Prior work on Private Inference (PI)--inferences performed directly on encrypted input--has focused on minimizing a network's ReLUs, which have been assumed to dominate PI latency rather than FLOPs. Recent work has shown that FLOPs for PI can no longer be ignored and have high latency penalties. In this paper, we develop DeepReShape, a network redesign technique that tailors architectures to PI constraints, optimizing for both ReLUs and FLOPs for the first time. The {\em key insight} is that a strategic allocation of channels such that the network's ReLUs are aligned in their criticality order simultaneously optimizes ReLU and FLOPs efficiency. DeepReShape automates network development with an efficient process, and we call generated networks HybReNets. We evaluate DeepReShape using standard PI benchmarks and demonstrate a 2.1\% accuracy gain with a 5.2$\times$ runtime improvement at iso-ReLU on CIFAR-100 and an 8.7$\times$ runtime improvement at iso-accuracy on TinyImageNet. Furthermore, we demystify the input network selection in prior ReLU optimizations and shed light on the key network attributes enabling PI efficiency.
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