Asymptotic Almost Sure Efficiency of Averaged Stochastic Algorithms

Pelletier, Mariane

doi:10.1137/s0363012998308169

Cited by 38 publications

(53 citation statements)

References 16 publications

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“…Table 1 gives the obtained values of the two-dimensional vectors θ * n andθ * n . In Figure 1, we test the robustness of both routines, for the computation ofθ * n , using the averaged algorithm "à la Ruppert & Poliak" (see, e.g., [19]) known to give optimal rate for convergence. We implement this averaged algorithm using both constrained and unconstrained procedures.…”

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confidence: 99%

Importance sampling and statistical Romberg method

Alaya¹,

Hajji²,

Kebaier³

2015

Bernoulli

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The efficiency of Monte Carlo simulations is significantly improved when implemented with variance reduction methods. Among these methods, we focus on the popular importance sampling technique based on producing a parametric transformation through a shift parameter θ. The optimal choice of θ is approximated using Robbins-Monro procedures, provided that a nonexplosion condition is satisfied. Otherwise, one can use either a constrained Robbins-Monro algorithm (see, e.g., Arouna (Monte Carlo Methods Appl. 10 (2004) 1-24) and Lelong (Statist. Probab. Lett. 78 (2008) 2632-2636)) or a more astute procedure based on an unconstrained approach recently introduced by Lemaire and Pagès in (Ann. Appl. Probab. 20 (2010) 1029-1067). In this article, we develop a new algorithm based on a combination of the statistical Romberg method and the importance sampling technique. The statistical Romberg method introduced by Kebaier in (Ann. Appl. Probab. 15 (2005) 2681-2705) is known for reducing efficiently the complexity compared to the classical Monte Carlo one. In the setting of discritized diffusions, we prove the almost sure convergence of the constrained and unconstrained versions of the Robbins-Monro routine, towards the optimal shift θ * that minimizes the variance associated to the statistical Romberg method. Then, we prove a central limit theorem for the new algorithm that we called adaptive statistical Romberg method. Finally, we illustrate by numerical simulation the efficiency of our method through applications in option pricing for the Heston model.Eψ(X T ) = Eg(θ, X T ).

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confidence: 99%

Importance sampling and statistical Romberg method

Alaya¹,

Hajji²,

Kebaier³

2015

Bernoulli

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“…One can observe that our stochastic Newton algorithm has the same asymptotic behavior as the averaged version of a stochastic gradient algorithm [5,7,13].…”

Section: Resultsmentioning

confidence: 99%

An Efficient Stochastic Newton Algorithm for Parameter Estimation in Logistic Regressions

Bercu¹,

Godichon²,

Portier³

2020

SIAM J. Control Optim.

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Logistic regression is a well-known statistical model which is commonly used in the situation where the output is a binary random variable. It has a wide range of applications including machine learning, public health, social sciences, ecology and econometry. In order to estimate the unknown parameters of logistic regression with data streams arriving sequentially and at high speed, we focus our attention on a recursive stochastic algorithm. More precisely, we investigate the asymptotic behavior of a new stochastic Newton algorithm. It enables to easily update the estimates when the data arrive sequentially and to have research steps in all directions. We establish the almost sure convergence of our stochastic Newton algorithm as well as its asymptotic normality. All our theoretical results are illustrated by numerical experiments.Our goal is the estimation of the vector of parameters θ. For that purpose, let G be the convex positive function defined, for all h ∈ R d+1 , by

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“…To overcome this classical problem, we introduce the empirical mean of the global algorithm implemented with a slowly decreasing step “à la Ruppert & Polyak” (see, e.g., Pelletier ). First, we write the global algorithm in a more synthetic way by setting for

n \geq 1

φ_{n} = (ξ_{n}, θ_{n}, C_{n}), φ_{0} = (ξ_{0}, θ_{0}, C_{0}),

…”

Section: Computational and Numerical Aspects Of Cvar Hedgingmentioning

confidence: 99%

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

Bardou

Frikha

Pagès

2013

Mathematical Finance

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In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.

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Asymptotic Almost Sure Efficiency of Averaged Stochastic Algorithms

Cited by 38 publications

References 16 publications

Importance sampling and statistical Romberg method

Importance sampling and statistical Romberg method

An Efficient Stochastic Newton Algorithm for Parameter Estimation in Logistic Regressions

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

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