finding a compromise between information and regret in...
TRANSCRIPT
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Finding a compromise between information and regretin clinical trials
Asya Metelkina and Luc Pronzato
AM: Azoth Systems LP: Laboratoire I3S (CNRS and Univ. de Nice Sophia Antipolis)
MASCOT-NUM Annual Conference 19 March 2019
1 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Luc Pronzato I know Luc since 2014...
Sophia Antipolis, I3S, Algorithmes: Luc works in I3S since 1992...
Luc, Antibes, sea: leasure or work?
2 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Plan
Optimal allocations in clinical studies...an optimal design problem...
� 1. Compromise between information and ethicsClinical studies.Objectives of adaptive allocation.Information, ethics and compromise criterion.
� 2. Allocation rules converging to the optimumEquivalence theorem: characterization of optimal allocation.Oracle allocation rule.Randomized oracle rule.
� 3. CARA and empirical process allocationCARA type allocation.Allocation based on empirical measures.Compromise procedures proposed in literature.
� 4. Conclusion and perspectives
3 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Optimal designs
Design space X
Parametric or non parametric model(s): f (x ,θ), x ∈ X
Criterion G to maximize or minimize
Choose measure ξ =
{w1 . . . wkx1 . . . xk
}to optimize G(ξ)
Generate a sequence {x1, . . . ,xn} to (quasi-)optimize G(x1, . . . ,xn)
4 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Clinical trials
n subjects. i-th subject:
Xi , Ti , Yicovariates, allocation, outcome
Assumptions:
� E(Yi |Xi = x ,Ti = k) = ηk (x ,θk ), Yi ∈ Y ⊂ R, k = 1, . . . ,K .
� Xi are i.i.d. ∼ µ, (Xi ∈ X ⊂ Rd , µ(x)≥ 0,∫
X µ(dx) = 1)
� covariate-adaptive allocations
P(Ti = k |Xi = x) = πk (x)
Example 1 : Logistic regression.Y = {0,1},X = [−1,1], K = 2P(Yi = 1|Xi = x ,Ti = k) = ηk (x ,θk )
ηk (x ,θk ) =1
1 + e−θk x
θ1 = 5,θ2 =−5.
5 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Clinical trials
n subjects. i-th subject:
Xi , Ti , Yicovariates, allocation, outcome
Assumptions:� E(Yi |Xi = x ,Ti = k) = ηk (x ,θk ),� Xi are i.i.d. ∼ µ,� covariate-adaptive allocations
P(Ti = k |Xi = x) = πk (x)
10 subjects6 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Clinical trials
n subjects. i-th subject:
Xi , Ti , Yicovariates, allocation, outcome
Assumptions:� E(Yi |Xi = x ,Ti = k) = ηk (x ,θk ),� Xi are i.i.d. ∼ µ,� covariate-adaptive allocations
P(Ti = k |Xi = x) = πk (x)
10 subjects allocated7 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Clinical trials
n subjects. i-th subject:
Xi , Ti , Yicovariates, allocation, outcome
Assumptions:
� E(Yi |Xi = x ,Ti = k) = ηk (x ,θk ),
� Xi are i.i.d. ∼ µ,
� covariate-adaptive allocations
P(Ti = k |Xi = x) = πk (x)
Example 2: Linear regression with i.i.d errors.Y = R,X = [0,1], K = 2Yi = ηk (Xi ,θk ) + εi if Ti = k , εi |= Xiηk (x ,θk ) = ak + bk xa1 = 0, b1 = 10, a2 = 5, b2 =−4.
8 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Clinical trials
n subjects. i-th subject:
Xi , Ti , Yicovariates, allocation, outcome
Assumptions:� E(Yi |Xi = x ,Ti = k) = ηk (x ,θk ),� Xi are i.i.d. ∼ µ,� covariate-adaptive allocations
P(Ti = k |Xi = x) = πk (x)
10 subjects allocated9 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Objectives of clinical trial
Objectives:
� Information: to better estimate θ = (θ1, . . . ,θK )
� Ethics: to favor allocation to best treatment η∗(Xi ) = maxk=1...,K
ηk (Xi ,θk )
Yi |Ti = k are i.i.d with E(Yi |Ti = k) =∫X
ηk (x ,θk )πk (x)µ(dx)︸ ︷︷ ︸ξk (dx)
ηnewk (x ,θk ) = ηk (x ,θk )πk (x) or ξ = (ξ1, . . . ,ξK ) ∈ Ξ(µ)
modified model allocation measure
Ξ(µ) = {ξ = (ξ1, . . . ,ξK ) | ξk is µ−a.c., ∑Kk=1 ξk = µ, ξk ≥ 0}
is a convex set.
