Pseudo-Iterate: Generalized Iteration Concepts

• Pseudo-iterate is a surrogate output or generalized iterate introduced when ordinary self-composition is unavailable or structurally unsuitable.
• It appears across domains such as convex optimization, chaotic iterations, bandits, and functional equations, where averaging or asynchronous updates improve convergence.
• Applications extend to fractional, mean-defined, and lifted iterates in cryptography, stochastic dynamics, and hybrid systems that require controlled, non-standard recursion.

Searching arXiv for recent and relevant papers on “pseudo-iterate” and related iteration notions across optimization, dynamics, and semantics. “Pseudo-iterate” is not a single standardized term across the arXiv literature. In some areas it denotes a surrogate output that replaces the raw last iterate; in others it denotes an asynchronously updated state, a generalized iterate obtained from a non-self-map, a fractional or mean-defined iterate, or an object that imitates genuine iteration only in law or on a cofinal class. This suggests a family of closely related ideas rather than a unique definition: the iterate-like object is introduced when ordinary self-composition, the actual terminal state, or a pathwise limit is unavailable, inconvenient, or structurally the wrong notion of output (Kornowski et al., 15 Apr 2026, Bahi et al., 2011, Goncharov, 2023).

1. Terminological scope

Several papers make explicit that the exact term is absent even when adjacent ideas are central. In convex Lipschitz optimization, “pseudo-iterate” does not appear; the closest notions are the average iterate, horizon-dependent outputs, and specially selected stopping times, all contrasted with the actual last iterate $x_T$ (Kornowski et al., 15 Apr 2026). In chaotic-iteration pseudorandom number generation, by contrast, the term is tied to a mechanism of asynchronous partial updates governed by a strategy sequence rather than to ordinary full synchronous iteration (Bahi et al., 2011). In iterative functional equations and generalized iteration theory, the relevant object may be an implicitly defined iterate, or an iterate of an associated self-map on a larger state space, rather than a direct self-composition of the original map (Draga et al., 2018, Parimoo, 2020).

Domain	Iterate-like object	Role
Convex optimization	average iterate, horizon-dependent output, selected stopping time	surrogate output distinct from the actual last iterate
Chaotic iterations / PRNGs	strategy-driven partial update $F_f(s,x)$, global map $G_f$	asynchronous update mechanism
Bandits and games	Bregman-divergence proxy, transformed played average $\bar x^t$	surrogate of last-iterate convergence
Functional and generalized iteration	mean-defined iterate, iterate of an associated self-map	extension beyond ordinary self-composition
Logic and stochastic dynamics	cofinally equivalent operators, occupation-measure limits	iterate-like behavior without a single ordinary iterate

A useful unifying distinction is between actual iterates and constructed iterate-like outputs. The former are the literal states generated by repeated application of an update rule. The latter include averages, selected subsequences, auxiliary state-space lifts, and coordinate systems such as Abel functions. The literature repeatedly treats these constructed objects as necessary precisely when naïve iteration loses desirable structural properties.

2. Surrogate outputs in optimization and online learning

In first-order convex optimization, the distinction between the actual last iterate and surrogate outputs is especially sharp. The setting is projected subgradient descent

$x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$

for convex $G$-Lipschitz $f$ on a closed convex set $X$ of diameter at most $D$, with last-iterate error

$f(x_T)-\min_{x\in X}f(x).$

The paper “Gradient Descent’s Last Iterate is Often (slightly) Suboptimal” proves that no stepsize sequence admits an anytime guarantee

$F_f(s,x)$0

so neither GD nor SGD can achieve the information-theoretically optimal $F_f(s,x)$1 last-iterate rate in an anytime fashion. By contrast, the average iterate attains the classical $F_f(s,x)$2 rate, standard anytime last-iterate schedules yield $F_f(s,x)$3, and Jain et al.’s horizon-dependent construction recovers $F_f(s,x)$4 only when $F_f(s,x)$5 is known in advance (Kornowski et al., 15 Apr 2026). In this setting, the pseudo-iterate idea is therefore naturally realized by averaging, modified output rules, or specially chosen stopping times rather than by the raw terminal state.

A related but distinct surrogate appears in stochastic bandits. “Last Iterate Analyses of FTRL in Stochastic Bandits” studies the actual probability iterate $F_f(s,x)$6 of $F_f(s,x)$7-Tsallis-INF, but the main theorem controls not simple regret directly but the expected Bregman divergence

$F_f(s,x)$8

showing

$F_f(s,x)$9

The paper explicitly presents this divergence as a proxy or surrogate of last-iterate convergence; the rigorous simple-regret consequence is only $G_f$0, even though logarithmic regret suggests a possible $G_f$1 law (Zhan et al., 26 Oct 2025). Here the pseudo-iterate is not a new state, but a surrogate geometry of the actual iterate.

