metelkina and Pronzato (2017) gave this compromise a precise optimal-design formulation. Writing ξk\xi_k for the sub-measure of μ\mu corresponding to subjects allocated to treatment kk, the normalized Fisher information matrix of the experiment is M(ξ)=∑k=1K∫Mk(x) dξk(x)M(\xi)=\sum_{k=1}^K\int M_k(x)\,d\xi_k(x), where Mk(x)M_k(x) is the elementary information matrix of one response under treatment kk; information is measured by ψ(ξ)=Φ[M(ξ)]\psi(\xi)=\Phi[M(\xi)] with Φ\Phi a classical design criterion, for example D-optimality (Φ(M)=log⁡det⁡M\Phi(M)=\log\det M, which maximizes the geometric mean of the estimation precisions). The ethical cost is the regret Rn=1n∑i=1n(η∗(Xi)−ηki(Xi,θki)),η∗(x)=max⁡kηk(x,θk),R_n=\frac{1}{n}\sum_{i=1}^n\big(\eta^*(X_i)-\eta_{k_i}(X_i,\theta_{k_i})\big),\qquad \eta^*(x)=\max_{k}\eta_k(x,\theta_k), the average response shortfall relative to an oracle that always assigns the best treatment. Optimal compromises maximize the compound criterion H(α)(ξ)=(1−α)ψ(ξ)+αφ(ξ)H^{(\alpha)}(\xi)=(1-\alpha)\psi(\xi)+\alpha\varphi(\xi) with φ(ξ)=−R(ξ)\varphi(\xi)=-R(\xi) the negative regret and α∈[0,1]\alpha\in[0,1]; by Lagrangian duality this is equivalent to maximizing information under a regret constraint R(ξ)≤τR(\xi)\le\tau, with Lagrange coefficient C=α/(1−α)C=\alpha/(1-\alpha). The paper proves an equivalence theorem (necessary and sufficient optimality conditions) for these compound designs, studies their uniqueness and continuity in α\alpha, and constructs a doubly-adaptive sequential allocation rule, rule (4.6): a biased-coin-type randomized rule (in the lineage of Efron's biased coin and the doubly-adaptive biased coin designs of Hu and Zhang (2004)) whose allocation probabilities depend on the current empirical design ξ^n\hat\xi^n, for which the criterion value H(α)(ξ^n)H^{(\alpha)}(\hat\xi^n) converges almost surely to the optimum (Theorem 4.1). For fixed α<1\alpha<1 the authors state that replacing the nominal parameter value by the estimate θ^n\hat\theta^n (a CARA implementation) preserves strong consistency and asymptotic normality of θ^n\hat\theta^n under suitable regularity assumptions, with detailed developments deferred to a follow-up paper that never appeared.

Two asymptotic gaps remain. First, for classical bandit problems without covariates the best achievable expected regret is E{Rn}=O(log⁡n/n)\mathrm{E}\{R_n\}=O(\log n/n) on the per-subject scale used here (cumulative regret O(log⁡n)O(\log n); Lai and Robbins (1985)), and one would like the trial to approach this rate by letting the compromise weight strengthen over time, αn↑1\alpha_n\uparrow1 equivalently Cn→∞C_n\to\infty, for instance through the Lagrangian recursion Cn+1=max⁡{0, Cn+γ[R(ξ^n;θ^n)−τ]}C_{n+1}=\max\{0,\,C_n+\gamma[R(\hat\xi^n;\hat\theta^n)-\tau]\} with γ>0\gamma>0, so that the trial explores early and exploits late. The danger is identifiability: if the allocation concentrates on the currently estimated best treatment too quickly, some parameters stop being estimated consistently, θ^n\hat\theta^n may become inconsistent, and the justification of the whole construction collapses. Which growth rates of CnC_n are safe, and whether any of them reaches the optimal regret decay, is open. Second, even at fixed α\alpha the existing functional central limit theorem (functional CLT) machinery for doubly-adaptive biased coin designs, which covers adaptation of the treatment proportions alone, does not extend to rule (4.6), where the whole information matrix M(ξ^n)M(\hat\xi^n) is adapted; no closed-form or qualitative characterization of the limiting covariance of the allocation proportions under (4.6), nor of which optimal design is approached when the maximizer of H(α)H^{(\alpha)} is not unique, is available (numerical evidence in the source suggests convergence to a unique limit).

Problem Statement

Within the framework of Metelkina and Pronzato (2017), that is, response models E{Yt∣X=x,t=k}=ηk(x,θk)\mathrm{E}\{Y_t\mid X=x,t=k\}=\eta_k(x,\theta_k) with covariates i.i.d. from a measure μ\mu under their regularity assumptions, information criterion ψ(ξ)=Φ[M(ξ)]\psi(\xi)=\Phi[M(\xi)] with Φ\Phi a classical design criterion applied to the normalized Fisher information matrix, and normalized regret Rn=1n∑i=1n(η∗(Xi)−ηki(Xi,θki)),η∗(x)=max⁡kηk(x,θk),R_n=\frac{1}{n}\sum_{i=1}^n\big(\eta^*(X_i)-\eta_{k_i}(X_i,\theta_{k_i})\big),\qquad \eta^*(x)=\max_{k}\eta_k(x,\theta_k), where kik_i is the treatment assigned to subject ii and η∗(x)\eta^*(x) is the response of the best treatment for covariates xx, consider covariate-adjusted response-adaptive implementations of the compound information-regret design — allocations whose target is the optimal decomposition of the compound criterion H(α)(ξ)=(1−α)ψ(ξ)+αφ(ξ)H^{(\alpha)}(\xi)=(1-\alpha)\psi(\xi)+\alpha\varphi(\xi) with φ(ξ)=−R(ξ)\varphi(\xi)=-R(\xi) the negative regret, computed at the current ML estimate θ^n\hat\theta^n — in which the compromise weight varies with the sample size: αn↑1\alpha_n\uparrow1, equivalently Cn=αn/(1−αn)→∞C_n=\alpha_n/(1-\alpha_n)\to\infty. Determine: (i) which increase rates of CnC_n preserve strong consistency of the ML estimator θ^n\hat\theta^n; (ii) whether any such rate allows the allocation to attain the Lai-Robbins benchmark rate E{Rn}=O(log⁡n/n)\mathrm{E}\{R_n\}=O(\log n/n) of the covariate-free problem, or the best achievable expected-regret decay rate in the presence of covariates (Lai-Robbins-type asymptotic efficiency), and which rates do; (iii) the companion fixed-α\alpha theory still missing from the source: the limiting covariance of the empirical allocation proportions under the doubly-adaptive sequential rule (4.6) of that paper, a biased-coin-type randomized rule whose allocation probabilities depend on the current empirical design ξ^n\hat\xi^n, in comparison with its oracle counterparts, and which optimal design is approached in the limit when the set of maximizers of H(α)H^{(\alpha)} is not unique. Items (i), (ii) and (iii) are independent of one another, and each is resolvable on its own. Each item is posed in the framework's stated generality, not for a single numerical example.

References

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