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.

翻译：我们首次在$L_2$与假设空间（视为向量值再生核希尔伯特空间）之间内插的连续范数尺度上，给出了无穷维向量值岭回归的最优收敛率。这些收敛率能够处理真实回归函数不包含于假设空间的设定失配情形。我们将假设空间容量的标准假设与向量值插值空间的新型张量积构造相结合，以刻画回归函数的平滑性。我们的上界不仅达到了与标量值核岭回归相同的收敛率，还去除了目标回归函数有界的假设。对于下界，我们通过投影论证将问题简化为标量情形。我们证明了这些收敛率在大多数情况下是最优的，且与输出空间的维度无关。我们以向量值索博列夫空间这一特例对结果进行了说明。