Generative artificial intelligence, large language models and ChatGPT in musculoskeletal Oncology: Current applications and future potential.

The optimization process in deep learning can vary significantly in terms of smoothness and convergence rate, depending on various factors such as the complexity of the model, the quality/quantity of the data or the loss landscape characteristics. However, for non-convex objectives this process has often been observed to be far from smooth and steady, often dominated by discrete, successive phases. Recent studies have shed light on several key aspects influencing these phases and the overall optimization dynamics (, ; ; ; ; ; ; ; ; ). The importance of gradient noise in escaping local optima of non-convex optimization has been explored, demonstrating its role in guaranteeing polynomial time convergence to a global optimum (). The authors of the same work suggest the existence of a phase transition for a perturbed gradient descent (GD) algorithm, from escaping local optima to converging to a global solution as the artificial noise decreases. In a later work, a phenomenon called “super-convergence” has been highlighted, where models trained with a two-phase cyclical learning rate may lead to improved regularization balance and generalization (). Furthermore, recent investigations have discovered a two-phase learning regime for full-batch GD, characterized by distinct behaviors (). During the “lazy” phase , the model behaves linearly about its initial parameters (). However, optimal performance is usually achieved during the “catapult” phase , which exhibits instabilities due to the increased curvature. The findings of this study are also supported using the NTK method (), which has been proven to be an effective tool for studying deep learning dynamics in the infinite width limit. Other researchers have demonstrated that neural networks tend towards a self-organized critical state, characterized by scale invariance, where both trainable and non-trainable parameters are driven to a stochastic equilibrium (), which is also present in many biological systems. An intriguing study showed empirically that contrary to the theoretical clues on stability, GD very often trains at the “edge of stability”, which defines a regime of hypercritical sharpness (max Hessian eigen-value), that hovers just above , during which there is non-monotonic convergence (), present also for adaptive optimizers (). The authors also posed questions regarding the mystery of progressive sharpening preceding the “edge of stability” phase. A follow-up paper explains this equilibrium state using the cubic Taylor expansion, showing that GD has an inherent self-stabilizing mechanism, which decreases the sharpness whenever extreme oscillations are present (). Analyzing the dynamics of popular optimizers, such as Adam (), has also revealed insights into the non-convex learning process. For instance, the concept of rotational equilibrium studies a steady state of angular network parameter updates, offering a deeper understanding of the phase transitions experienced by the weight vector during optimization (). Additionally, the “double-descent” phenomenon has emerged as a notable discovery, revealing a critical range of network width that negatively impacts generalization performance, with implications for the success of large language models (), while this transition can also be met epoch-wise. Collectively, these findings emphasize the complexity of deep learning optimization and the existence of discrete learning phases, offering valuable insights for improving optimization algorithms and understanding deep neural network behavior. A reasonable speculation could be that most of these phase transitions largely coincide, but this investigation is beyond the scope of our study.