  Post-training quantization (PTQ) has been gaining popularity for the
deployment of deep neural networks on resource-limited devices since unlike
quantization-aware training, neither a full training dataset nor end-to-end
training is required at all. As PTQ schemes based on reconstructing each layer
or block output turn out to be effective to enhance quantized model
performance, recent works have developed algorithms to devise and learn a new
weight-rounding scheme so as to better reconstruct each layer or block output.
In this work, we propose a simple yet effective new weight-rounding mechanism
for PTQ, coined \emph{FlexRound}, based on element-wise division instead of
typical element-wise addition such that FlexRound enables jointly learning a
common quantization grid size as well as a different scale for each pre-trained
weight. Thanks to the reciprocal rule of derivatives induced by element-wise
division, FlexRound is inherently able to exploit pre-trained weights when
updating their corresponding scales, and thus, flexibly quantize pre-trained
weights depending on their magnitudes. We empirically validate the efficacy of
FlexRound on a wide range of models and tasks. To the best of our knowledge,
our work is the first to carry out comprehensive experiments on not only image
classification and natural language understanding but also natural language
generation. Moreover, we demonstrate, for the first time, that large language
models can be efficiently quantized, with only a negligible impact on
performance compared to half-precision baselines, achieved by reconstructing
the output in a block-by-block manner. Our code is available at
\url{https://github.com/onliwad101/FlexRound_LRQ}.
