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Training deep neural networks (DNNs) under weak supervision has attracted increasing research attention as it can significantly reduce the annotation cost. However, labels from weak supervision can be noisy, and the high capacity of DNNs enables them to easily overfit the label noise, resulting in poor generalization. Recent methods leverage self-training to build noise-resistant models, in which a teacher trained under weak supervision is used to provide highly confident labels for teaching the students. Nevertheless, the teacher derived from such frameworks may have fitted a substantial amount of noise and therefore produce incorrect pseudo-labels with high confidence, leading to severe error propagation. In this work, we propose Meta Self-Refinement (MSR), a noise-resistant learning framework, to effectively combat label noise from weak supervision. Instead of relying on a fixed teacher trained with noisy labels, we encourage the teacher to refine its pseudo-labels. At each training step, MSR performs a meta gradient descent on the current mini-batch to maximize the student performance on a clean validation set. Extensive experimentation on eight NLP benchmarks demonstrates that MSR is robust against label noise in all settings and outperforms state-of-the-art methods by up to 11.4% in accuracy and 9.26% in F1 score.