This paper presents a method to leverage arbitrary neural network architecture for control variates. Control variates are crucial in reducing the variance of Monte Carlo integration, but they hinge on finding a function that both correlates with the integrand and has a known analytical integral. Traditional approaches rely on heuristics to choose this function, which might not be expressive enough to correlate well with the integrand. Recent research alleviates this issue by modeling the integrands with a learnable parametric model, such as a neural network. However, the challenge remains in creating an expressive parametric model with a known analytical integral. This paper proposes a novel approach to construct learnable parametric control variates functions from arbitrary neural network architectures. Instead of using a network to approximate the integrand directly, we employ the network to approximate the anti-derivative of the integrand. This allows us to use automatic differentiation to create a function whose integration can be constructed by the antiderivative network. We apply our method to solve partial differential equations using the Walk-on-sphere algorithm. Our results indicate that this approach is unbiased and uses various network architectures to achieve lower variance than other control variate methods.

翻译：本文提出了一种利用任意神经网络架构构建控制变量的方法。控制变量在降低蒙特卡洛积分方差方面至关重要，但其关键在于找到一个既与被积函数相关又具有已知解析积分的函数。传统方法依赖启发式规则选择该函数，其表达能力可能不足以与被积函数良好相关。近期研究通过使用可学习的参数模型（如神经网络）对被积函数进行建模，缓解了这一问题。然而，如何构建具有已知解析积分且表达能力强的参数模型仍是挑战。本文提出了一种新颖方法，可从任意神经网络架构构建可学习的参数化控制变量函数。我们并非直接使用网络逼近被积函数，而是利用网络逼近被积函数的原函数。这使得我们能够通过自动微分构造一个函数，其积分可由原函数网络构建。我们将该方法应用于使用Walk-on-sphere算法求解偏微分方程。结果表明，该方法具有无偏性，并能利用多种网络架构实现比其他控制变量方法更低的方差。