Stability Analysis for Regularized Least Squares Regression
<|reference_start|>Stability Analysis for Regularized Least Squares Regression: We discuss stability for a class of learning algorithms with respect to noisy labels. The algorithms we consider are for regression, and they involve the minimization of regularized risk functionals, such as L(f) := 1/N sum_i (f(x_i)-y_i)^2+ lambda ||f||_H^2. We shall call the algorithm `stable' if, when y_i is a noisy version of f*(x_i) for some function f* in H, the output of the algorithm converges to f* as the regularization term and noise simultaneously vanish. We consider two flavors of this problem, one where a data set of N points remains fixed, and the other where N -> infinity. For the case where N -> infinity, we give conditions for convergence to f_E (the function which is the expectation of y(x) for each x), as lambda -> 0. For the fixed N case, we describe the limiting 'non-noisy', 'non-regularized' function f*, and give conditions for convergence. In the process, we develop a set of tools for dealing with functionals such as L(f), which are applicable to many other problems in learning theory.<|reference_end|>
@article{rudin2005stability,
    title={Stability Analysis for Regularized Least Squares Regression},
    author={Cynthia Rudin},
    journal={arXiv preprint arXiv:cs/0502016},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0502016},
    primaryClass={cs.LG}
}