Training a machine learning (ML) model is an optimization process that typically requires a set of examples with ground-truth labels beforehand. Obtaining these ground truths usually requires laborious human effort and can be costly. The scalability of human labeling becomes a concern when there are a large number of data points that need to be labeled. To address this practical issue, we consider an optimization setting where a subset of samples has ground-truth labels, while the remaining data points were labeled by an ML predictor, which could potentially assign incorrect labels to those points. We propose both first-order and zeroth-order stochastic optimization methods for training on datasets consisting of a mix of true-labeled and ML-labeled samples. We further conduct comprehensive empirical studies from training a linear regression model to fine-tuning a large language model (LLM) to showcase the effectiveness of the proposed algorithms for training with ML-labeled data.

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