We propose a cumulative oversampling (CO) method for online learning. Our key idea is to sample parameter estimations from the updated belief space once in each round (similar to Thompson Sampling), and utilize the cumulative samples up to the current round to construct optimistic parameter estimations that asymptotically concentrate around the true parameters as tighter upper confidence bounds compared to the ones constructed with standard UCB methods. We apply CO to a novel budgeted variant of the Influence Maximization (IM) semi-bandits with linear generalization of edge weights, whose offline problem is NP-hard. Combining CO with the oracle we design for the offline problem, our online learning algorithm simultaneously tackles budget allocation, parameter learning, and reward maximization. We show that for IM semi-bandits, our CO-based algorithm achieves a scaled regret comparable to that of the UCB-based algorithms in theory, and performs on par with Thompson Sampling in numerical experiments.