Method, Apparatus and Computer Program for generating robust automatic learning systems and testing trained automatic learning systems

A method for training an automated learning system, the method comprising:

• processing a first training input with a first neural network of the automated learning system, the first neural network having an input layer and an output layer, the input layer receiving the first training input, the output layer outputting a first training output;
• processing the first training output with a second neural network of the automated learning system, the second neural network including an input layer and an output layer, the output layer of the first neural network corresponding to the input layer of the second neural network, the input layer of the first neural network corresponding to the output layer of the second neural network, the output layer of the second neural network outputting a second training output;
• and
• determining an objective function depending on the second training output, the first training input, and a predetermined modification magnitude,
• wherein the objective function includes at least a first term and a second term, the first term corresponding to a product of the second training output multiplied by the first training input, the second term corresponding to a dual norm of the first training output weighted by the predetermined modification magnitude, and
• wherein at least one of the first term and the second term of the objective function is approximated using a random Cauchy projection, the random Cauchy projection being determined by passing a random Cauchy matrix through the second neural network.