Minimax-optimal Halpern iterations for Lipschitz maps

TL;DR

This work extends Halpern iterations to general ρ-Lipschitz maps in normed spaces, deriving tight non-asymptotic residual bounds and introducing a minimax-optimal Halpern scheme, m-opt-halpern, whose parameters are computed via a recursion $R_n^*=V_ρ(R_{n-1}^*)$ and $β_n^*=B_ρ(R_{n-1}^*)$. It uncovers regime-specific behavior: for ρ<1 a Halpern phase precedes a Banach–Picard phase, while for ρ≥1 the optimal method remains Halpern and converges toward the minimal displacement bound $r_ρ=1-1/ρ$ on bounded domains. The paper also proposes adaptive anchoring (ada-halpern) that can outperform the minimax bounds in practice, and develops alternative bounds for unbounded domains via $\flat$-opt-halpern and affine-map results ($\text{aff}$-halpern). Together, these results provide a unified, minimax-optimal framework for fixed-point iterations in general normed spaces, with concrete algorithmic prescriptions and practical variants for affine and nonlinear operators. The findings offer acceleration over classical BP methods in near-nonexpansive settings and deliver theoretically tight benchmarks for residual performance across contractions, expansions, and unbounded domains.

Abstract

This paper investigates the minimax-optimality of Halpern fixed-point iterations for Lipschitz maps in general normed spaces. Starting from an a priori bound on the orbit of iterates, we derive non-asymptotic estimates for the fixed-point residuals. These bounds are tight, meaning that they are attained by a suitable Lipschitz map and an associated Halpern sequence. By minimizing these tight bounds we identify the minimax-optimal Halpern scheme. For contractions, the optimal iteration exhibits a transition from an initial Halpern phase to the classical Banach-Picard iteration and, as the Lipschitz constant approaches one, we recover the known convergence rate for nonexpansive maps. For expansive maps, the algorithm is purely Halpern with no Banach-Picard phase; moreover, on bounded domains, the residual estimates converge to the minimal displacement bound. Inspired by the minimax-optimal iteration, we design an adaptive scheme whose residuals are uniformly smaller than the minimax-optimal bounds, and can be significantly sharper in practice. Finally, we extend the analysis by introducing alternative bounds based on the distance to a fixed point, which allow us to handle mappings on unbounded domains; including the case of affine maps for which we also identify the minimax-optimal iteration.

Figures (4)

• Figure 1: Behavior of $r\mapsto V_{\!\rho}(r)$ for different values of $\rho$. The dashed lines illustrate the convergence of the iterates $r_{n}=V_{\!\rho}(r_{n-1})$ started from $r_0=1$, towards $r_{\!\rho}\triangleq\max\{0,1-1/\rho\}\in[0,1)$, the unique solution of $r_\rho=V_{\!\rho}(r_\rho)$ (fat dot). The plot on the left is for $\rho=3/4$ with $r_{\!\rho}=0$ and a linear regime for $r\leq 1/3$ (brown), whereas the right plot is for $\rho=3/2$ with $r_{\!\rho}=1/3$ and no change of regime. In this latter case the curve $V_{\!\rho}(r)$ is tangent to the diagonal at $r_{\!\rho}$.
• Figure 2: Monotone convergence of $Q_n(\rho)$ towards $Q_\infty(\rho)$ ($n=0,1,2,3,4,6,9,13,19,32,64$). Each map $\rho\mapsto Q_n(\rho)$ increases to a maximum and then falls below 4 as $\rho\uparrow 1$. The limit $Q_\infty(\rho)$ increases throughout $[0,1)$ and converges to $e^2$ when $\rho\uparrow 1$.
• Figure 3: Comparison of m-opt-halpern and ada-halpern with banach-picard (in log-scale).
• Figure 4: Comparison of aff-halpern and $\flat$-opt-halpern with banach-picard and m-opt-halpern, for the map $T_d$ in dimension $d=100$, starting from a random initial point $x^0$ with $x^0_i\sim U[-1,1]$. Multiple runs with $\rho=0.98$ show that the residual achieved by $\flat$-opt-halpern is consistently 2 orders of magnitude smaller than banach-picard, with even larger speedups for $\rho$ closer to 1. Initially m-opt-halpern converges slightly faster than $\flat$-opt-halpern, but later the order is eventually reversed (for $\rho>1$ this occurs for $n$ larger than the 200 iterations shown in the plot).

Theorems & Definitions (50)

• Remark 1
• Lemma 1
• proof
• Proposition 1
• proof
• Remark 2
• Lemma 2
• proof
• Proposition 2
• proof