We consider the distributed SGD problem, where a main node distributes gradient calculations among $n$ workers. By assigning tasks to all the workers and waiting only for the $k$ fastest ones, the main node can trade-off the algorithm's error with its runtime by gradually increasing $k$ as the algorithm evolves. However, this strategy, referred to as adaptive $k$-sync, neglects the cost of unused computations and of communicating models to workers that reveal a straggling behavior. We propose a cost-efficient scheme that assigns tasks only to $k$ workers, and gradually increases $k$. We introduce the use of a combinatorial multi-armed bandit model to learn which workers are the fastest while assigning gradient calculations. Assuming workers with exponentially distributed response times parameterized by different means, we give empirical and theoretical guarantees on the regret of our strategy, i.e., the extra time spent to learn the mean response times of the workers. Furthermore, we propose and analyze a strategy applicable to a large class of response time distributions. Compared to adaptive $k$-sync, our scheme achieves significantly lower errors with the same computational efforts and less downlink communication while being inferior in terms of speed.