We study influence maximization in general stochastic models, the observed cascades model, and the independent cascade (IC) model. For general stochastic models with only black-box sample access, we introduce a low-adaptivity optimization framework that improves sample complexity and running time over Sadeh et al. (2020) and is instrumental to all our results. We further introduce an adaptive algorithm guided by empirical variance, avoiding pessimistic worst-case bounds. Combining our optimization framework with sketching, we obtain the first algorithm with provable guarantees and nearly-linear running time for influence maximization on observed cascades, optimal up to logarithmic factors. For IC, we prove a novel tail bound replacing a factor $n$ with $τ$ (the number of diffusion steps) in sample complexity, improving over prior work when $τ$ is small, as is common due to small-world phenomena. Experiments confirm substantial speedups while maintaining solution quality.

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