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Low-resolution neural networks represent both weights and activations with few bits, drastically reducing the multiplication complexity. Nonetheless, these products are accumulated using high-resolution (typically 32-bit) additions, an operation that dominates the arithmetic complexity of inference when using extreme quantization (e.g., binary weights). To further optimize inference, we propose a method that adapts neural networks to use low-resolution (8-bit) additions in the accumulators, achieving classification accuracy comparable to their 32-bit counterparts. We achieve resilience to low-resolution accumulation by inserting a cyclic activation layer, as well as an overflow penalty regularizer. We demonstrate the efficacy of our approach on both software and hardware platforms.