When it comes to choosing between particular algorithms, I'd typically approach a new problem starting with a very simple hypothesis space -- for example, simple logistic regression or softmax regression. (Of course, this comes all after exploring and getting familiar with the dataset.) I'd use my initial model as a benchmark and try another bunch of simple classifiers with piece-wise or non-linear hypothesis spaces like decision trees, random forests, and (Rbf kernel) SVMs. If these don't cut it, I'd explore further options including MLPs, RNNs, and ConvNets if appropriate.
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