Pushing the Complexity Boundaries of Fixed-Point Equations: Adaptation to Contraction and Controlled Expansion

Fixed-point equations with Lipschitz operators are central to areas such as optimization, game theory, economics, and dynamical systems. When the operator is contractive or nonexpansive (i.e., when its Lipschitz constant is γ ≤ 1), decades of work have established efficient algorithms with tight oracle complexity guarantees. In sharp contrast, even mildly expansive operators (γ > 1) render fixed-point computation intractable in the worst case, with exponential oracle lower bounds. This dichotomy leaves open a fundamental question: are there intermediate regimes where efficient approximation is still possible? Our main contribution is in establishing such regimes. First, we show that a seemingly misguided idea -- running Halpern iteration with a fixed step size -- provably finds ε-approximate fixed points at near-optimal oracle complexity for contractive and nonexpansive operators, and it succeeds for mildly expansive operators up to the hardness frontier. Building on this insight, we design the Gradual Halpern Algorithm (GHAL) and its parameter-free variant AdaGHAL, which adapt automatically to the operator's Lipschitz constant and recover optimal complexity across the contractive and nonexpansive regimes, while extending guarantees into the mildly expansive case. Finally, we introduce the class of gradually expansive operators, permitting constant expansion (up to ≈ 1.4), and prove that AdaGHAL finds ε-approximate fixed points in O(1/ε) iterations -- demonstrating for the first time that efficient fixed-point computation is possible under controlled expansion significantly beyond the nonexpansive regime. Our results apply in general normed vector spaces -- including infinite-dimensional Banach spaces, and generalize to non-positively curved geodesic metric spaces.