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Sequential operator learning under dependent data

This paper establishes time-uniform self-normalized concentration bounds for stochastic processes in Hilbert spaces to provide regression-error guarantees for learning linear and nonlinear operators from dependent, sequentially collected data without requiring independence or mixing assumptions.

Original authors: Rafael Oliveira

Published 2026-08-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Rafael Oliveira

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 vast landscape of modern science, researchers often face a puzzle that looks deceptively simple: how do you learn the rules of a system when the very act of observing it changes what you see next? This question sits at the heart of adaptive learning, a field where machines do not just passively absorb static data but interact with a changing world. Imagine a scientist trying to understand the flow of a river. If they simply drop sensors at random spots, they get a scattered picture. But if they use a model to decide where to place the next sensor based on what the previous ones found, the data becomes a connected story. This is the essence of sequential learning. However, this approach introduces a mathematical headache. Most traditional learning theories assume that every piece of data is independent, like rolling a die where the next roll has no memory of the last. In the real world, especially when dealing with complex, continuous systems like weather patterns or fluid dynamics, data points are deeply linked to one another. They form a dependent chain where the past constantly influences the future, and standard tools for measuring confidence in a model often break down.

This is the specific terrain Rafael Oliveira from CSIRO Technology in Sydney has mapped out in a new study. The research tackles the problem of learning "operators," which are essentially mathematical machines that transform one entire function or shape into another. Think of an operator not as a simple calculator that turns a number into a number, but as a device that turns a whole weather map into a prediction of tomorrow's weather map. While modern artificial intelligence has made great strides in learning these complex transformations, the guarantees that these models are actually correct have largely relied on the assumption that the training data was collected independently. Oliveira's work removes that crutch. The paper provides a rigorous mathematical framework that proves these learning models can be trusted even when the data is collected in a messy, dependent sequence, where future observations are chosen based on what was learned from the past.

The core achievement of this work is the development of a new way to measure uncertainty that holds true over time, regardless of how the data is gathered. In simpler terms, the researchers derived a set of rules that act like a safety net for learning algorithms. These rules ensure that even as the algorithm learns from a stream of connected, dependent observations, it can still calculate a precise bound on how far off its predictions might be. This is a significant leap forward because it allows for "time-uniform" guarantees. Instead of just saying a model is accurate on average, the new method ensures that the model's error stays within a known, safe range at every single step of the learning process, from the first observation to the thousandth. This is crucial for applications like adaptive experimental design, where a robot might be tasked with finding the best conditions for a chemical reaction by constantly adjusting its inputs based on immediate results. Without these guarantees, the robot might wander into dangerous or unproductive territory, convinced by faulty math that it is on the right track.

The study addresses two main types of learning scenarios. First, it looks at linear relationships, which are the straight-line connections between inputs and outputs in a high-dimensional space. The researchers showed that their new method works even when the true relationship is so complex that it cannot be perfectly represented by the mathematical space the algorithm is using. This is a common real-world problem where the model is an approximation, and the new math proves that the error can still be tightly controlled. Second, the paper extends these findings to nonlinear models, which are the complex, curved relationships often found in neural networks and deep learning. By applying their new concentration bounds to these models, the author demonstrated that even when the learning process involves complex, non-linear adjustments and regularizers (mathematical penalties that keep the model from becoming too wild), the error remains predictable and bounded.

What makes this work particularly robust is that it does not rely on the data being "mixed" or random in a statistical sense. Many previous theories required the data to eventually lose its memory of the past, a condition known as mixing, which rarely happens in truly adaptive systems. Oliveira's results work without this assumption. They hold true for any predictable sequence of data, meaning the inputs and the way they are observed can depend arbitrarily on everything that happened before. This opens the door to learning from stochastic dynamical data, such as the chaotic evolution of a storm system, where the future state is a direct, dependent consequence of the current state. The paper explicitly rules out the need for independence, showing that the old requirement for random, unconnected data points is not necessary for convergence.

The researchers built their argument on a foundation of advanced probability theory, specifically extending a concept known as self-normalized concentration. In everyday terms, this is a method for measuring how much a random process deviates from its expected path, but with a twist: the measurement scale adjusts itself based on the data seen so far. By adapting this concept to infinite-dimensional spaces and vector-valued noise, the team created a tool that can handle the complexity of continuous functions. They proved that for both linear and nonlinear operators, the error in the learned model shrinks at a predictable rate as more data is collected, provided the data collection process is sufficiently informative. This means that as an adaptive system gathers more information, it becomes mathematically certain that its model is getting closer to the truth, and the bounds on its uncertainty become tighter.

The implications of this work are most immediate for fields that rely on active learning and Bayesian optimization, where the goal is to find the best possible outcome with the fewest number of experiments. In these scenarios, every data point is expensive or time-consuming to obtain, so the ability to choose the next input intelligently is paramount. The new guarantees provide the theoretical backing needed to trust these adaptive strategies in high-stakes environments. Whether it is designing a new material, optimizing a climate model, or controlling a robotic system, the ability to learn from dependent, sequential data with rigorous error bounds transforms these tasks from risky guesses into mathematically grounded procedures. The paper does not claim to have solved every problem in operator learning, nor does it suggest that these models are perfect. Instead, it offers a solid, proven framework that removes a major theoretical barrier, allowing scientists to move forward with confidence that their adaptive learning systems are behaving as expected, even in the most complex and dependent environments.