In games, the surrogate can be made algorithmic. “From Average-Iterate to Last-Iterate Convergence in Games” introduces a reduction A2L in which an internal learner produces iterates $G_f$2, while the algorithm actually plays

$G_f$3

Theorem 1 shows that for the transformed dynamics

$G_f$4

so the played last iterate of the new dynamics is exactly the average iterate of the original one. Applied to OMWU, this yields an $G_f$5 last-iterate rate under gradient feedback and a $G_f$6 rate under bandit feedback in the stated zero-sum settings (Cai et al., 4 Jun 2025). This is one of the clearest constructions of a deliberately engineered pseudo-iterate: the reported iterate is a transformed history of an internal sequence.

3. Chaotic iterations and asynchronous pseudo-iterates

In cryptographic pseudorandom number generation, the term acquires a more literal dynamical meaning. “Class of Trustworthy Pseudo-Random Number Generators” studies discrete chaotic iterations on $G_f$7, where a Boolean state $G_f$8 evolves by updating only one coordinate at each time according to a strategy sequence $G_f$9. For

$\bar x^t$0

the one-coordinate update map is

$\bar x^t$1

and the iteration is

$\bar x^t$2

The induced global map on strategies and states is

$\bar x^t$3

Here the pseudo-iterative aspect is that the system does not synchronously apply $\bar x^t$4 to all coordinates; it repeatedly applies a partial update rule determined by $\bar x^t$5 (Bahi et al., 2011).

The paper’s main structural criterion is graph-theoretic. If $\bar x^t$6 is the iteration graph whose vertices are Boolean states and whose labeled arcs record one-bit transitions induced by $\bar x^t$7, then

$\bar x^t$8

Thus the admissible class of iterate functions is all $\bar x^t$9 with strongly connected iteration graph. This generalizes the earlier exclusive use of vectorial Boolean negation.

The paper then shows that strong connectivity is sufficient for topological chaos but not for statistical quality. Candidate functions are generated by removing edges from $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$0 while preserving strong connectivity, with Tarjan’s algorithm used as a practical SCC test. Statistical suitability is then evaluated by deviation from uniformity and by NIST SP 800-22. Among the listed $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$1 examples, only $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$2, $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$3, and $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$4 pass all $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$5 NIST tests, and these are exactly the functions with deviation less than $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$6 (Bahi et al., 2011). In this literature, “pseudo-iterate” therefore denotes an asynchronous update mechanism whose credibility depends jointly on graph-theoretic chaos and distributional mixing.

4. Generalized, mean-defined, and fractional iteration

A different tradition treats pseudo-iteration as a way of extending iteration beyond ordinary self-maps. “Means of iterates” studies continuous bijections $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$7 such that the $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$8-th iterate is the quasi-arithmetic mean of the finite orbit segment: $x_{t+1}=\operatorname{Proj}_X(x_t-\eta_t g_t)$9 After conjugation by $G$0, this becomes

$G$1

Under $G$2 and excluding only the case where both $G$3 and $G$4 are even, the continuous surjective solutions are classified as translations, affine maps, or three-piece affine maps; in arithmetic coordinates every solution has at most three affine pieces (Draga et al., 2018). The iterate is genuine, but it is prescribed implicitly by a mean relation rather than direct composition.

For higher-arity maps $G$5, ordinary self-composition is not type-correct, so a state-space lift is required. “Iteration of Functions $G$6 and their Periodicity” defines a self-map $G$7 recursively by

$G$8

and defines $G$9 as the $f$0-fold iterate of this associated self-map. This encodes recurrences

$f$1

The paper proves that if $f$2 is $f$3-involutory in each of its $f$4 arguments, then $f$5 is $f$6-involutory (Parimoo, 2020). Here the pseudo-iterate is the iterate of a canonically associated state update on $f$7.

A formal version of continuous iteration appears in “Tetration: an iterative approach”. For formal series with zero constant term, Bell matrices satisfy

$f$8

and for general series $f$9, Carleman matrices satisfy

$X$0

Noninteger iterates are then defined by matrix powers, giving an order-by-order route to continuous iteration and tetration (Aldrovandi, 2014). This is an explicitly linearized pseudo-iteration: composition is transported into matrix multiplication, and arbitrary iterate index is recovered by matrix functional calculus.

A computationally different route to fractional iterates uses Abel functions. “Half-Iterates of $X$1, $X$2 and $X$3” studies Abel equations

$X$4

and compositional square roots

$X$5

Once $X$6 is known, one obtains

$X$7

with the appropriate sign convention. The paper compares the Écalle–Jagy method and the Mavecha–Laohakosol method, showing that they may produce Abel functions differing by an additive constant $X$8, while the induced fractional iterate is unchanged when the normalization is used consistently. In the examples treated, nonzero constants $X$9 and $D$0 arise for sine and the exponential/logarithmic pair, while $D$1 for the logistic/radical pair (Finch, 9 Jun 2025). Fractional iteration is thus canonical only modulo translation in Abel coordinates.