10 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Information
Objectives:
� Information: to better estimate θ = (θ1, . . . ,θK )
Maximize Ψ(M(ξ,θ)):
Ψ strictly concave, Lowener increasing, differentiable
M(ξ,θ) = ∑Kk=1
∫X Mk (x ,θk )ξk (dx) Fisher information.
Example: Ψ(M) = logdet(M), Ψ(M) =−tr(M−q), q ∈ (0,+∞).
Motivation:Asymptotic normality of ML estimator θ̂n,
√n(θ̂n−θ)→N (0,M(ξ,θ)−1)
Unbiased estimators: Cramer-Rao bound for variance.
11 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Information
Objectives:
� Information: to better estimate θ = (θ1, . . . ,θK ) ⇒ maxξ∈Ξ(µ)
Ψ(M(ξ,θ)).
Ψ concave, Ξ(µ) convex set⇒ ξ∗(x ,θ) = arg max
ξ∈Ξ(µ)Ψ(M(ξ,θ))
Example 1: logistic regressionY ∈ {0,1}, X = [−1,1], µ = U([−1,1]),θk ∈ R, no covariates in common.
⇒ M(ξ,θ) has a block structure:
M(ξ,θ) =
1∫0
x2
η(x ,θ1)(1−η(x ,θ1))ξ1(dx) 0
01∫0
x2
η(x ,θ2)(1−η(x ,θ2))ξ2(dx)
Ψ(M(ξ,θ)) = log(det(M(ξ,θ))) =
2∑
k=1log
(1∫0
x2πk (x)ηk (x ,θk )(1−ηk (x ,θk ))
dx
)
12 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Information
Objectives:
� Information: to better estimate θ = (θ1, . . . ,θK ) ⇒ maxξ∈Ξ(µ)
Ψ(M(ξ,θ)).
Ψ concave, Ξ(µ) convex set⇒ ξ∗(x ,θ) = arg max
ξ∈Ξ(µ)Ψ(M(ξ,θ))
Example 2: linear regression with i.i.d. N(0,1) errors.µ = U([0,1]). ηk (x ,θk ) = ak + bk x , no common parameters.⇒ M(ξ,θ) has a block structure:
M(ξ) =
1∫0
ξ1(dx)1∫0
xξ1(x)dx 0 0
1∫0
xξ1(dx)1∫0
x2ξ1(dx) 0 0
0 01∫0
ξ2(dx)1∫0
xξ2(dx)
0 01∫0
xξ2(dx)1∫0
x2ξ2(dx)
13 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Information
Objectives:
� Information: to better estimate θ = (θ1, . . . ,θK ) ⇒ maxξ∈Ξ(µ)
Ψ(M(ξ,θ)).
Ψ concave, Ξ(µ) convex set⇒ ξ∗(x ,θ) = arg max
ξ∈Ξ(µ)Ψ(M(ξ,θ))
Example 2: linear regression with i.i.d. N(0,1) errors.µ = U([0,1]). ηk (x ,θk ) = ak + bk x , no common parameters.⇒ M(ξ,θ) has a block structure:
M(ξ) =
m1 m2 0 0m2 m3 0 00 0 1−m1
12 −m2
0 0 12 −m2
13 −m3
m1 =
1∫0
π1(x)dx ,
m2 =1∫0
xπ1(x)dx ,
m3 =1∫0
x2π1(x)dx .
Ψ(M(ξ)) = logdet(M(ξ)) = log(m1 ·m3−m2
2
)+ log
((1−m1)( 1
3 −m3)− ( 12 −m2)2
)
14 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Information
Objectives:
� Information: to better estimate θ = (θ1, . . . ,θK ) ⇒ maxξ∈Ξ(µ)
Ψ(M(ξ,θ)).
Ψ concave, Ξ(µ) convex set⇒ ξ∗(x ,θ) = arg max
ξ∈Ξ(µ)Ψ(M(ξ,θ))
Example 3: linear regression with i.i.d. N(0,1) errors.µ = U([−1,1]). ηk (x ,θk ) = a + bk x , with one common parameter.
M(ξ) =
1 m2 −m2m2 m3 0−m2 0 1
3 −m3
m2 = 12
1∫−1
xπ1(x)dx ,
m3 = 12
1∫0
x2π1(x)dx .
