Tracking Approximate Solutions of Parameterized Optimization Problems over Multi-Dimensional (Hyper-)Parameter Domains

Many machine learning methods are given as parameterized optimization problems. Important examples of such parameters are regularizationand kernel hyperparameters. These parameters have to be tuned carefully since the choice of their values can have a significant impact on the statistical performance of the learning methods. In most cases the parameter space does not carry much structure and parameter tuning essentially boils down to exploring the whole parameter space. The case when there is only one parameter received quite some attention over the years. First, algorithms for tracking an optimal solution for several machine learning optimization problems over regularizationand hyperparameter intervals had been developed, but since these algorithms can suffer from numerical problems more robust and efficient approximate path tracking algorithms have been devised and analyzed recently. By now approximate path tracking algorithms are known for regularizationand kernel hyperparameter paths with optimal path complexities that depend only on the prescribed approximation error. Here we extend the work on approximate path tracking algorithms with approximation guarantees to multi-dimensional parameter domains. We show a lower bound on the complexity of approximately exploring a multidimensional parameter domain that is the product of the corresponding path complexities. We also show a matching upper bound that can be turned into a theoretically and practically efficient algorithm. Experimental results for kernelized support vector machines and the elastic net confirm the theoretical complexity analysis. Proceedings of the 32 International Conference on Machine Learning, Lille, France, 2015. JMLR: W&CP volume 37. Copyright 2015 by the author(s).