Some embodiments of the invention provide a novel method for training a multi-layer node network that mitigates against overfitting the adjustable parameters of the network for a particular problem. During training, the method of some embodiments adjusts the modifiable parameters of the network by iteratively identifying different interior-node, influence-attenuating masks that effectively specify different sampled networks of the multi-layer node network. An interior-node, influence-attenuating mask specifies attenuation parameters that are applied (1) to the outputs of the interior nodes of the network in some embodiments, (2) to the inputs of the interior nodes of the network in other embodiments, or (3) to the outputs and inputs of the interior nodes in still other embodiments. In each mask, the attenuation parameters can be any one of several values (e.g., three or more values) within a range of values (e.g., between 0 and 1).