Technical Summary: Sequential Operator Learning under Dependent Data

Problem Formulation
This work addresses the challenge of learning operators $F^\star: \mathcal{U} \to \mathcal{V}$ between Hilbert spaces from sequentially collected data where observations are dependent. Unlike standard operator learning settings that assume independent and identically distributed (i.i.d.) training data, this framework considers scenarios where inputs $u_t$ and observation operators $D_t$ are predictable with respect to a filtration $\{\mathcal{F}_t\}_{t \ge 0}$ . This encompasses adaptive experimental design, active learning, Bayesian optimization, and dynamical system modeling where future sensing or forcing conditions depend on preceding data. The observation model is given by:
$y_t = D_t F^\star(u_t) + \xi_t \in \mathcal{Y}_t$
where $\xi_t$ represents conditionally sub-Gaussian noise. The setting allows for infinite-dimensional inputs and outputs, partial observations (via $D_t$ ), and structured noise, without requiring independence or mixing assumptions on the data trajectory.

Methodology
The core methodological contribution is the derivation of time-uniform self-normalized concentration bounds for stochastic processes in general Hilbert spaces with vector-valued noise.

Concentration Inequality: The author extends classical self-normalized concentration results (e.g., de la Peña et al., 2009; Abbasi-Yadkori et al., 2011) and recent vector-valued extensions to general Hilbert-valued increments. They prove that for a sequence of adapted, conditionally $\Sigma_t$ -sub-Gaussian random variables $\{\eta_t\}$ in a separable Hilbert space $\mathcal{H}$ , the self-normalized sum satisfies a high-probability bound involving the Fredholm determinant of the accumulated covariance operators. Specifically, for any boundedly invertible positive-definite operator $U$ :
$P\left[ \forall t \in \mathbb{N}, \left\| \sum_{i=1}^t \eta_i \right\|^2_{(U+V_t)^{-1}} \le 2 \log \left( \frac{\det(I + U^{-1/2}V_t U^{-1/2})^{1/2}}{\delta} \right) \right] \ge 1-\delta$
where $V_t = \sum_{i=1}^t \Sigma_i$ . The proof utilizes finite-rank approximations and the continuity of quadratic forms and Fredholm determinants.
Linear Regression: Applying the concentration bound to linear operator learning, the author derives error guarantees for estimating a bounded linear operator $F^\star$ using a Hilbert-Schmidt estimator. Crucially, the error is measured via dual pairings with trace-class test operators $\Phi$ , allowing the target $F^\star$ to lie outside the Hilbert estimation space (i.e., $F^\star$ need not be Hilbert-Schmidt). The error bound depends on the regularization parameter, the trace norm of the inverse covariance applied to the test operator, and the log-determinant term from the concentration inequality.
Nonlinear Parametric Regression: The framework is extended to nonlinear parametric models (e.g., neural operators) trained with strongly convex losses and regularizers. By leveraging an operator-valued extension of model-based RKHS constructions, the author provides convergence guarantees for the parameter estimates. The result relies on the global optimality of the solution within the parametric class and the self-normalized control of the cumulative gradient noise.

Key Results

Theoretical Bounds: The paper establishes simultaneous, time-uniform confidence bounds for both linear and nonlinear operator learning under dependent, adaptive data collection.
Relaxed Assumptions: Unlike prior work requiring trace-class kernels or i.i.d. data, this approach permits non-trace-class kernels provided the noise covariance structure ensures sufficient spectral decay (specifically, that induced proxies $M_t \Sigma_t M_t^*$ are trace class).
Convergence Rates: In a finite-dimensional persistent-excitation setting, the linear error bound yields a convergence rate of $O(\sqrt{\log t / t})$ . The author notes that general rates in infinite dimensions depend on controlling the variance-like terms and log-determinants, potentially via operator-valued analogues of maximum information gain.
Nonlinear Guarantees: For nonlinear models, the error bound includes a term dependent on the initialization distance and a stochastic term controlled by the determinant of the accumulated information matrix.

Significance and Claims
The paper claims to provide a common concentration framework for analyzing operator learning from dependent trajectories and adaptively collected observations. Its primary significance lies in:

Bridging the Gap: It addresses a gap in existing literature where theoretical guarantees for operator learning typically assume independent data, failing to cover adaptive settings common in scientific machine learning (e.g., weather forecasting, dynamical system modeling).
Infinite-Dimensional Flexibility: It allows for infinite-dimensional inputs and outputs without independence or mixing assumptions, a major step toward convergence guarantees for adaptive operator learning.
Generalization of Noise Models: By controlling Hilbert-valued noise through operator-valued sub-Gaussian covariance proxies, the results relax restrictions found in scalar-proxy analyses, accommodating scenarios like conditional mean embedding regression where Gram operators are not trace class.

The author positions these results as foundational for future work in adaptive experimental design, Bayesian optimization with neural operators, and learning guarantees from stochastic dynamical data, while acknowledging that the nonlinear results are preliminary regarding pointwise confidence bounds which require further control of variance-like evaluation terms.

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