This paper proposes a machine learning method to predict the solutions of related nonlinear optimal control problems given some parametric input, such as the initial state. The map between problem parameters to optimal solutions is called the problem-optimum map, and is often discontinuous due to nonconvexity, discrete homotopy classes, and control switching. This causes difficulties for traditional function approximators such as neural networks, which assume continuity of the underlying function. This paper proposes a mixture of experts (MoE) model composed of a classifier and several regressors, where each regressor is tuned to a particular continuous region. A novel training approach is proposed that trains classifier and regressors independently. MoE greatly outperforms standard neural networks, and achieves highly reliable trajectory prediction (over 99.5 accuracy) in several dynamic vehicle control problems.
The capacity of a neural network to absorb information is limited by its number of parameters. Conditional computation, where parts of the network are active on a per-example basis, has been proposed in theory as a way of dramatically increasing model capacity without a proportional increase in computation. In practice, however, there are significant algorithmic and performance challenges. In this work, we address these challenges and finally realize the promise of conditional computation, achieving greater than 1000x improvements in model capacity with only minor losses in computational efficiency on modern GPU clusters. We introduce a Sparsely-Gated Mixture-of-Experts layer (MoE), consisting of up to thousands of feed-forward sub-networks. A trainable gating network determines a sparse combination of these experts to use for each example. We apply the MoE to the tasks of language modeling and machine translation, where model capacity is critical for absorbing the vast quantities of knowledge available in the training corpora. We present model architectures in which a MoE with up to 137 billion parameters is applied convolutionally between stacked LSTM layers. On large language modeling and machine translation benchmarks, these models achieve significantly better results than state-of-the-art at lower computational cost.  Given a supervised learning context, the mixture of experts approach uses several neural networks in parallel to provide a modular solution to the overall problem.. Under the mixtures of experts architecture a method for 'designing' the number of experts and assigning local 'regions' of the input space to individual experts is investigated. Classification performance and transparency of the scheme is found to be significantly better than that using a standard mixtures of experts. We present a tree-structured architecture for supervised learning. The statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models (GLIM's). Learning is treated as a maximum likelihood problem; in particular, we present an Expectation-Maximization (EM) algorithm for adjusting the parameters of the architecture. We also develop an on-line learning algorithm in which the parameters are updated incrementally. Comparative simulation results are presented in the robot dynamics domain.