Functions with average smoothness: structure, algorithms, and learning
This work addresses the challenge of improving generalization in machine learning by refining smoothness assumptions, offering a novel framework that could benefit algorithms in metric space learning, though it appears incremental in its technical extensions.
The authors tackled the problem of learning real-valued functions on metric spaces by introducing an average smoothness measure based on local slopes, which can be much smaller than the Lipschitz constant, leading to sharper generalization bounds. They provided distribution-sensitive bounds and efficient algorithms for smoothing labels, with guarantees for extending smoothness from samples to the whole space.
We initiate a program of average smoothness analysis for efficiently learning real-valued functions on metric spaces. Rather than using the Lipschitz constant as the regularizer, we define a local slope at each point and gauge the function complexity as the average of these values. Since the mean can be dramatically smaller than the maximum, this complexity measure can yield considerably sharper generalization bounds -- assuming that these admit a refinement where the Lipschitz constant is replaced by our average of local slopes. Our first major contribution is to obtain just such distribution-sensitive bounds. This required overcoming a number of technical challenges, perhaps the most formidable of which was bounding the {\em empirical} covering numbers, which can be much worse-behaved than the ambient ones. Our combinatorial results are accompanied by efficient algorithms for smoothing the labels of the random sample, as well as guarantees that the extension from the sample to the whole space will continue to be, with high probability, smooth on average. Along the way we discover a surprisingly rich combinatorial and analytic structure in the function class we define.