We propose two novel stochastic gradient descent algorithms, ByGARS and ByGARS++, for distributed machine learning in the presence of any number of Byzantine adversaries. In these algorithms, reputation scores of workers are computed using an auxiliary dataset at the server. This reputation score is then used for aggregating the gradients for stochastic gradient descent. The computational complexity of ByGARS++ is the same as the usual distributed stochastic gradient descent method with only an additional inner product computation in every iteration. We show that using these reputation scores for gradient aggregation is robust to any number of multiplicative noise Byzantine adversaries and use two-timescale stochastic approximation theory to prove convergence for strongly convex loss functions. We demonstrate the effectiveness of the algorithms for non-convex learning problems using MNIST and CIFAR-10 datasets against almost all state-of-the-art Byzantine attacks. We also show that the proposed algorithms are robust to multiple different types of attacks at the same time.