In this paper, we propose a deep learning framework based on randomized neural network. In particular, inspired by the principles of Random Vector Functional Link (RVFL) network, we present a deep RVFL network (dRVFL) with stacked layers. The parameters of the hidden layers of the dRVFL are randomly generated within a suitable range and kept fixed while the output weights are computed using the closed form solution as in a standard RVFL network. We also propose an ensemble deep network (edRVFL) that can be regarded as a marriage of ensemble learning with deep learning. Unlike traditional ensembling approaches that require training several models independently from scratch, edRVFL is obtained by training a single dRVFL network once. Both dRVFL and edRVFL frameworks are generic and can be used with any RVFL variant. To illustrate this, we integrate the deep learning networks with a recently proposed sparse-pretrained RVFL (SP-RVFL). Extensive experiments on benchmark datasets from diverse domains show the superior performance of our proposed deep RVFL networks.
Typically, Eq. can be solved via a closed form solution using either ridge regression (i.e. @math ) or Moore-Penrose pseudoinverse (i.e. @math ) @cite_4 . Using Moore-Penrose pseudoinverse, the solution is given by: @math while using the regularized least squares (or ridge regression), the closed form solution is given by:
@cite_4
2024591553
We study the approximation of Ef(X-T) by a Monte Carlo algorithm, where X is the solution of a stochastic differential equation and f is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps delta and delta(beta), 0 < beta < 1. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal beta (which turns out to be beta = 1 2), we establish a central limit: type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.