Sound Mixed Fixed-Point Quantization of Neural Networks

Neural networks are increasingly being used as components in safety-critical applications, for instance, as controllers in embedded systems. Their formal safety verification has made significant progress but typically considers only idealized real-valued networks. For practical applications, such neural networks have to be quantized, i.e., implemented in finite-precision arithmetic, which inevitably introduces roundoff errors. Choosing a suitable precision that is both guaranteed to satisfy a roundoff error bound to ensure safety and that is as small as possible to not waste resources is highly nontrivial to do manually. This task is especially challenging when quantizing a neural network in fixed-point arithmetic, where one can choose among a large number of precisions and has to ensure overflow-freedom explicitly. This paper presents the first sound and fully automated mixed-precision quantization approach that specifically targets deep feed-forward neural networks. Our quantization is based on mixed-integer linear programming (MILP) and leverages the unique structure of neural networks and effective over-approximations to make MILP optimization feasible. Our approach efficiently optimizes the number of bits needed to implement a network while guaranteeing a provided error bound. Our evaluation on existing embedded neural controller benchmarks shows that our optimization translates into precision assignments that mostly use fewer machine cycles when compiled to an FPGA with a commercial HLS compiler than code generated by (sound) state-of-the-art. Furthermore, our approach handles significantly more benchmarks substantially faster, especially for larger networks.