Smooth globally PLI functions are nonlinear least-squares, and so are their gradient-dominated cousins

What the math gives to ML

The paper provides a nonstandard structural result: under a semiglobal Polyak–Łojasiewicz inequality, a smooth objective can still be reduced to nonlinear least squares even when the usual global PŁI fails at large loss values. The key quantitative condition is that the gradient lower bound \(\|\nabla f(x)\|\geq \alpha(f(x)-f^*)\) may weaken far from the optimum but must satisfy \(\alpha(s)\geq c\sqrt{s}\) near \(s=0\). This explains a two-phase optimization profile observed in continuous-time LQR: approximately linear loss decrease far away, followed by exponential convergence near the optimum. A direct neural-network transfer is a semiglobal-PŁ monitor and optimizer scheduler that detects this phase transition from the empirical gradient-to-loss ratio and changes learning-rate, stopping, or noise policies accordingly.