Lipschitz Continuity Assumption

The Lipschitz continuity assumption, a common constraint in optimization and machine learning, posits that a function's change in output is bounded proportionally to its input change. Recent research focuses on relaxing this assumption, exploring scenarios with heavy-tailed gradients, locally-defined Lipschitz constants, or even abandoning Lipschitz continuity altogether in favor of Hölder continuity or other weaker conditions. This shift is driven by the need to handle more complex, real-world data and model architectures, including deep neural networks and stochastic algorithms. Overcoming limitations imposed by strict Lipschitz assumptions is crucial for improving the robustness and efficiency of various machine learning methods and broadening their applicability.