---
title: Minimax-Optimal Halpern Scheme
url: https://www.emergentmind.com/topics/minimax-optimal-halpern-scheme
type: topic
---

# Minimax-Optimal Halpern Scheme

The minimax-optimal Halpern scheme refers to a class of fixed-point iterative algorithms for nonexpansive or Lipschitz operators that provably attain the optimal worst-case convergence rate for driving the fixed-point residual to zero. This paradigm generalizes Halpern’s classical anchoring approach, and exact characterizations exist both for nonexpansive maps in Hilbert/normed spaces and for operators relevant to monotone inclusions, variational inequalities, and convex-concave minimax optimization. The minimax-optimality is established via tight non-asymptotic bounds, algebraic invariant theory (H-invariance), and lower bounds showing that no first-order scheme can improve upon the obtained rates under black-box oracle models.

## 1. Halpern Iteration Fundamentals and Minimax Rate

Halpern’s iteration for finding $x^* = T(x^*)$ with $T$ nonexpansive ($\|T x - T y\| \le \|x-y\|$) in a normed space is anchored to an initial point $x^0$ and uses a sequence of weights $(\alpha_k)$:
\[
x^{k+1} = \alpha_{k+1} x^0 + (1 - \alpha_{k+1}) T x^k
\]
The optimal choice for minimax convergence in the nonexpansive ($\rho = 1$) case is $\alpha_{k+1} = \frac{1}{k+2}$, yielding
\[
\|x^{N} - T x^{N}\|^2 \le \frac{4 \|x^0 - x^*\|^2}{N^2}
\]
and no deterministic first-order method can improve on the $O(1/N^2)$ bound in this setting [2511.14915].

In the general Lipschitz regime ($\|T x - T y\| \le \rho \|x-y\|$, $\rho > 0$), the minimax-optimal sequence $(\beta_n^*)$ results from recursively minimizing a tight, quadratic upper bound on $\|x^n - T x^n\|$:
\[
x^n = (1-\beta_n) x^0 + \beta_n T x^{n-1}, \quad \beta_0 = 0, \quad x^0 \text{ given}
\]
with the recursion for $\beta_n^*$:
\[
\beta_n^* = \min \left\{ 1, \frac{1}{2} \left( \frac{1}{\rho} + 1 - R_{n-1}^* \right) \right\}
\]
and residual (for $\kappa = \sup_{m,n} \|T x^m - T x^n\|$):
\[
R_n^* = 1 - \beta_n^* + \rho (\beta_n^*)^2 + \rho \beta_n^* (R_{n-1}^* - 1)
\]
[2601.15996]. This recursion is tight: for each $n$, there exists a $\rho$-Lipschitz $T$ and initialization for which equality holds.

## 2. H-Invariance Theory and Complete Characterization

The exhaustive algebraic theory of minimax acceleration for nonexpansive fixed-point algorithms is provided via H-invariance [2511.14915]. Any algorithm admitting the lower-triangular "moment-mixing" form
\[
y_{k+1} = y_k - \sum_{j=0}^k h_{k+1, j+1} (y_j - T y_j)
\]
is described by its H-matrix. The family of minimax-optimal algorithms is precisely those with H-invariants:
\[
P(N-1, m; H) = \frac{1}{N} \binom{N}{m+1}, \quad m = 1, \ldots, N-1
\]
and nonnegative H-certificates $\lambda_{k,j}^*(H) \ge 0$ (unique solution to a linear-quadratic identity). Only methods with these invariants and certificates attain the guaranteed $O(1/N^2)$ rate. The classical Halpern method (OHM) and its H-dual are extremal cases.

## 3. Behavior under Contractive and Expansive Operators

For $\rho < 1$ (contractions), the minimax-optimal Halpern sequence transitions from a sublinear Halpern phase to geometric Banach–Picard iteration, as $\beta_n^*$ reaches $1$ and stays there (from the index $n_0$ for which $R_{n_0}^* \le 1/\rho - 1$). This yields rapid geometric decay $\|x^n - T x^n\| \le \kappa \rho^{n-n_0} R_{n_0}^*$. As $\rho \uparrow 1$, the Halpern phase length diverges and recovers the $O(1/n)$ rate for nonexpansive maps.

