Exponentially Increasing the Capacity-to-Computation Ratio for Conditional Computation in Deep Learning

Many state-of-the-art results obtained with deep networks are achieved with the largest models that could be trained, and if more computation power was available, we might be able to exploit much larger datasets in order to improve generalization ability. Whereas in learning algorithms such as decision trees the ratio of capacity (e.g., the number of parameters) to computation is very favorable (up to exponentially more parameters than computation), the ratio is essentially 1 for deep neural networks. Conditional computation has been proposed as a way to increase the capacity of a deep neural network without increasing the amount of computation required, by activating some parameters and computation “on-demand”, on a per-example basis. In this note, we propose a novel parametrization of weight matrices in neural networks which has the potential to increase up to exponentially the ratio of the number of parameters to computation. The proposed approach is based on turning on some parameters (weight matrices) when specific bit patterns of hidden unit activations are obtained. In order to better control for the overfitting that might result, we propose a parametrization that is tree-structured, where each node of the tree corresponds to a prefix of a sequence of sign bits, or gating units, associated with hidden units. 1 Conditional Computation for Deep Nets Deep learning is about learning hierarchically-organized representations, with higher levels corresponding to more abstract concepts automatically learned from data, either in a supervised, unsupervised, semi-supervised way, or via reinforcement learning (Mnih et al., 2013). See Bengio et al. (2013b) for a recent review. There have been a number of breakthroughs in the application of deep learning, e.g., in speech (Hinton et al., 2012a) and computer vision (Krizhevsky et al., 2012). Most of these involve deep neural networks that have as much capacity (the number of units and parameters) as possible, given the constraints on training and test time that made these experiments reasonably feasible. It has recently been reported that bigger models could yield better generalization on a number of datasets (Coates et al., 2011; Hinton et al., 2012b; Krizhevsky et al., 2012; Goodfellow et al., 2013) provided appropriate regularization such as dropout (Hinton et al., 2012b) is used. These experiments however have generally been limited by training time in which the amount of training data that could be exploited. An important factor in these recent breakthroughs has been the availability of GPUs which have allowed training deep nets at least 10 times faster, often more (Raina et al., 2009). However, whereas the task of recognizing handwritten digits, traffic signs (Ciresan et al., 2012) or faces (?) is solved to the point of achieving roughly human-level performance, this is far from true for other tasks