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A Banach-Space Theory of Markovian Halpern Iteration for Non-Expansive Maps

This paper introduces a variance-reduced Markovian PAGE-Halpern method for finding fixed points of non-expansive operators in general finite-dimensional Banach spaces, achieving an $\tilde O(\epsilon^{-3})$ sample complexity and high-probability guarantees by leveraging Poisson equation analysis and norm smoothing techniques.

Original authors: Ege C. Kaya, Arda Fazla, M. Berk Sahin, Abolfazl Hashemi

Published 2026-08-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ege C. Kaya, Arda Fazla, M. Berk Sahin, Abolfazl Hashemi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of computer learning, machines often try to find a stable answer by repeatedly guessing and correcting themselves. Imagine a hiker trying to find the bottom of a valley in thick fog. If the ground slopes steadily downward, the hiker can simply keep walking in the direction of the steepest drop and will eventually reach the bottom. This is how many learning algorithms work when the problem is straightforward: every step brings them closer to a single, unique solution. However, many real-world learning tasks are not like a simple valley. Sometimes the ground is flat, or it has many different low spots, or the path forward is blocked by noise that doesn't fade away. In these difficult situations, the standard "keep walking downhill" approach can get stuck or wander aimlessly. To solve this, mathematicians developed a specific strategy called Halpern iteration. Instead of just reacting to the immediate slope, this method keeps a fixed reference point in mind—a starting anchor—and constantly pulls the current guess back toward it. This simple act of remembering where you started helps the algorithm navigate flat or tricky terrain and guarantees it will eventually settle on a specific, correct answer.

The challenge arises when the information the computer receives is not perfect. In many practical applications, like training a robot to walk or a program to play a game, the data comes from a continuous, moving sequence of events rather than a clean, random list of facts. This is known as a Markovian trajectory, where the next piece of information depends heavily on the one that came just before it. When researchers tried to apply the Halpern strategy to this kind of noisy, dependent data, they found it worked, but it was incredibly slow. To get a precise answer, the computer had to process a massive amount of data, making the method impractical for complex problems. The researchers in this study set out to fix this speed problem without losing the reliability of the method. They wanted to know if they could make the algorithm smarter about how it uses the data it already has, specifically when that data comes from a single, unbroken stream of events.

The team discovered that by changing how the algorithm estimates the next step, they could dramatically reduce the amount of data needed. Instead of treating every new piece of information as a completely fresh start, they designed a system that looks at the difference between two very similar guesses made using the exact same piece of data. Think of it like checking your speed: if you know your speed at one moment and your speed a split second later, you can calculate how much you accelerated without needing to know your exact position on the map. By focusing on these small changes rather than rebuilding the entire picture from scratch every time, the algorithm can learn much faster. The researchers proved mathematically that this approach, which they call a variance-reduced method, allows the computer to reach a precise answer with far fewer data points than before.

This improvement is significant because it works even when the mathematical rules governing the problem are complex and do not follow the simple, smooth geometry of a standard valley. In many advanced learning tasks, such as those involving maximum values or specific types of averages, the rules are "non-smooth," meaning the ground can have sharp edges or flat spots that confuse standard methods. The researchers showed that their new technique works in these difficult, jagged environments as well. They demonstrated that by measuring the progress of the algorithm in a way that respects these sharp edges, the method remains stable and efficient. This is a crucial step because it means the theory can be applied to the messy, real-world problems found in robotics and game-playing AI, where the rules are often defined by maximums and minimums rather than smooth curves.

To test their ideas, the researchers ran simulations using a simple model of a robot moving around a small, eight-state world. They compared their new, fast method against the older, slower approach. In the tests, the new method reached the desired level of accuracy using significantly fewer steps. In one scenario, the older method failed to reach a high level of precision within the time limit, while the new method succeeded every time. In another test with a more difficult, "slow-moving" environment, the new method was able to find the solution with a fraction of the data required by the old method. The results confirmed that the strategy of reusing the same data point to measure changes is not just a theoretical trick, but a practical way to make learning algorithms much more efficient.

The study also addressed a common concern in computer science: how to be sure the algorithm will work reliably, not just on average. In the real world, a single unlucky run of bad data could cause a standard algorithm to fail. The researchers proved that their method provides a strong guarantee that the algorithm will succeed with very high probability, even in the presence of noise. They achieved this by using a special mathematical tool that smooths out the rough edges of the data just enough to make the analysis possible, without changing the actual problem the computer is trying to solve. This ensures that the fast performance is not a fluke but a consistent feature of the method.

