Successful compressed-sensing reconstruction often involves tuning one or more regularization weights. However, tuning the regularization weights is a subject-specific, task-dependent and non-trivial task. Recent studies have proposed to determine the weights by minimizing the statistical risk of removing significant coefficients using line searches across a range of parameters. However, the line-search procedures lead to prolonged reconstruction times. Here, we propose a new self-tuning approach generalized for multi-coil, multi-acquisition CS reconstructions that leverage projection onto epigraph sets of l1 and TV balls. The proposed method yields 7 to 9-fold gain in computational efficiency over conventional methods while enabling further improved image quality.