In neural machine translation, cross entropy (CE) is the standard loss function in two training methods of auto-regressive models, i.e., teacher forcing and scheduled sampling. In this paper, we propose mixed cross entropy loss (mixed CE) as a substitute for CE in both training approaches. In teacher forcing, the model trained with CE regards the translation problem as a one-to-one mapping process, while in mixed CE this process can be relaxed to one-to-many. In scheduled sampling, we show that mixed CE has the potential to encourage the training and testing behaviours to be similar to each other, more effectively mitigating the exposure bias problem. We demonstrate the superiority of mixed CE over CE on several machine translation datasets, WMT16 Ro-En, WMT16 Ru-En, and WMT14 En-De in both teacher forcing and scheduled sampling setups. Furthermore, in WMT14 En-De, we also find mixed CE consistently outperforms CE on a multi-reference set as well as a challenging paraphrased reference set. We also found the model trained with mixed CE is able to provide a better probability distribution defined over the translation output space. Our code is available at https://github.com/haorannlp/mix.
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