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We propose Label Confidence Weighted Learning (LCWL), a novel approach that incorporates a label confidence weighting scheme in the training loss of the encoder-decoder model, setting it apart from existing confidence-weighting methods primarily designed for classification. Experimentation on English grade-level simplification dataset shows that LCWL outperforms state-of-the-art unsupervised baselines. Fine-tuning the LCWL model on in-domain data and combining with Symmetric Cross Entropy (SCE) consistently delivers better simplifications compared to strong supervised methods. 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