Noise on Gradient Systems with Forgetting

Abstract

In this paper, we study the effect of noise on a gradient system with forgetting. The noise include multiplicative noise, additive noise and chaotic noise. For multiplicative or additive noise, the noise is a mean zero Gaussian noise. It is added to the state vector of the system. For chaotic noise, it is added to the gradient vector. Let \({\mathbf x}\) be the state vector of a system, \(S_b\) be the variance of the Gaussian noise, \(\kappa '\) is average noise level of the chaotic noise, \(\lambda \) is a positive constant, \(V({\mathbf x})\) be the energy function of the original gradient system, \(V_{\otimes }({\mathbf x})\), \(V_{\oplus }({\mathbf x})\) and \(V_{\odot }({\mathbf x})\) be the energy functions of the gradient systems, if multiplicative, additive and chaotic noises are introduced. Suppose \(V({\mathbf x}) = F({\mathbf x}) + \lambda \Vert {\mathbf x}\Vert ^2_2\). It is shown that \(V_{\otimes }({\mathbf x}) = V({\mathbf x}) + (S_b/2) \sum _{j=1}^n (\partial ^2 F({\mathbf x})/\partial x_j^2) x_j^2 - S_b \sum _{j=1}^n \int x_j (\partial ^2 F({\mathbf x})/\partial x_j^2) dx_j\), \(V_{\oplus }({\mathbf x}) = V({\mathbf x}) + (S_b/2) \sum _{j=1}^n \partial ^2 F({\mathbf x})/\partial x_j^2\), and \(V_{\odot }({\mathbf x}) = V({\mathbf x}) + \kappa '\sum _{i=1}^n x_i\). The first two results imply that multiplicative or additive noise has no effect on the system if \(F({\mathbf x})\) is quadratic. While the third result implies that adding chaotic noise can have no effect on the system if \(\kappa '\) is zero. As many learning algorithms are developed based on the method of gradient descent, these results can be applied in analyzing the effect of noise on those algorithms.