On incremental and semi-global exponential stability of gradient flows satisfying generalized Åojasiewicz inequalities
Provides theoretical convergence rates in state space for nonconvex gradient flows, addressing a gap in existing Łojasiewicz-based analyses that only give objective-value convergence.
This paper uses contraction theory to derive state-space convergence guarantees for gradient flows satisfying generalized Łojasiewicz inequalities, showing semi-global exponential stability for strongly convex objectives and global incremental exponential stability under curvature-based conditions.
The Åojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly give rates of convergence in the state. In this paper, we use contraction theory to derive state-space guarantees for gradient systems satisfying generalized Åojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Åojasiewicz inequality implies semi-global exponential stability; on arbitrary compact subsets, this yields exponential stability. We then give two curvature-based sufficient conditions, together with constraints on the Åojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable.