We present a generic strategy improvement algorithm (GSIA) to find an optimal strategy of simple stochastic games (SSG). We prove the correctness of GSIA, and derive a general complexity bound, which implies and improves on the results of several articles. First, we remove the assumption that the SSG is stopping, which is usually obtained by a polynomial blowup of the game. Second, we prove a tight bound on the denominator of the values associated to a strategy, and use it to prove that all strategy improvement algorithms are in fact fixed parameter tractable in the number r of random vertices. All known strategy improvement algorithms can be seen as instances of GSIA, which allows to analyze the complexity of converge from below by Condon [14] and to propose a class of algorithms generalising Gimbert and Horn's algorithm [16, 17]. These algorithms terminate in at most r! iterations, and for binary SSGs, they do less iterations than the current best deterministic algorithm given by Ibsen-Jensen and Miltersen [18].