optimization loss due to misperceived probability

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Suppose $a$ is chosen to maximize the expected value of $u(a,x)$ under a probability measure of $x$. Image the true distribution is $P(x)$, but the optimization may be conducted under a misperceived distribution $Q(x)$. We denote the optimal action under $P$ and $Q$ as $a^*(P),a^*(Q)$ respectively,
beginalign
a^*(P)&=textargmax_a int u(a,x)d P(x),\
a^*(Q)&=textargmax_a int u(a,x)d Q(x).
endalign



Now we investigate the loss of the misoptimization
beginalign
Delta(P,Q) = int left[uleft(a^*(P),xright) - uleft(a^*(Q),xright) right]d P(x).
endalign



My question: Is there a good bound or approximation for $Delta(P,Q)$ in general, in terms of some function (or value function) of $u$ and some metric to measure the distance between two distributions $P$ and $Q$? It's fine to assume good properties for $u$, say smooth and bounded and assume $P$ and $Q$ are close in some intuitive sense.



It would be nice to also allow the possibility that $P$ and $Q$ are indeed Dirac measure placed at $p$ and $q$ with $p,q$ being close to each other.



It seems like a well-motivated question, but I failed to find any literature on this. Thanks in advance for discussions or pointing me to some extant results.



For an ideal solution, it's better not to use KL-divergence, because it's infinite between Dirac $P$ and Dirac $Q$. Wasserstein metric appears more likely to be related. I guess some form of envelope theorem might be useful, since an optimization is involved.




probability-theory optimization information-theory operations-research optimal-transport




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  • Comment to point me to other formulation of this question or anything helpful is welcomed!
    – Sean
    7 hours ago














up vote
2
down vote

favorite












Suppose $a$ is chosen to maximize the expected value of $u(a,x)$ under a probability measure of $x$. Image the true distribution is $P(x)$, but the optimization may be conducted under a misperceived distribution $Q(x)$. We denote the optimal action under $P$ and $Q$ as $a^*(P),a^*(Q)$ respectively,
beginalign
a^*(P)&=textargmax_a int u(a,x)d P(x),\
a^*(Q)&=textargmax_a int u(a,x)d Q(x).
endalign



Now we investigate the loss of the misoptimization
beginalign
Delta(P,Q) = int left[uleft(a^*(P),xright) - uleft(a^*(Q),xright) right]d P(x).
endalign



My question: Is there a good bound or approximation for $Delta(P,Q)$ in general, in terms of some function (or value function) of $u$ and some metric to measure the distance between two distributions $P$ and $Q$? It's fine to assume good properties for $u$, say smooth and bounded and assume $P$ and $Q$ are close in some intuitive sense.



It would be nice to also allow the possibility that $P$ and $Q$ are indeed Dirac measure placed at $p$ and $q$ with $p,q$ being close to each other.



It seems like a well-motivated question, but I failed to find any literature on this. Thanks in advance for discussions or pointing me to some extant results.



For an ideal solution, it's better not to use KL-divergence, because it's infinite between Dirac $P$ and Dirac $Q$. Wasserstein metric appears more likely to be related. I guess some form of envelope theorem might be useful, since an optimization is involved.




probability-theory optimization information-theory operations-research optimal-transport




share|cite|improve this question



















  • Comment to point me to other formulation of this question or anything helpful is welcomed!
    – Sean
    7 hours ago












up vote
2
down vote

favorite









up vote
2
down vote

favorite











Suppose $a$ is chosen to maximize the expected value of $u(a,x)$ under a probability measure of $x$. Image the true distribution is $P(x)$, but the optimization may be conducted under a misperceived distribution $Q(x)$. We denote the optimal action under $P$ and $Q$ as $a^*(P),a^*(Q)$ respectively,
beginalign
a^*(P)&=textargmax_a int u(a,x)d P(x),\
a^*(Q)&=textargmax_a int u(a,x)d Q(x).
endalign



Now we investigate the loss of the misoptimization
beginalign
Delta(P,Q) = int left[uleft(a^*(P),xright) - uleft(a^*(Q),xright) right]d P(x).
endalign



My question: Is there a good bound or approximation for $Delta(P,Q)$ in general, in terms of some function (or value function) of $u$ and some metric to measure the distance between two distributions $P$ and $Q$? It's fine to assume good properties for $u$, say smooth and bounded and assume $P$ and $Q$ are close in some intuitive sense.



It would be nice to also allow the possibility that $P$ and $Q$ are indeed Dirac measure placed at $p$ and $q$ with $p,q$ being close to each other.



It seems like a well-motivated question, but I failed to find any literature on this. Thanks in advance for discussions or pointing me to some extant results.



For an ideal solution, it's better not to use KL-divergence, because it's infinite between Dirac $P$ and Dirac $Q$. Wasserstein metric appears more likely to be related. I guess some form of envelope theorem might be useful, since an optimization is involved.




probability-theory optimization information-theory operations-research optimal-transport




share|cite|improve this question











Suppose $a$ is chosen to maximize the expected value of $u(a,x)$ under a probability measure of $x$. Image the true distribution is $P(x)$, but the optimization may be conducted under a misperceived distribution $Q(x)$. We denote the optimal action under $P$ and $Q$ as $a^*(P),a^*(Q)$ respectively,
beginalign
a^*(P)&=textargmax_a int u(a,x)d P(x),\
a^*(Q)&=textargmax_a int u(a,x)d Q(x).
endalign



Now we investigate the loss of the misoptimization
beginalign
Delta(P,Q) = int left[uleft(a^*(P),xright) - uleft(a^*(Q),xright) right]d P(x).
endalign



My question: Is there a good bound or approximation for $Delta(P,Q)$ in general, in terms of some function (or value function) of $u$ and some metric to measure the distance between two distributions $P$ and $Q$? It's fine to assume good properties for $u$, say smooth and bounded and assume $P$ and $Q$ are close in some intuitive sense.



It would be nice to also allow the possibility that $P$ and $Q$ are indeed Dirac measure placed at $p$ and $q$ with $p,q$ being close to each other.



It seems like a well-motivated question, but I failed to find any literature on this. Thanks in advance for discussions or pointing me to some extant results.



For an ideal solution, it's better not to use KL-divergence, because it's infinite between Dirac $P$ and Dirac $Q$. Wasserstein metric appears more likely to be related. I guess some form of envelope theorem might be useful, since an optimization is involved.





probability-theory optimization information-theory operations-research optimal-transport





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asked 7 hours ago









Sean

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  • Comment to point me to other formulation of this question or anything helpful is welcomed!
    – Sean
    7 hours ago
















  • Comment to point me to other formulation of this question or anything helpful is welcomed!
    – Sean
    7 hours ago















Comment to point me to other formulation of this question or anything helpful is welcomed!
– Sean
7 hours ago




Comment to point me to other formulation of this question or anything helpful is welcomed!
– Sean
7 hours ago















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