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This paper introduces a novel method for accelerating the optimization of high-dimensional functions through a deep neural network (DNN) that incorporates contextual features in its first layer. By optimizing DNN parameters using standard gradient descent while keeping the input-feature basis fixed, the authors achieve significant efficiency gains, particularly through the use of a randomized tensor decomposition strategy that reduces storage costs by at least eight orders of magnitude. The proposed approach enables the effective training of models across dimensions ranging from 5 to 40, showcasing its potential for practical applications in high-dimensional function learning.
Achieving an unprecedented reduction in storage costs for high-dimensional function optimization, this method allows for efficient training of deep neural networks in dimensions previously deemed impractical.
In this work we present a method to accelerate the optimization of learning high dimensional functions using deep neural network (DNN). This optimization procedure introduces contextual features into the first layer of a DNN. The parameters of DNN are optimized via standard gradient descent while keeping the input-feature basis fixed. After optimization of the DNN parameters, the feature layer is provided a chance to update and change before DNN optimization resumes. The feature layer has two types of functions: those that can be evaluated quickly in a matrix-free way on the domain (i.e. rank-1 features) and more complex features that must first be decomposed using tensor network (TN) decomposition strategies (tensor features). In particular, we study the effect of adding features which distill pretrained DNN into TNs using a discretize and decompose strategy. To efficiently decompose high-dimensional functions constructed from discretized DNN, we leverage a randomized tensor decomposition strategy. Using randomization, we are able to reduce the storage cost of decomposing high dimensional functions by at least 8 orders of magnitude. Using this approach, we are able to efficiently train models between 5 and 40 dimensions.
