Title: Polygonal Unadjusted Langevin Algorithms: Creating stable and efficient adaptive algorithms for neural networks. Abstract: We present a new class of Langevin based algorithms, which overcomes many of the known shortcomings of popular adaptive optimizers that are currently used for the fine-tuning of deep learning models. Its underpinning theory relies on recent advances of  Euler's polygonal approximations for stochastic differential equations (SDEs) with monotone coefficients. As a result, it inherits the stability properties of tamed algorithms, while it addresses other known issues, e.g. vanishing gradients in neural networks. In particular, we provide a nonasymptotic analysis and full theoretical guarantees for the convergence properties of an algorithm of this novel class, which we named TH$\varepsilon$O POULA (or, simply, TheoPouLa). Finally, several experiments are presented with different types of deep learning models, which show the superior performance of TheoPouLa over many popular adaptive optimization algorithms.