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Applied mathematics and machine computations have raised a lot of hope since the recent success of supervised learning. Many practitioners in industries have been trying to switch from their old paradigms to machine learning. Interestingly, those data scientists spend more time scrapping, annotating and cleaning data than fine-tuning models. This thesis is motivated by the following question: can we derive a more generic framework than the one of supervised learning in order to learn from clutter data? This question is approached through the lens of weakly supervised learning, assuming that the bottleneck of data collection lies in annotation. We model weak supervision as giving, rather than a unique target, a set of target candidates. We argue that one should look for an ``optimistic'' function that matches most of the observations. This allows us to derive a principle to disambiguate partial labels. We also discuss the advantage to incorporate unsupervised learning techniques into our framework, in particular manifold regularization approached through diffusion techniques, for which we derived a new algorithm that scales better with input dimension then the baseline method. Finally, we switch from passive to active weakly supervised learning, introducing the ``active labeling'' framework, in which a practitioner can query weak information about chosen data. Among others, we leverage the fact that one does not need full information to access stochastic gradients and perform stochastic gradient descent.