15 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Regret and reward
Objectives:
� Ethics: to favor allocation to best treatment η∗(Xi ) = maxk=1...,K
ηk (Xi ,θk )
Minimize the regret R(ξ,θ) =K∑
k=1
∫X
(η∗(x ,θ)−ηk (x ,θ))ξk (dx)
Maximize the reward Φ(ξ,θ) =K∑
k=1
∫X
ηk (x ,θ)ξk (dx)
Best treatment allocation: dξkdµ (x ,θ) = πk (x ,θ) =
1, if ηk (x ,θk ) = η∗(x ,θ)1m , if m ties
0, otherwise
16 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Information, ethics and compromise
Problem: information OR ethics? May be contradictory!
Example: Yi ∈ {0,1}, ηi (x ,θ) = θi , no covariates
Information Neyman allocation: P(Ti = 1) =
√θ1(1−θ1)√
θ1(1−θ1)+√
θ2(1−θ2):
maximizes the power of Wald’s test(
with statistics θ̂n−θ
V(θ̂n)
).
favors inferior treatment if θ1 + θ2 > 1.
Ethics Best treatment allocation: P(Ti = 1) = δθ1≥θ2 no information about the worthtreatment parameters.
Solution: Compromise criterion = convex combination of information and reward:
Maximize (1−α) ·Ψ(M(ξ,θ)) + α ·Φ(ξ,θ), α ∈ (0,1)
Equivalent to
{maxΨ(M(ξ,θ))
Φ(ξ,θ)≥ τ(α,θ).
17 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Optimal allocation measure
K = 2. Optimal allocation measure: ξ∗α
(θ) = arg maxξ∈Ξ(µ)
Hα(ξ,θ)
Asymptotic optimality criterion:
Hα(ξ,θ) = (1−α)Ψ
(2
∑k=1
Mk (ξk ,θ)
)︸ ︷︷ ︸
information
+α
2
∑k=1
φk (ξk ,θ)︸ ︷︷ ︸reward
.
Mk (θ,ξk ) =∫
X Mk (x ,θ)ξk (dx) and φk (θ,ξk ) =∫
X ηk (x ,θ)ξk (dx).
Convex set of allocation measures
Ξ(µ) = {ξ = (ξ1,ξ2) ∈M 2X | ξ1 + ξ2 = µ,ξ1 ≥ 0,ξ2 ≥ 0, ξ1,ξ2 a.c. wrt µ}.
Maximize concave function Hα(ξ,θ) on a convex set Ξ(µ)⇒ Equivalence Theorem
18 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Equivalence Theorem
Under differentiability conditions:ξ∗α
(θ) can be characterized through directional derivatives of Hα.
Equivalence theorem
ξ∗α
= arg maxξ∈Ξ(µ)
Hα(ξ,θ) is characterized by the function ∆12(ξ,x ,θ):
1 ∆12(ξ,x ,θ)≥ 0 ξ∗1-a.s., ∆12(ξ,x ,θ)≤ 0 ξ∗2-a.s.
2 There exist X1,X2 ⊂ X such thatξ∗1 = µ on X1, ξ∗2 = µ on X2,infx∈X1 ∆12(ξ,x ,θ)≥ 0 , supx∈X2
∆12(ξ,x ,θ)≤ 0∆12(ξ,x ,θ) = 0, if x ∈ X \(X1
⋂X2).
� Here ∆12(ξ,x ,θ) = G1(ξ,x ,θ)−G2(ξ,x ,θ).
� Directional derivatives (in direction of point mass at x)
G1(ξ,x ,θ) = limγ→0+
γ−1[Hα(ξ + γ(δx ,0),θ)−Hα(ξ,θ)]
G2(ξ,x ,θ) = limγ→0+
γ−1[Hα(ξ + γ(0,δx ),θ)−Hα(ξ,θ)]
19 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Logistic regression example
X = [0,1] and µ = U([0,1]). Ψ(M) = logdet(M).Logistic regression ηk (x ,θ) = 0.25 + 0.5 · 1
1+exp(−(θk ,1−x)θk ,2).
θ = (θ1,θ2), θk ∈ R2, k = 1,2.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η1(x)
η2(x)
x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
x
G(α
)1
(ξ∗,x
)−G
(α)
2(ξ
∗,x
)A B C
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α
1
2
1
2
Figure : Left: Statistical model of response η1(x ,θ1) et η2(x ,θ2).Middle: Function ∆12(ξ,x ,θ) from Equivalence Theorem for α = 0.7.Right: Ensembles X1 (treatment 1) et X2 (treatment 2) as function of α.
