Multilingual machine translation (MMT) benefits from cross-lingual transfer but is a challenging multitask optimization problem. This is partly because there is no clear framework to systematically learn language-specific parameters. Self-supervised learning (SSL) approaches that leverage large quantities of monolingual data (where parallel data is unavailable) have shown promise by improving translation performance as complementary tasks to the MMT task. However, jointly optimizing SSL and MMT tasks is even more challenging. In this work, we first investigate how to utilize intra-distillation to learn more *language-specific* parameters and then show the importance of these language-specific parameters. Next, we propose a novel but simple SSL task, concurrent denoising, that co-trains with the MMT task by concurrently denoising monolingual data on both the encoder and decoder. Finally, we apply intra-distillation to this co-training approach. Combining these two approaches significantly improves MMT performance, outperforming three state-of-the-art SSL methods by a large margin, e.g., 11.3\% and 3.7\% improvement on an 8-language and a 15-language benchmark compared with MASS, respectively
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