Training neural networks using monotone variational inequality
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Despite the vast empirical success of neural networks, theoretical understanding of the training procedures remains limited, especially in providing performance guarantees of testing performance due to the non-convex nature of the optimization problem. Inspired by a recent work of (Juditsky Nemirovsky, 2019), instead of using the traditional loss function minimization approach, we reduce the training of the network parameters to another problem with convex structure – to solve a monotone variational inequality (MVI). The solution to MVI can be found by computationally efficient procedures, and importantly, this leads to performance guarantee of ℓ_2 and ℓ_∞ bounds on model recovery accuracy and prediction accuracy under the theoretical setting of training one-layer linear neural network. In addition, we study the use of MVI for training multi-layer neural networks and propose a practical algorithm called stochastic variational inequality (SVI), and demonstrates its applicability in training fully-connected neural networks and graph neural networks (GNN) (SVI is completely general and can be used to train other types of neural networks). We demonstrate the competitive or better performance of SVI compared to the stochastic gradient descent (SGD) on both synthetic and real network data prediction tasks regarding various performance metrics.
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