For $\rho > 1$ (expansive), the sequence $\beta_n^*$ remains strictly less than $1$, and the residual converges to $r_\rho = 1 - 1/\rho > 0$, matching the minimal displacement on bounded domains. The minimax scheme is purely Halpern throughout.

## 4. Potential-Based Analysis and Lower Bounds

The minimax-optimal rates are validated through tight potential-based analysis [2002.08872]. For monotone inclusions $0 \in F(u^*)$ (with cocoercive $F$), setting step-weight $\lambda_k = 1/(k+1)$ and backtracking on the operator constant yields
\[
\|F(u_k)\| \le O\left( \frac{L \|u_0 - u^*\|}{k} \right)
\]
The "potential" function is quadratic-minus-linear in the operator evaluations and telescopes over the run.

Lower bounds are established via reductions: for any first-order method, problems exist for which $O(1/k)$ is unimprovable (e.g., convex-concave saddle problems in $d = \Omega(k)$ dimensions) [2002.08872]. The Halpern scheme matches these bounds up to polylogarithmic factors and is parameter-free.

## 5. Splitting Schemes and Extensions

Halpern-type minimax algorithms have been extended to monotone splitting and saddle-point settings [2110.08150]. For $0 \in A(x) + B(x)$ (maximally monotone $A$, Lipschitz $B$), Halpern–Popov variants give
\[
\|G_\gamma(x_k)\|^2 = O\left( \frac{1}{k} \right)
\]
where $G_\gamma$ is the forward–backward residual. Two splitting schemes are constructed:
- Extra-Anchored Gradient-Splitting (EAG-Split): two resolvent calls per iteration.
- Past-Extra-Anchored Gradient-Splitting (PEAG-Split): only one computation each of $J_{\gamma A}$, $J_{\gamma B}$, and $B$.

Application to convex–concave minimax problems ($\min_z \max_w \{ f(z) + L(z, w) - g(w) \}$) leverages the Halpern scheme for the skew-gradient optimality operator, establishing $O(1/k)$ convergence of the gradient norm, matching known lower bounds.

## 6. Adaptive Variants and Practical Refinements

Adaptive Halpern schemes, inspired by minimax theoretical bounds, track empirical residuals $\widehat{\kappa}_n$ to compute step sizes $\beta_n$ that can outperform the non-adaptive minimax bound in practical settings [2601.15996]. The update is
\[
\beta_n = \min \left\{ 1, \frac{1}{2} \left( \frac{1}{\rho} + 1 - R_{n-1} \right) \right\}
\]
with $R_n = 1 - \beta_n + \beta_n \frac{\|T x^n - T x^{n-1}\|}{\widehat{\kappa}_n}$. This process is universally better or equal compared to the minimax rates.

Extensions further cover unbounded domains (via displacement bounds based on fixed-point distance), affine maps (closed-form for optimal $\beta_n$), and strongly monotone operators (with restarted Halpern strategies).

## 7. Comparison to Alternative Acceleration and Practical Implications

Nesterov’s acceleration for monotone inclusions yields $O(1/\sqrt{k})$ last-iterate rates unless extra structure (e.g., cocoercivity) is present. Anchored Extragradient schemes match the Halpern $O(1/k)$ rate but require twice as many operator calls. The minimax-optimal Halpern schemes uniquely combine single oracle usage, explicit algebraic invariant theory, and optimality under black-box oracle models and mere monotonicity/Lipschitz assumptions [2110.08150], [2511.14915], [2601.15996].

Halpern-type minimax schemes unify approaches for splitting, inclusion, variational inequality, and convex–concave min–max, always matching fundamental lower bounds up to log-factors. The certificate-based H-invariance characterization catalogues every possible optimal algorithm in this class. These results constitute the canonical foundation for minimax-optimal first-order fixed-point algorithms.

Source: https://www.emergentmind.com/topics/minimax-optimal-halpern-scheme