eep earning in this we show how one can construct rich class of approximating functions called neural networks the learners belonging to the neural-network class of functions have attractive properties that have made them ubiquitous in modern machine learning applications -their training is computationally feasible and their complexity is easy to control and fine-tune introduction in we described the basic supervised learning tasknamelywe wish to predict random output from random input xusing prediction function that belongs to suitably chosen class of approximating functions more generallywe may wish to predict vector-valued output using prediction function from class in this denotes the vector-valued output for given input this differs from our previous use ( in table )where denotes vector of scalar outputs in the machine learning contextthe class is sometimes referred to as the hypothesis space or the universe of possible modelsand the representational capacity of hypothesis representational capacity space is simply its complexity suppose that we have class of functions gl indexed by parameter that controls the complexity of the classso that gl gl+ gl+ in selecting suitable class of functionswe have to be mindful of the approximation-estimation tradeoff on the one handthe class gl must be complex (richenough to accurately represent the optimal unknown prediction function gwhich may require very large on the other handthe learners in the class gl must be simple enough to train with small estimation error and with minimal demands on computer memorywhich may necessitate small in balancing these competing objectivesit helps if the more complex class gl+ is easily constructed from an already existing and simpler gl the simpler class of functions gl may itself be constructed by modifying an even simpler class gl- and so on class of functions that permits such natural hierarchical construction is the class of neural networks conceptuallya neural network with layers is nonlinear parametric neural networks regression model whose representational capacity can easily be controlled by