Convergence bounds for fixed, adaptive, and stochastic gradient methods under Polyak–Łojasiewicz geometry

The Polyak–Łojasiewicz (PL) inequality offers a useful way to study gradient-based optimization beyond the setting of strong convexity, while still allowing linear convergence under suitable smoothness conditions. This study develops a unified descent-based analysis for fixed-step gradient descent, safeguarded objective-gap step rules, Armijo backtracking, and mini-batch stochastic gradient descent. For the deterministic methods, the analysis provides explicit contraction bounds that show how convergence depends on smoothness, gradient domination, and step-selection parameters. In the stochastic setting, conditional unbiasedness together with a bounded conditional variance leads to a finite-time expectation bound and an upper estimate for the asymptotic residual neighborhood. The analysis is further extended to generalized gradient inequalities. Smooth explicit gradient descent yields linear or polynomial convergence depending on the exponent, whereas superlinear behavior for exponents below one half is established separately for an implicit decrease sequence. This distinction avoids transferring continuous-time behavior directly to explicit discrete gradient iterations. For continuously differentiable squared-error models, uniform spectral bounds on the Jacobian Gram matrix are used to derive regional gradient domination relative to the zero-loss target. A complementary path-length argument gives a sufficient condition for trajectory containment and establishes convergence to zero loss under the stated regional assumptions. Numerical illustrations based on quadratic objectives, power functions, implicit recurrences, stochastic models, and a constant-kernel least-squares example are used to examine the theoretical bounds, accepted Armijo steps, backtracking costs, convergence regimes, and stochastic uncertainty.