We investigate the test risk of continuous-time stochastic gradient flow
dynamics in learning theory. Using a path integral formulation we provide, in
the regime of a small learning rate, a general formula for computing the
difference between test risk curves of pure gradient and stochastic gradient
flows. We apply the general theory to a simple model of weak features, which
displays the double descent phenomenon, and explicitly compute the corrections
brought about by the added stochastic term in the dynamics, as a function of
time and model parameters. The analytical results are compared to simulations
of discrete-time stochastic gradient descent and show good agreement.

我们使用路径积分形式，对学习理论中连续时间随机梯度流动力学的测试风险进行了研究。在学习速率较小的情况下，我们提供了一个通式，用于计算纯梯度流动和随机梯度流动的测试风险曲线之间的差异。我们将该通式应用于一个简单的弱特征模型，显示了双峰下降现象，并明确计算了动力学中添加的随机项随时间和模型参数的修正。分析结果与离散时间随机梯度下降的模拟结果进行了比较，显示出良好的一致性。