  Tensor network contraction is central to problems ranging from many-body
physics to computer science. We describe how to approximate tensor network
contraction through bond compression on arbitrary graphs. In particular, we
introduce a hyper-optimization over the compression and contraction strategy
itself to minimize error and cost. We demonstrate that our protocol outperforms
both hand-crafted contraction strategies in the literature as well as recently
proposed general contraction algorithms on a variety of synthetic and physical
problems on regular lattices and random regular graphs. We further showcase the
power of the approach by demonstrating approximate contraction of tensor
networks for frustrated three-dimensional lattice partition functions, dimer
counting on random regular graphs, and to access the hardness transition of
random tensor network models, in graphs with many thousands of tensors.