5. Pseudo-limits, evitable iterates, and eventual regularity

In stochastic dynamics, the pseudo-iterate may be an iterate-like limit that does not exist as a path-valued object. “Iterating Brownian motions, ad libitum” studies

$D$2

for independent two-sided Brownian motions. The sequence $D$3 does not converge in any usual functional sense and is not tight in compact-uniform topology, yet finite-dimensional marginals converge, and the occupation measures

$D$4

converge in distribution to a random probability measure $D$5. That limit almost surely has a compactly supported continuous density $D$6, Hölder $D$7 for every $D$8, interpreted as the local time process of the infinitely iterated Brownian motion (Curien et al., 2011). The infinite iterate therefore exists only as a pseudo-limit described by finite-dimensional laws and occupation measures.

In arithmetic logic, the pseudo-iterate is an operator that imitates genuine iterates without being determined by a single iterate. “Evitable iterates of the consistency operator” fixes a sound recursive extension $D$9 of $f(x_T)-\min_{x\in X}f(x).$0 and studies recursive monotone operators on the Lindenbaum algebra. For every $f(x_T)-\min_{x\in X}f(x).$1, the paper constructs a recursive monotone $f(x_T)-\min_{x\in X}f(x).$2-valued operator $f(x_T)-\min_{x\in X}f(x).$3 such that on cofinally many true $f(x_T)-\min_{x\in X}f(x).$4,

$f(x_T)-\min_{x\in X}f(x).$5

while on another cofinal class,

$f(x_T)-\min_{x\in X}f(x).$6

These are evitable iterates: they are cofinally as strong as $f(x_T)-\min_{x\in X}f(x).$7 and cofinally as weak as $f(x_T)-\min_{x\in X}f(x).$8, so cofinal comparison does not uniquely characterize higher iterates of the consistency operator (Walsh, 2022).

In algebraic geometry, a related phenomenon is “eventual regularity.” “Rational self-maps of projective surfaces with a regular iterate” studies dominant rational self-maps $f(x_T)-\min_{x\in X}f(x).$9 for which some iterate $F_f(s,x)$00 is regular. If some iterate is regular and non-invertible, then one can choose such an $F_f(s,x)$01 with

$F_f(s,x)$02

and the bound is sharp on $F_f(s,x)$03. The paper’s classification shows that the only genuine pseudo-iterate phenomena for non-invertible iterates occur on toric surfaces. In the birational case on $F_f(s,x)$04, if $F_f(s,x)$05 preserves no non-constant rational fibration and some iterate is regular, then $F_f(s,x)$06 itself is regular (Saleh, 5 Sep 2025). Here the iterate becomes more regular than the original map, but only under tightly constrained geometry.

6. Abstract semantics and conceptual synthesis

Category-theoretic semantics treats pseudo-iteration as a controlled weakening or generalization of ordinary iteration. “Shades of Iteration: from Elgot to Kleene” distinguishes Elgot iteration

$F_f(s,x)$07

from Kleene iteration

$F_f(s,x)$08

It introduces while-monads with an operator

$F_f(s,x)$09

and proves that, under the stated expressive decision hypothesis, Elgot monads and while-monads are equivalent. The translations are explicit: $F_f(s,x)$10 and

$F_f(s,x)$11

Kleene monads are then characterized as Elgot monads with join-semilattice enrichment, the normalization

$F_f(s,x)$12

and strong uniformity (Goncharov, 2023). In this abstract setting, a pseudo-iterate is a lawful loop or recursion operator that exists before, or even without, full Kleene-star algebra.

“A Semantics for Hybrid Iteration” gives a concrete semantic realization of this idea for hybrid systems. Guardedness is interpreted as progressiveness in time, so recursive unfolding is allowed only when it is temporally productive. The modified hybrid monad $F_f(s,x)$13 supports guarded iteration on progressive maps as the least solution of

$F_f(s,x)$14

while the larger monad $F_f(s,x)$15 supports total Elgot iteration via an iteration-congruent retraction. This framework admits open trajectories, finite-time divergence, and Zeno behavior; total hybrid iteration decomposes as

$F_f(s,x)$16

that is, into progressive iteration plus a singular zero-duration component (Goncharov et al., 2018). The hybrid case makes especially explicit that pseudo-iteration is often an admissibility-controlled replacement for naïve unrestricted recursion.

This suggests that “pseudo-iterate” is best understood as a family-resemblance concept. In optimization it is a surrogate output replacing a provably suboptimal last iterate; in chaotic PRNGs it is an asynchronous partial update; in generalized iteration it is a lifted, mean-defined, or fractional extension of composition; in logic and probability it is an iterate-like object that exists only cofinally or in distribution; and in categorical semantics it is a guarded or while-based recursion operator that relaxes the requirements of total Kleene-style iteration.