H-invariance theory: A complete characterization of minimax optimal fixed-point algorithms

Abstract: For nonexpansive fixed-point problems, Halpern's method with optimal parameters, its so-called H-dual algorithm, and in fact, an infinite family of algorithms containing them, all exhibit the exactly minimax optimal convergence rates. In this work, we provide a characterization of the complete, exhaustive family of distinct algorithms using predetermined step-sizes, represented as lower triangular H-matrices, which attain the same optimal convergence rate. The characterization is based on polynomials in the entries of the H-matrix that we call H-invariants, whose values stay constant over all optimal H-matrices, together with H-certificates, of which nonnegativity precisely specifies the region of optimality within the common level set of H-invariants. The H-invariance theory we present offers a novel view of optimal acceleration in first-order optimization as a mathematical study of carefully selected invariants, certificates, and structures induced by them.

• The paper introduces an algebraic framework where H-invariants and H-certificates fully characterize minimax optimality for fixed-point algorithms using lower-triangular H-matrices.
• It rigorously establishes necessary and sufficient conditions, revealing an infinite family of algorithms that achieve the optimal worst-case rate of O(1/N²).
• The study distinguishes the unique anytime optimality of the Optimal Halpern Method and demonstrates non-duality in H-invariance theory through explicit counterexamples.

Comprehensive Characterization of Minimax Optimal Fixed-Point Algorithms via H-Invariance Theory

Introduction: Minimax Optimality in Fixed-Point Algorithms

The problem of designing minimax optimal first-order algorithms for nonexpansive fixed-point problems is central to continuous optimization and monotone inclusions. Minimax optimality refers to algorithms that minimize the worst-case convergence rate over a designated problem class. For nonexpansive mappings, recent advances showed the classical Halpern iteration, the so-called Optimal Halpern Method (OHM), achieves an exact optimal rate—with squared residual decaying at $O(1/N^2)$—but follow-up work established that this optimality is not unique; infinitely many distinct algorithms can attain the same minimax rate. The paper "H-invariance theory: A complete characterization of minimax optimal fixed-point algorithms" (2511.14915) presents a rigorous and comprehensive characterization of all minimax optimal fixed-point algorithms expressible via predetermined step-size schemes, formalized as lower-triangular H-matrices.

H-Invariants and Certificates: Algebraic Structures Encoding Exact Optimality

The core contribution is an algebraic framework defining H-invariants and H-certificates, which together specify necessary and sufficient conditions for minimax optimality. Any fixed-point algorithm using a predetermined, lower-triangular H-matrix $H$ in its update rule,

$$y_{k+1} = y_k - \sum_{j=0}^k h_{k+1,j+1}(y_j - T(y_j)),$$

is characterized by a vector of polynomial invariants $P(N-1,m;H)$, where $m=1,\dots,N-1$ and $N$ is the number of operator evaluations. For minimax optimality, these must satisfy

$$P(N-1,m;H) = \frac{1}{N}\binom{N}{m+1}\qquad m=1,\dots,N-1,$$

which forms a system of homogeneous, degree-$m$ polynomial equations in the entries of $H$.

The second component, H-certificates $\lambda_{k,j}^\star(H)$, are polynomial multipliers coupling inequalities in the convergence analysis. Minimax optimality is certified if these quantities are nonnegative for all relevant indices,

$$\lambda_{k,j}^\star(H)\geq 0,\qquad k=2,\dots,N,\quad j=1,\dots,k-1.$$

Negativity in any certificate directly implies suboptimality, allowing for an explicit construction of adversarial instances.

Main Result: Necessary and Sufficient Conditions for Minimax Optimality

The main theorem establishes if and only if conditions for a fixed-point algorithm with H-matrix representation to achieve the exact optimal worst-case rate ($4R^2/N^2$ for initial distance $R$ to fixed point), rendering the search for minimax optimal schemes a purely algebraic question about H-invariants and nonnegativity of H-certificates.

Geometry and Non-Uniqueness: Convex Structure of Optimal Schemes

A notable implication is the existence of an infinite family of distinct, nonexpansive minimax optimal algorithms parametrized by the solution set of the H-invariance polynomial equations and nonnegative certificates. The optimal set forms an intersection of algebraically-defined level sets, further restricted by certificate inequalities, yielding a convex or polytopal region in the space of H-matrices.

This geometry is visually depicted in the case $N=4$, where the set of optimal H-matrices, parametrized by their independent entries, forms a convex polyhedron.

Figure 1: The optimal H-matrix set $_\star$ for $N=4$, parametrized by $(h_{1,1},h_{2,1},h_{2,2})$ and bounded by positivity constraints in H-certificates.

Special Cases: Uniqueness of Anytime Optimality and Boundary Algorithms

Among all minimax optimal algorithms, only the OHM possesses the anytime optimality property, i.e., matches the lower bound at every iterate $k$ rather than solely at terminal points. This uniqueness is rigorously proven via a recursive argument based on the structure of H-invariants. In contrast, other optimal algorithms may only achieve optimality after a fixed number of operator calls.

The framework also exposes new optimal methods with sparse certificates, leading to simple convergence proofs and extremal schemes (e.g., self-H-dual algorithms and algorithms without dual optimality). The full structure of the boundary and extremal algorithms in the feasible H-matrix set is characterized analytically.

Contradictory Claim: Non-Duality in H-Invariance Theory

A strong claim demonstrated in the paper is that the H-dual of an optimal algorithm is not necessarily optimal. Explicit counterexamples are constructed, showing that the nonnegativity of H-certificates is not preserved under anti-diagonal reflection, refuting previous conjectures that optimality would be dual-invariant.

Broader Implications and Future Directions

The theoretical framework introduced offers a mathematical lens for the study of optimal acceleration in first-order algorithms, replacing ad hoc algorithmic design with a systematic invariance-based approach. This has practical significance for computer-assisted design of optimization algorithms: instead of numerically optimizing performance, one can algebraically enumerate all optimal H-matrices for a given problem size and directly select schemes based on additional criteria (robustness, simplicity, computational efficiency, etc).

The authors suggest possible generalizations to other problem classes, including nonsmooth convex minimization and composite optimization, where non-uniqueness also occurs. Extensions to adaptive or closed-loop step-size selection, dynamic optimality (e.g., subgame perfect equilibrium notions), and broader monotone inclusion frameworks are plausible and represent natural future directions for theoretical AI and optimization research.

Conclusion

This paper delivers a definitive algebraic characterization of all minimax optimal fixed-point algorithms representable via step-size matrices, resolving the non-uniqueness puzzle and providing structural insight into optimality landscapes in first-order methods. The H-invariance framework elevates the analysis of fixed-point schemes to the study of symmetry, invariance, and certificates, offering a generalizable blueprint for identifying and classifying optimal algorithms in high-dimensional settings. The results have immediate impact on theoretical optimization and the design of robust, structure-aware AI and numerical methods.