Assuming local Lipschitz continuity with polynomial growth for stochastic gradients has significant implications for optimization algorithms like TheoPouLa. By assuming this type of continuity, we relax the requirement for global Lipschitz continuity on gradients while still ensuring stability and convergence properties of the algorithm. The polynomial growth condition allows us to handle situations where gradients may grow rapidly without being globally bounded or Lipschitz continuous. This assumption enables more flexibility in handling complex optimization problems involving neural networks where traditional assumptions may not hold.