Abstract: We investigate the effects of post-training quantization and quantization-aware training on the generalization of Transformer language models. We present a new method called self-distilled quantization (SDQ) that minimizes accumulative quantization errors and outperforms baselines. We apply SDQ to multilingual models XLM-R$_{\text{Base}}$ and InfoXLM$_{\text{Base}}$ and demonstrate that both models can be reduced from 32-bit floating point weights to 8-bit integer weights while maintaining a high level of performance on the XGLUE benchmark. Our results also highlight the challenges of quantizing multilingual models, which must generalize to languages they were not fine-tuned on.