Ultimately, this work bridges a gap between elegant mathematical theory and the messy reality of continuous data streams. It shows that by carefully analyzing how errors accumulate and by using the structure of the data stream itself, we can build learning systems that are both robust and efficient. The findings suggest that for problems where the data comes from a continuous flow, such as monitoring a sensor or playing a game in real-time, there is no need to wait for massive amounts of data to get a good answer. With the right approach, the computer can learn effectively from a single, ongoing journey, making it possible to solve complex problems that were previously too slow or unstable to tackle.

Technical Summary: A Banach-Space Theory of Markovian Halpern Iteration for Non-Expansive Maps

1. Problem Statement

The paper addresses the stochastic approximation of fixed points for a non-expansive operator $T$ when the oracle samples originate from a continuing Markovian trajectory. Unlike contractive operators, where fixed points are unique and errors geometrically attenuate, non-expansive operators present challenges including non-unique fixed points, the absence of strict contraction drift, and the persistence of sampling errors without geometric decay.

The specific setting involves a Markov chain $(Y_t)_{t \ge 0}$ generating samples for an operator $H(x, y)$ , where the mean operator $Tx = \mathbb{E}_\pi[H(x, Y)]$ is non-expansive in a finite-dimensional normed space $(\mathbb{R}^d, \|\cdot\|)$ . The goal is to find an approximate fixed point $x$ such that $\|x - Tx\|$ is small, using a single, uninterrupted Markovian trajectory. The analysis must account for the martingale component of the Markovian error and its conditional bias at block boundaries, without assuming the chain restarts or reaches stationarity between iterations.

2. Methodology

2.1 Baseline: Markovian Block Halpern Iteration

The authors first analyze a direct block-minibatch implementation of Halpern iteration. Given an anchor $x_0$ and step sizes $\beta_n = n/(n+1)$ , the update is:
$x_n = (1-\beta_n)x_0 + \beta_n G_n$
where $G_n$ is a block average of $k_n$ samples collected from the Markov chain.

Analysis: Using a perturbation-level bound, the authors show that to achieve an expected last-iterate residual of order $O(\log N / N)$ , the block sizes must grow as $k_n \sim n^4$ to control the Markovian bias and variance.
Complexity: This results in a sample complexity of $\tilde{O}(\epsilon^{-5})$ .

2.2 Variance Reduction: Markovian PAGE-Halpern

To improve the sample complexity, the paper introduces a variance-reduced method based on the PAGE (Probabilistic Adaptive Gradient Estimator) framework, adapted for Markovian trajectories.

Mechanism: The estimator $v_n$ for the operator value (either $Tx_n$ or $F(x_n) = (I-T)x_n$ ) is updated recursively. At each iteration, with probability $p_n$ , a "refresh block" re-estimates the operator at the current point. With probability $1-p_n$ , a "difference block" uses the same Markov states to evaluate the operator at two consecutive query points ( $x_n$ and $x_{n-1}$ ), estimating the difference $Tx_n - Tx_{n-1}$ (or $F(x_n) - F(x_{n-1})$ ).
Poisson Decomposition: The analysis relies on the Poisson equation decomposition of Markovian noise. This allows the authors to control the conditional bias and second moments of block averages without discarding burn-in segments, treating the boundary terms as $O(k^{-1})$ .

2.3 Hilbert Space vs. Banach Space Approaches

The methodology diverges based on the geometry of the space:

Hilbert Spaces: The authors leverage the cocoercivity of $F = I - T$ . In Hilbert spaces, non-expansiveness of $T$ implies that $F$ is monotone and $1/2$ -cocoercive. This allows the fixed-point residual $\|x - Tx\|$ to be controlled directly via the operator norm $\|F(x)\|$ , enabling the use of standard PAGE residual bounds.
Banach Spaces: In general Banach spaces (e.g., $\ell_\infty$ or block-sup norms common in Reinforcement Learning), the inner-product structure and cocoercivity are absent. The authors introduce a displacement-level bound. Instead of estimating $F(x)$ , they estimate $T$ directly. The convergence analysis relies on controlling the increments of the scaled Halpern displacement ( $\Delta_n = \beta_n U_n - \beta_{n-1} U_{n-1}$ ) in the original non-expansiveness norm. This avoids the need for a hidden inner-product geometry.

