This paper considers a stochastic control framework, in which the residual
model uncertainty of the dynamical system is learned using a Gaussian Process
(GP). In the proposed formulation, the residual model uncertainty consists of a
nonlinear function and state-dependent noise. The proposed formulation uses a
posterior-GP to approximate the residual model uncertainty and a prior-GP to
account for state-dependent noise. The two GPs are interdependent and are thus
learned jointly using an iterative algorithm. Theoretical properties of the
iterative algorithm are established. Advantages of the proposed state-dependent
formulation include (i) faster convergence of the GP estimate to the unknown
function as the GP learns which data samples are more trustworthy and (ii) an
accurate estimate of state-dependent noise, which can, e.g., be useful for a
controller or decision-maker to determine the uncertainty of an action.
Simulation studies highlight these two advantages.

该论文考虑了使用高斯过程学习动态系统的残差模型不确定性的随机控制框架，其中残差模型不确定性包括非线性函数和状态相关噪声，并提出了一种后验高斯过程来近似残差模型不确定性和一种先验高斯过程来解释状态相关噪声，这两种高斯过程相互依赖并使用迭代算法共同学习，该方法有快速收敛和准确的状态相关噪声估计的优势。

状态相关噪声的高斯过程用于随机控制

Gaussian Processes with State-Dependent Noise for Stochastic Control

Stochastic gradient descent (SGD) and its variants are mainstream methods to
train deep neural networks. Since neural networks are non-convex, more and more
works study the dynamic behavior of SGD and the impact to its generalization,
especially the escaping efficiency from local minima. However, these works take
the over-simplified assumption that the covariance of the noise in SGD is (or
can be upper bounded by) constant, although it is actually state-dependent. In
this work, we conduct a formal study on the dynamic behavior of SGD with
state-dependent noise. Specifically, we show that the covariance of the noise
of SGD in the local region of the local minima is a quadratic function of the
state. Thus, we propose a novel power-law dynamic with state-dependent
diffusion to approximate the dynamic of SGD. We prove that, power-law dynamic
can escape from sharp minima exponentially faster than flat minima, while the
previous dynamics can only escape sharp minima polynomially faster than flat
minima. Our experiments well verified our theoretical results. Inspired by our
theory, we propose to add additional state-dependent noise into (large-batch)
SGD to further improve its generalization ability. Experiments verify that our
method is effective.

通过研究具有状态相关噪声的随机梯度下降的动态行为，我们证明了幂律动态可以比之前的动态更快地从锐化极小值中逃脱，从而提出了一种新方法来进一步提高其概括能力。