Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm
This work provides foundational theoretical guarantees for vector-valued machine learning, with potential impact in multi-output regression tasks, though it is incremental in extending existing scalar results to vector settings.
The paper tackles the problem of deriving optimal convergence rates for vector-valued ridge regression in infinite-dimensional reproducing kernel Hilbert spaces, including misspecified cases where the true function is not in the hypothesis space, achieving rates that match real-valued kernel ridge regression and are independent of output dimension.
We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between $L_2$ and the hypothesis space, which we consider as a vector-valued reproducing kernel Hilbert space. These rates allow to treat the misspecified case in which the true regression function is not contained in the hypothesis space. We combine standard assumptions on the capacity of the hypothesis space with a novel tensor product construction of vector-valued interpolation spaces in order to characterize the smoothness of the regression function. Our upper bound not only attains the same rate as real-valued kernel ridge regression, but also removes the assumption that the target regression function is bounded. For the lower bound, we reduce the problem to the scalar setting using a projection argument. We show that these rates are optimal in most cases and independent of the dimension of the output space. We illustrate our results for the special case of vector-valued Sobolev spaces.