A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$-Superlinear Convergence for Stochastic Games on Domains

Reisinger, Christoph; Zhang, Yufei

doi:10.1007/s10208-020-09460-1

Cited by 24 publications

(12 citation statements)

References 42 publications

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“…We then extend the residual based method for scalar PDEs in [44,31] to the coupled PDE system (4.3).…”

Section: Implementation Of the Fipde Methods Via Residual Approximationmentioning

confidence: 99%

A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems

Reisinger¹,

Stockinger²,

Zhang³

2021

Preprint

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A PDE-based accelerated gradient algorithm is proposed to seek optimal feedback controls of McKean-Vlasov dynamics subject to nonsmooth costs, whose coefficients involve mean-field interactions both on the state and action. It exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is realized via a particle approximation, and the required gradient is evaluated through a coupled system of nonlocal linear PDEs. The latter is solved by finite difference approximation or neural network-based residual approximation, depending on the state dimension. Exhaustive numerical experiments for low and high-dimensional meanfield control problems, including sparse stabilization of stochastic Cucker-Smale models, are presented, which reveal that our algorithm captures important structures of the optimal feedback control, and achieves a robust performance with respect to parameter perturbation.

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“…We then extend the residual based method for scalar PDEs in [44,31] to the coupled PDE system (4.3).…”

Section: Implementation Of the Fipde Methods Via Residual Approximationmentioning

confidence: 99%

A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems

Reisinger¹,

Stockinger²,

Zhang³

2021

Preprint

Self Cite

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“…Recent years have seen progress, in particular in the context of option pricing for Black-Scholes-type models, for DNN-based numerical approximation of diffusion models on possibly large baskets (see e.g. Berner et al [9], Elbrächter et al [22] and Ito et al [34], Reisinger and Zhang [45] for game-type options). These references prove that DNN-based approximations of option prices on possibly large baskets of risky assets can overcome the so-called curse of dimensionality in the context of affine diffusion models for the dynamics of the (log-)prices of the underlying risky assets.…”

Section: Introductionmentioning

confidence: 99%

Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

Gonon¹,

Schwab

2021

Finance Stoch

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We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of $d$ d risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process $X$ X that ensure $\varepsilon $ ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to ${\mathcal{O}}(\varepsilon ^{-1})$ O ( ε − 1 ) , and with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) which grow polynomially in $d$ d , thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps.In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide $\varepsilon $ ε expression error in DNN size ${\mathcal{O}}(|\log (\varepsilon )|^{a})$ O ( | log ( ε ) | a ) with exponent $a$ a proportional to $d$ d , but with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) growing exponentially with respect to $d$ d . Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.

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“…We refer, for instance, to [43,44] for approximation methods for semilinear parabolic PDEs based on standard Monte Carlo approximations for nested conditional expectations. We refer, for instance, to [5,12,13,16,17,21,33,34,39,41] and the references therein for deep learning-based approximation methods for high-dimensional PDEs. We refer, for instance, to [14,15,29] for full-history recursive multilevel Picard approximation methods for semilinear parabolic PDEs (in the following we abbreviate full-history recursive multilevel Picard by MLP).…”

Section: Introductionmentioning

confidence: 99%

Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations

Hutzenthaler¹,

Jentzen²,

Kuckuck³

et al. 2021

Preprint

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Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general time horizons and Lipschitz continuous nonlinearities. However, each of the error analyses for MLP approximation schemes in the existing literature studies the L 2 -root-mean-square distance between the exact solution of the PDE under consideration and the considered MLP approximation and none of the error analyses in the existing literature provides an upper bound for the more general L p -distance between the exact solution of the PDE under consideration and the considered MLP approximation. It is the key contribution of this article to extend the L 2 -error analysis for MLP approximation schemes in the literature to a more general L p -error analysis with p ∈ (0, ∞). In particular, the main result of this article proves that the proposed MLP approximation scheme indeed overcomes the curse of dimensionality in the numerical approximation of highdimensional semilinear PDEs with the approximation error measured in the L p -sense with p ∈ (0, ∞).

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A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$-Superlinear Convergence for Stochastic Games on Domains

Cited by 24 publications

References 42 publications

A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems

A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems

Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations

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