We analyze the generalization gap (gap between the training and test errors) when training a potentially over-parametrized model using a Markovian stochastic training algorithm, initialized from some distribution $\theta_0 \sim p_0$. We focus on Langevin dynamics with a positive temperature $\beta^{-1}$, i.e. gradient descent on a training loss $L$ with infinitesimal step size, perturbed with $\beta^{-1}$-variances Gaussian noise, and lightly regularized or bounded. There, we bound the generalization gap, at any time during training, by $\sqrt{(\beta\mathbb{E} L (\theta_0) + \log(1/\delta))/N}$ with probability $1-\delta$ over the dataset, where $N$ is the sample size, and $\mathbb{E} L (\theta_0) =O(1)$ with standard initialization scaling. In contrast to previous guarantees, we have no dependence on either training time or reliance on mixing, nor a dependence on dimensionality, gradient norms, or any other properties of the loss or model. This guarantee follows from a general analysis of any Markov process-based training that has a Gibbs-style stationary distribution. The proof is surprisingly simple, once we observe that the marginal distribution divergence from initialization remains bounded, as implied by a generalized second law of thermodynamics.

翻译：我们分析了使用马尔可夫随机训练算法（从分布 $\theta_0 \sim p_0$ 初始化）训练一个可能过参数化模型时的泛化间隙（训练误差与测试误差之间的差距）。我们重点关注具有正温度 $\beta^{-1}$ 的朗之万动力学，即在训练损失 $L$ 上进行梯度下降，采用无穷小步长，并受到 $\beta^{-1}$ 方差高斯噪声扰动，同时施加轻微正则化或有界约束。在此框架下，我们以 $1-\delta$ 的概率（关于数据集）将任意训练时刻的泛化间隙上界约束为 $\sqrt{(\beta\mathbb{E} L (\theta_0) + \log(1/\delta))/N}$，其中 $N$ 为样本量，且 $\mathbb{E} L (\theta_0) =O(1)$ 符合标准初始化缩放设定。与先前理论保证不同，我们的结果既不依赖于训练时间或混合性假设，也不依赖于维度、梯度范数或损失函数及模型的任何其他性质。这一保证源于对任何具有吉布斯式平稳分布的马尔可夫过程训练方法的通用分析。证明过程出人意料地简洁，其关键在于我们观察到边际分布与初始分布的散度始终保持有界，这可由广义热力学第二定律推导得出。