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Quadratic Łojasiewicz loss landscapes and application to trust-region algorithms
Abstract
Classical optimization algorithms can see their local convergence rates deteriorate when the Hessian at the optimum is singular. The latter is inescapable when the optima are non-isolated. Yet, several algorithms behave perfectly nicely even when optima form a continuum (e.g., due to overparameterization). This has been explained under various structural assumptions, including the quadratic Łojasiewicz inequality. I will present its relationships to other standard conditions in the optimization literature, and the strong properties it implies on the landscape of the cost function. Then, I will discuss applications to trust-region methods, for which we can secure local super-linear rates.
