TL;DR#
Many machine learning problems involve predicting vector-valued outputs, such as in multi-task learning. Existing theoretical understanding of algorithms for these problems, particularly regarding their efficiency and behavior in high- or infinite-dimensional settings, has been limited. This is especially true for cases where the true regression function is not contained within the model’s hypothesis space (misspecified scenario).
This paper addresses these gaps by providing rigorous theoretical analysis of a broad class of regularized algorithms, including kernel ridge regression and gradient descent. The authors rigorously confirm the saturation effect for ridge regression and provide a novel upper bound on finite-sample risk for general spectral algorithms, applicable to both well-specified and misspecified settings. Notably, this upper bound is shown to be minimax optimal in various settings and explicitly considers the case of infinite-dimensional output variables.
Key Takeaways#
Why does it matter?#
This paper is crucial for researchers working on vector-valued regression and spectral learning algorithms. It provides rigorous theoretical guarantees for a broad class of algorithms, addressing a gap in existing research. The results are minimax optimal in various settings, including infinite-dimensional outputs, with implications for many practical applications.
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Full paper#
