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We study the robustness of ridge regression, lasso regression, and of a neural network, when the training set has been randomly corrupted and in response to this corruption the training-size is reduced in order to remove the corrupted data. While the neural network appears to be the most robust method among these three, nevertheless lasso regression appears to be the method of choice since it suffers less loss both when the full information is available to the learner, as well as when a significant amount of the original training set has been rendered useless because of random data corruption.