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Taylor-based pseudo-metrics for random process fitting in dynamic programming.

Sylvain Gelly 1 Jérémie Mary 1 Olivier Teytaud 1
1 TANC - Algorithmic number theory for cryptology
Inria Saclay - Ile de France, LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
Abstract : Stochastic optimization is the research of $x$ optimizing $E\ C(x,A)$, the expectation of $C(x,A)$, wher e $A$ is a random variable. Typically $C(x,a)$ is the cost related to a strategy $x$ which faces the reali zation $a$ of the random process. Many stochastic optimization problems deal with multiple time steps, leading to computationally difficu lt problems ; efficient solutions exist, for example through Bellman's optimality principle, but only provided that the random process is represented by a well structured process, typically an inhomogeneous Markovian process (hopefully with a finite number of states) or a scenario tree. The problem is that in the general case, $A$ is far from b eing Markovian. So, we look for $A'$, "looking like $A$", but belonging to a given family $\A'$ which do es not at all contain $A$. The problem is the numerical evaluation of "$A'$ looks like $A$". A classical method is the use of the Kantorovitch-Rubinstein distance or other transportation metrics \c ite{Pflug}, justified by straightforward bounds on the deviation $|E C(x,A)-E C(x,A')|$ through the use of the Kantorovitch-Rubinstein distance and uniform lipschitz conditions. These approaches might be bett er than the use of high-level statistics \cite{Keefer}. We propose other (pseudo-)distances, based upon refined inequalities, guaranteeing a good choice of $A'$. Moreover, as in many cases, we indeed prefer t he optimization with risk management, e.g. optimization of $EC(x,noise(A))$ where $noise(.)$ is a random noise modelizing the lack of knowledge on the precise random variables, we propose distances which can deal with a user-defined noise. Tests on artificial data sets with realistic loss functions show the rel evance of the method.
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https://hal.inria.fr/inria-00000217
Contributor : Olivier Teytaud <>
Submitted on : Tuesday, September 13, 2005 - 6:52:08 PM
Last modification on : Thursday, December 3, 2020 - 4:56:03 PM
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Sylvain Gelly, Jérémie Mary, Olivier Teytaud. Taylor-based pseudo-metrics for random process fitting in dynamic programming.. PDMIA, 2005, Lille, 16 p. ⟨inria-00000217⟩

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