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PIM (processing-in-memory) based hardware accelerators have shown great potentials in addressing the computation and memory access intensity of modern CNNs (convolutional neural networks). While adopting NVM (non-volatile memory) helps to further mitigate the storage and energy consumption overhead, adopting quantization, e.g., shift-based quantization, helps to tradeoff the computation overhead and the accuracy loss, integrating both NVM and quantization in hardware accelerators leads to sub-optimal acceleration. In this paper, we exploit the natural shift property of DWM (domain wall memory) to devise DWMAcc, a DWM-based accelerator with asymmetrical storage of weight and input data, to speed up the inference phase of shift-based CNNs. DWMAcc supports flexible shift operations to enable fast processing with low performance and area overhead. We then optimize it with zero-sharing , input-reuse , and weight-share schemes. Our experimental results show that, on average, DWMAcc achieves 16.6× performance improvement and 85.6× energy consumption reduction over a state-of-the-art SRAM based design.