20 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Oracle allocation rule
� Rule 1. Oracle allocation rule(necessitates the knowledge of µ, θ and construction of ξ
∗α
(θ) )
P(Tn+1 = 1|Xn+1 = x) = π∗α(x ,θ)
with π∗α(x ,ξ,θ) =dξ∗1,α
dµ (x ,θ).
Remark: π∗α(x ,θ) = 1 on a subset x ∈ X0 ⊂ X is possible.
Example:P(Ti = 2[Xi = x) = 1 if x ∈ [0,A]∩ (B,C),P(Ti = 1[Xi = x) = 1 if x ∈ (A,B)∩ (C,1].
21 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Empirical information and reward
Empirical compromise criterion:
(1−α) ·Ψ(Mn(θ)) + α · (Φn(θ)), α ∈ (0,1)
Empirical reward :
Φn(θ) =1n
2
∑k=1
n
∑i=1,Ti =k
ηk (Xi ,θ)
Empirical information
Ψ(Mn(θ)) with Mn(θ) =1n
2
∑k=1
n
∑i=1,Ti =k
Mk (Xi ,θ)
� δTi =1Xi are i.i.d. ∼ ξ∗1(θ).
� Empirical allocations ξ̂n
= 1n
n∑
i=1(δTi =1,δTi =2) are asymptotically optimal:
Hα(ξ̂n,θ)→ Hα(ξ
∗α
(θ),θ) a.s.
22 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Randomized oracle randomization
π∗α(x ,θ) = 1 for x ∈ X0 with µ(X0) > 0⇒ prediction bias.
Optimal allocation measure under a constraint of randomization level β ∈ (0,1) :
ξ∗α,β
(θ) = βξµ + ξ̃α,β
(θ) with
ξµ
= ( µ2 ,
µ2 ) and ξ̃
α,β(θ) = arg max
ξ∈Ξ(µ(1−β))Hα
(βξ
µ+ ξ,θ
).
Randomized Oracle Rule 1.
P(Tn+1 = 1|Xn+1 = x) =β
2+ π∗α,β(x ,θ)
with πα,β(x ,θ) =d ξ̃1,α,β
dµ (x ,θ).
23 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Example of logistic regression
X = [0,1]. θ = (θ1,θ2), θk ∈ R2, k = 1,2.Logistic regression ηk (x ,θ) = ak + 0.5 · 1
1+exp(−θk ,1−θk ,2x).
a1 = 0.1, a2 = 0.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η1(x)
η2(x)
x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α
1
2
1
2
1
2
Figure : Left : Outcome models.Right: Randomization level β = 0.2, deterministic part of allocation function ξ̃
α,β(θ) as fonction
of α.
24 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Randomized oracle rule
α,β ∈ (0,1). Asymptotic properties of:
Empirical allocations ξ̂n = 1n ∑
ni=1 Ti Xi converge to ξ
∗α,β
Compromise criterion is asymptotically optimal: H(ξ̂n,θ)−−−→n→∞
H(ξ∗α,β
,θ).
Allocation proportions ρn
= 1n
n∑
i=1Ti are consistent and asymptotically normal:
ρn−−−→n→∞
ρ∗k (θ) where ρ
∗k (θ) := ξ
∗k (X ,θ).
√n(
ρn−ρ∗k (θ)
)d−−−→
n→∞N (0,Σ∗) Σ∗ = diag(ρ
∗k (θ))−ρ
∗k (θ)ρ
∗k (θ)t
Bounds on regret(reward) and information.
25 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Adaptive sequential allocations:
Subjects arrive sequentially⇒ can use past covariates, allocations, outcomesinformation.
Allocation functions: P(Tn = k |Xn = x ,Fn)where Fn = σ(X1, . . . ,Xn−1,T1, . . . ,Tn−1,Y1, . . . ,Yn−1)
The sequence (T1, . . . ,Tn) is a stochastic process.
Motivation:
K = 2, Y = {0,1}, ηk (x) = θk , θk ∈ [0,1], Nn,1 := ∑ni=1(Ti = 1).
variance of (Nn,1− n2 ) is of order n for P(Tn = 1) = 1
2 ,bounded Biased Coin Design BCD(p), p ∈ ( 1
2 ,1]
P(Tn+1 = 1|Fn) =
p, 2Nn,1 < n12 , 2Nn,1 = n(1−p), 2Nn,1 > n
[Efron1971],[Markaryan,Rosenberger 2010]26 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Covariate Adjusted Response Adaptive designs
µ known, θ unknown⇒, use θ̂n (well defined n > n0, β > 0) [Zhang et al.’07]
� Rule 2 Covariate-Adjusted Response Adaptive targeting ξ∗α
(θ):
P(Tn+1 = 1|Xn+1 = x ,F n) = π(x , θ̂n)
here π is computed from ξ∗α,β
(θ̂n).
