We propose to add independent pseudo quantization noise to model parameters during training to approximate the effect of a quantization operator. This method, DiffQ, is differentiable both with respect to the unquantized parameters, and the number of bits used. Given a single hyper-parameter expressing the desired balance between the quantized model size and accuracy, DiffQ can optimize the number of bits used per individual weight or groups of weights, in a single training. We experimentally verify that our method outperforms state-of-the-art quantization techniques on several benchmarks and architectures for image classification, language modeling, and audio source separation. For instance, on the Wikitext-103 language modeling benchmark, DiffQ compresses a 16 layers transformer model by a factor of 8, equivalent to 4 bits precision, while losing only 0.5 points of perplexity. Code is available at: https://github.com/facebookresearch/diffq
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