2.4 High-Probability Guarantees

To establish high-probability bounds in non-smooth Banach spaces, the authors employ an auxiliary smooth norm $\|\cdot\|_s$ . The recursive estimator is controlled in this smooth norm (where martingale concentration inequalities hold), and the result is mapped back to the working norm using norm-comparison constants. This handles the non-smoothness of sup-norms without altering the fixed-point geometry.

3. Key Contributions

Perturbation and Displacement Bounds: The authors prove two types of inexact Halpern residual bounds. The first uses the magnitude of oracle errors (suitable for block methods), while the second uses increments of scaled errors (essential for variance-reduced PAGE methods).
Markovian Block Baseline: They establish that a block-minibatch Halpern method achieves an expected residual of $O(\log N / N)$ but incurs a sample complexity of $\tilde{O}(\epsilon^{-5})$ .
Hilbert-Space PAGE-Halpern: They construct a PAGE-Halpern method for Hilbert spaces using refresh and same-state difference blocks. By utilizing the Poisson equation and cocoercivity, they achieve a sample complexity of $O(\epsilon^{-3})$ without restarting the chain.
Banach-Space Extension: The primary theoretical contribution is extending the PAGE construction to general finite-dimensional Banach spaces. By shifting the analysis from the operator $F=I-T$ to the displacement of $T$ , they achieve $\tilde{O}(\epsilon^{-3})$ sample complexity in the original non-expansiveness norm, covering sup and block-sup geometries.
High-Probability Guarantees: They provide high-probability bounds for the Banach result by measuring the estimator in an auxiliary smooth norm, showing that the leading accuracy dependence remains $\tilde{O}(\epsilon^{-3})$ with only logarithmic dimension dependence.

4. Results

Theoretical Complexity

Baseline: $\tilde{O}(\epsilon^{-5})$ samples.
PAGE-Halpern (Hilbert): $O(\epsilon^{-3})$ samples.
PAGE-Halpern (Banach): $\tilde{O}(\epsilon^{-3})$ samples.
The improvement from $\epsilon^{-5}$ to $\epsilon^{-3}$ closes the gap toward the known $\Omega(\epsilon^{-3})$ lower bound for linear-span oracle models, though the paper notes the lower bound applies to a specific oracle model that is weaker than the multi-point access used by PAGE.

Experimental Validation

The authors validate the theory using two controlled examples based on an 8-state Markov chain:

Hilbert Case (Policy Evaluation): A linear, non-expansive operator in the Euclidean norm.
Banach Case (Average-Reward Control): A nonlinear operator involving a maximization step, non-expansive only in the sup norm.

Findings:

In both fast-mixing and slow-mixing scenarios, the PAGE-Halpern methods significantly outperform the block Halpern baseline.
For the Banach control problem, the Banach PAGE method reached a residual of 0.05 in every repetition within the budget, whereas the block method succeeded in only 8 out of 20 repetitions.
The experiments confirm that same-state differences become increasingly valuable as the target accuracy increases or the mixing time of the chain grows, as they avoid the high cost of reconstructing operator values from scratch.

5. Significance and Claims

The paper claims to provide a finite-sample theory for stochastic Halpern iteration driven by a single continuing Markovian trajectory. Its significance lies in:

Bridging Geometry Gaps: It extends variance-reduced fixed-point iteration from Hilbert spaces to general Banach spaces, specifically addressing the non-smooth geometries (sup-norms) prevalent in Reinforcement Learning and distributional RL.
Handling Markovian Dependence: It explicitly accounts for the conditional bias and martingale components of Markovian errors via the Poisson equation, avoiding the need for stationarity assumptions or chain restarts.
Practical Efficiency: The results demonstrate that Halpern anchoring combined with same-state variance reduction provides robust last-iterate guarantees in geometries natural to RL, improving transition efficiency by orders of magnitude compared to block methods.
Explicit Information Structure: The work clarifies that the required information structure involves stability of iterates, Poisson-based bias control, and the ability to reuse a single Markov state for multi-point oracle queries.

The authors modestly note that while their upper bound of $\tilde{O}(\epsilon^{-3})$ matches the order of the known lower bound for a specific oracle model, the lower bound does not constitute a matched minimax statement for their specific multi-point information model. However, the construction provides a concrete, implementable online procedure that achieves this improved rate.