Asymptotic properties:
Consistency and asymptotic normality of θ̂n:
Asymptotic optimality of ξ̂n
= 1n
n∑
n=1(δTi =1δXi (.),δTi =2δXi (.))
Asymptotic properties of proportions.
27 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Allocation based on empirical process.
� θ known, µ unknown⇒ replace ξ∗α
(θ) by ξ̂n
.
� Rule 3. Allocation based on empirical process
P(Tn+1 = 1|T n,Xn,Xn+1 = x) = πn(x ,θ)
with π(x , ξ̃,θ) = d ξ̃1dµ (x ,θ) et ξ̂
n= 1
n
n∑1
(δTi =1δXi ,δTi =2δXi ).
� Asymptotic optimality: Hα(ξ̂n,θ)→ Hα(ξ
∗α,β
(θ),θ) a.s.
� Simulations: ⇒ allocation proportions ρn
very less that for rules 1 and 2.
� Randomization β > 0, θ and µ are unknown.
Rule 4. Empirical procedure of CARA type use both ξ̂n
and θ̂n
P(Tn+1 = 1|F n,Xn+1 = x) = πn(x , θ̂n).
28 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Logistic regression example
β = 0, X = [0,1]. θ = (θ1,θ2), θk ∈ R2, k = 1,2.Logistic regression ηk (x ,θ) = 1
1+exp(−θk ,1−θk ,2x), θk = (2,10) .
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.5
1
1.5
n1/2(Nn,1/n− ρ∗1)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.5
1
1.5
n1/2(Nn,2/n− ρ∗2)
Figure : Variability of treatments proportions:Up: treatment 1. Down: treatment 2.dotted = rule 3, red = rule 1, blue = limit law for rule 1.
29 / 32
OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Comparison with procedures proposed in literature
Ad-hoc rules proposed in literature (logistic regression)
� Allocation based on covariates-adjusted odds ratio [Rosenberger et al.’01]
P(Tn+1 = 1 | Fn,Xn+1 = x) = η1(x ,θ̂n)(1−η2(x ,θ̂n))
(1−η1(x ,θ̂n))η2(x ,θ̂n)odds ratio.
� Modification of allocation rule from [Hu&Zhu&Hu’15]
P(Tn+1 = 1 | Fn,Xn+1 = x)da
1 ·eb1
da1 ·eb
1 + da2 ·eb
2
witha > 0,b > 0.Information dk = dk (ξ
n,x ,θ) = tr(M−1(ξ
n,θ)Mk (x ,θ))
Inverse of reward ek = ek (x ,θ) = 1/(1−ηk (x ,θ)) .
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OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Example: Logistic regression.
0 0.01 0.02 0.03 0.04 0.05 0.06−20.5
−20
−19.5
−19
−18.5
−18
−17.5
α = 0.95
α = 0
α = 1
R(ξ∗)
ψ(ξ
∗)
Figure : Red curve Ψ(M(ξ∗α
)) vs. R(ξ∗α
) parametrized by α.? = allocation based on odd rations [Rosenberger et al.’01].O = modified procedure [Hu&Zhu&Hu’15] (from left to right a = 1, b = 10,6,4,3.)
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OptimalAllocation
AsyaMetelkinaand LucPronzato
Compromisebetweeninformationand ethics
Allocationrulesconverging tothe optimum
CARA andempiricalprocessallocation
Conclusion and perspectives
Fo more details see: [A. Metelkina, L. Pronzato “Information-regret compromise incovariate-adaptive treatment allocation” AoS’17]
Perspectives:
� Adaptation of efficient allocation procedure [Zhang&Hu AMJCU’09].
� Other forms of regret/reward, e.g. quadratic in (η∗−ηk ).
� Approximate computation of θ 7→∆12(x ,ξ∗α
(θ),θ).
� α = α(τ′) for R(ξ∗α
)≤ τ′, so Φ(ξ,θ)≥ τ(τ′,θ).
� Sequence of αn tending to 0. At which speed?
� Bayesian approach (wrt to θ)
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