Scale invariant process regression: Towards Bayesian ML with minimal assumptions
This work addresses the need for less arbitrary regularization methods in machine learning, offering a foundational approach that could impact various ML areas beyond regression.
The paper tackles the problem of regularization in machine learning by deriving a novel stochastic process from minimal invariance assumptions, resulting in a Bayesian regression method that performs equally to Gaussian processes while eliminating kernel choices and improving extrapolation.
Current methods for regularization in machine learning require quite specific model assumptions (e.g. a kernel shape) that are not derived from prior knowledge about the application, but must be imposed merely to make the method work. We show in this paper that regularization can indeed be achieved by assuming nothing but invariance principles (w.r.t. scaling, translation, and rotation of input and output space) and the degree of differentiability of the true function. Concretely, we derive a novel (non-Gaussian) stochastic process from the above minimal assumptions, and we present a corresponding Bayesian inference method for regression. The mean posterior turns out to be a polyharmonic spline, and the posterior process is a mixture of t-processes. Compared with Gaussian process regression, the proposed method shows equal performance and has the advantages of being (i) less arbitrary (no choice of kernel) (ii) potentially faster (no kernel parameter optimization), and (iii) having better extrapolation behavior. We believe that the proposed theory has central importance for the conceptual foundations of regularization and machine learning and also has great potential to enable practical advances in ML areas beyond regression.