Over the years, many frameworks and optimization techniques have been proposed to accelerate graph neural networks (GNNs). Compared to the optimizations explored in these systems, we observe that different matrix re-associations of GNN computations lead to novel input-sensitive performance behavior. We leverage this observation to propose SENSEi, a system that exposes different sparse and dense matrix primitive compositions based on different matrix re-associations of GNN computations and selects the best among them based on input attributes. SENSEi executes in two stages: (1) an offline compilation stage that enumerates all valid re-associations leading to different sparse-dense matrix compositions and uses input-oblivious pruning techniques to prune away clearly unprofitable candidates and (2) an online runtime system that explores the remaining candidates and uses light-weight cost models to select the best re-association based on the input graph and the embedding sizes on a given hardware platform. On a wide range of configurations, SENSEi achieves speedups of up to $2.012\times$ and $1.85\times$ on graph convolutional networks and up to $6.294\times$ and $16.274\times$ on graph attention networks, on GPUs and CPUs respectively. We also show that its technique generalizes to GNN variants, including those that require sampling. Furthermore, we show that SENSEi's techniques are agnostic to the underlying GNN system, and can be used to yield synergistic improvements across a diverse set of implementations.

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