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Stochastic Matrix Factorization Max Welling SMF • Last time: The SVD can do a matrix factorization of the user-item-rating matrix. • Main question to answer: how do we ignore the majority of the entries that have no ratings? Set of observed user-item pairs!! Gradient Descent Compute the direction of steepest descent and take small steps in that direction (show youtube demo) Steepest decent for SMF • Choose a single observed user-item pair (u,i). • Compute the following gradients: • Perform these updates: (for just this (u,i) pair). Stepsize • The stepsize needs to be tuned to be of the right size. • Too small: progress is too slow • Too large: the value for A,B will explode • You can measure you progress by monitoring C(A,B) as you keep updating. • C should go down on average. • Note that we don’t have to fill in values for the unobserved ratings! • Predictions for the unobserved rating are very simple: Regularization We keep A,B terms small so that we do not fit too much to the current data. If we fit too much we would overfit and that might not generalize well to new data. Extra terms Some Extra Bias Terms. New Cost: Homework Compute the derivatives: What are the new update rules?
