To Drop or Not to Drop: Robustness, Consistency and Differential Privacy Properties of Dropout

Training deep belief networks (DBNs) requires optimizing a non-convex function with an extremely large number of parameters. Naturally, existing gradient descent (GD) based methods are prone to arbitrarily poor local minima. In this paper, we rigorously show that such local minima can be avoided (upto an approximation error) by using the dropout technique, a widely used heuristic in this domain. In particular, we show that by randomly dropping a few nodes of a one-hidden layer neural network, the training objective function, up to a certain approximation error, decreases by a multiplicative factor. On the flip side, we show that for training convex empirical risk minimizers (ERM), dropout in fact acts as a "stabilizer" or regularizer. That is, a simple dropout based GD method for convex ERMs is stable in the face of arbitrary changes to any one of the training points. Using the above assertion, we show that dropout provides fast rates for generalization error in learning (convex) generalized linear models (GLM). Moreover, using the above mentioned stability properties of dropout, we design dropout based differentially private algorithms for solving ERMs. The learned GLM thus, preserves privacy of each of the individual training points while providing accurate predictions for new test points. Finally, we empirically validate our stability assertions for dropout in the context of convex ERMs and show that surprisingly, dropout significantly outperforms (in terms of prediction accuracy) the L2 regularization based methods for several benchmark datasets.

Real time clustering of time series using triangular potentials

Motivated by the problem of computing investment portfolio weightings we investigate various methods of clustering as alternatives to traditional mean-variance approaches. Such methods can have significant benefits from a practical point of view since they remove the need to invert a sample covariance matrix, which can suffer from estimation error and will almost certainly be non-stationary. The general idea is to find groups of assets which share similar return characteristics over time and treat each group as a single composite asset. We then apply inverse volatility weightings to these new composite assets. In the course of our investigation we devise a method of clustering based on triangular potentials and we present associated theoretical results as well as various examples based on synthetic data.