Learning Stochastic Graph Neural Networks with Constrained Variance

TL;DR

This paper introduces a variance-constrained optimization framework for stochastic graph neural networks (SGNNs) to improve their robustness by balancing expected performance and output deviation, supported by theoretical analysis and simulations.

Contribution

It proposes a novel variance-constrained learning method for SGNNs, with an alternating primal-dual algorithm and theoretical guarantees on convergence and performance.

Findings

The variance-constrained approach reduces output deviation in SGNNs.

Theoretical analysis reveals a trade-off between robustness and discrimination.

Numerical simulations confirm improved performance with controllable variance.

Abstract

Stochastic graph neural networks (SGNNs) are information processing architectures that learn representations from data over random graphs. SGNNs are trained with respect to the expected performance, which comes with no guarantee about deviations of particular output realizations around the optimal expectation. To overcome this issue, we propose a variance-constrained optimization problem for SGNNs, balancing the expected performance and the stochastic deviation. An alternating primal-dual learning procedure is undertaken that solves the problem by updating the SGNN parameters with gradient descent and the dual variable with gradient ascent. To characterize the explicit effect of the variance-constrained learning, we conduct a theoretical analysis on the variance of the SGNN output and identify a trade-off between the stochastic robustness and the discrimination power. We further analyze the duality gap of the variance-constrained optimization problem and the converging behavior of the primal-dual learning procedure. The former indicates the optimality loss induced by the dual transformation and the latter characterizes the limiting error of the iterative algorithm, both of which guarantee the performance of the variance-constrained learning. Through numerical simulations, we corroborate our theoretical findings and observe a strong expected performance with a controllable standard deviation.