Before revision:  We answer this question in the affirmative by designing a computationally efficient  algorithm with near-optimal label complexity of O\big({s \log ^3 d \cdot \log^3\frac{1{\epsilon}}  } \big) and noise tolerance \eta = \Omega(\epsilon), where \epsilon \in (0, 1) is the target error rate .
After revision:  We answer this question in the affirmative by designing a computationally efficient algorithm with near-optimal label complexity of O\big({s \log {\epsilon}} } \big) and noise tolerance \eta = \Omega(\epsilon), where \epsilon \in (0, 1) is the target error rate .