While deep neural networks (DNNs) have demonstrated impressive performance in solving many challenging tasks, they are limited to resource-constrained devices owing to their demand for computation power and storage space. Quantization is one of the most promising techniques to address this issue by quantizing the weights and/or activation tensors of a DNN into lower bit-width fixed-point numbers. While quantization has been empirically shown to introduce minor accuracy loss, it lacks formal guarantees on that, especially when the resulting quantized neural networks (QNNs) are deployed in safety-critical applications. A majority of existing verification methods focus exclusively on individual neural networks, either DNNs or QNNs. While promising attempts have been made to verify the quantization error bound between DNNs and their quantized counterparts, they are not complete and more importantly do not support fully quantified neural networks, namely, only weights are quantized. To fill this gap, in this work, we propose a quantization error bound verification method (QEBVerif), where both weights and activation tensors are quantized. QEBVerif consists of two analyses: a differential reachability analysis (DRA) and a mixed-integer linear programming (MILP) based verification method. DRA performs difference analysis between the DNN and its quantized counterpart layer-by-layer to efficiently compute a tight quantization error interval. If it fails to prove the error bound, then we encode the verification problem into an equivalent MILP problem which can be solved by off-the-shelf solvers. Thus, QEBVerif is sound, complete, and arguably efficient. We implement QEBVerif in a tool and conduct extensive experiments, showing its effectiveness and efficiency.
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