Staggered Integral Online Conformal Prediction for Safe Dynamics Adaptation with Multi-Step Coverage Guarantees

arXiv:2604.06058 · eess.SY, cs.RO, cs.SY · Submitted 2026-04-07 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Staggered Integral Online Conformal Prediction for Safe Dynamics Adaptation with Multi-Step Coverage Guarantees".

Dev: The gist: Safety-critical control of uncertain, adaptive systems often relies on conservative, worst-case uncertainty bounds that limit closed-loop performance, and this research proposes Staggered Integral Online Conformal Prediction (SI-OCP),

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: Let's start by looking at the title and who wrote this stuff: "Staggered Integral Online Conformal Prediction for Safe Dynamics Adaptation with Multi-Step Coverage Guarantees" by Daniel M. Cherenson and Dimitra Panagou.

Dev: It suggests they're doing two main things here, first, using this staggered integral prediction method to handle the adaptation part of the system, and second, ensuring we get coverage guarantees over multiple steps rather than just one step at a time.

Taro: So when you hear "staggered," what does that mean for us in terms of how we're using this online prediction? Is it like they are predicting things in a sequence, but adjusting the threshold at each stage?

Rosa: That’s right. They update the predictions in a staggered manner as those delayed labels start to come through, and at each time step, SI-OCP calibrates a robustness margin for the future horizon.

Dev: And they provide what they call asymptotic long-run multi-step coverage of the true disturbance when that conformalized margin is applied to a robust predictive safety critical controller.

Taro: That sounds like it moves beyond just getting a good prediction right now, aiming for a kind of guaranteed performance over the whole time horizon.

Rosa: It's about achieving asymptotic long-run safety when you couple this SI-OCP framework with controllers like robust tube model predictive control. This is a big step because it ties uncertainty quantification directly into the safety mechanism itself.

Dev: What I find interesting is that they keep it agnostic to the underlying adaptation law, meaning it should work whether the system is using classical linear models or a complex deep neural network for its learning.

The paper's summary: Rosa: So what's the core of what they actually did in "Staggered Integral Online Conformal Prediction for Safe Dynamics Adaptation with Multi-Step Coverage Guarantees"?

Dev: At its heart, they developed a conformal uncertainty quantification framework specifically for online dynamics adaptation without needing measurements of the state derivatives or directly revealed residual labels.

Taro: So they figured out a way to measure how poorly the model is doing even when we can't see the true error immediately after a control action? That's where this integral non-conformity score comes in.

Rosa: Precisely. They use this integral non-conformity score to quantify the lumped effect of both the disturbance and any errors that come from how the system is learning its dynamics.

Dev: They define it as the supremum integral score over a rolling window of measured state, input, and parameter trajectories. This essentially creates a true label for a prediction made by one thread at time tk minus Tp.

Taro: That rolling window idea sounds like they are trying to build a more complete picture of what's happening in the system dynamics over that short interval.

Rosa: Yeah, and they maintain an array of P independent non-conformity thresholds, updating them according to the active thread index, which is k modulo P.

Dev: They are essentially managing multiple perspectives on the uncertainty by staggering how they update those thresholds based on when the delayed labels arrive.

The paper's improvements: Rosa: Now let's talk about what makes this work better than what was out there before, looking at the specific improvements in "Staggered Integral Online Conformal Prediction for Safe Dynamics Adaptation with Multi-Step Coverage Guarantees."

Dev: The main improvement is extending uncertainty quantification from just one step or a single prediction to something that provides multi-step coverage guarantees.

Taro: That’s what I mean, because in real control, you usually need to look ahead over a few steps to make sure you don't crash later, not just the very next moment.

Rosa: Right. They do this by updating the threshold recursively using a formula that incorporates step sizes ηk and a failure probability α.

Dev: If they use a non-increasing sequence for those step sizes, they get empirical coverage error that decreases with iterations k, which leads to what they call the asymptotic long-run coverage guarantee of one minus alpha <ref:2604.06058#pg1>.

Taro: That mathematical guarantee is important because it tells us that if we keep running the algorithm, we can trust that the system will eventually be safe in a probabilistic sense.

Rosa: It’s about getting that confidence level back when you're dealing with unknown dynamics and disturbances, which is a major practical advantage for safety critical control.

Conclusion: Dev: So to wrap up, this paper introduces the Staggered Integral Online Conformal Prediction for Safe Dynamics Adaptation with Multi-Step Coverage Guarantees as a way to quantify uncertainty without needing state derivatives or immediate residual labels.

Rosa: The key is using that integral non-conformity score and staggering the updates across P threads to get multi-step coverage guarantees, which leads to asymptotic long-run safety when combined with controllers like robust tube model predictive control.

Taro: From my view, the fact that it works for all types of online dynamics predictors, from simple models to deep neural networks, makes this framework quite versatile for real-world use cases.

Dev: It’s about integrating uncertainty quantification into the safety loop in a way that accounts for how the system is learning its own behavior in real time.

Rosa: This research shows a pathway toward more reliable control of uncertain systems by moving away from purely conservative bounds toward methods that can adapt their safety margins based on what they are actually seeing and learning.

Dev: We're looking forward to seeing how this approach handles the latency and the computational load in actual, high-speed control loops.

Taro: I just hope future work looks at ways to reduce those assumptions about the derivative of the disturbance, because that’s where we can really push this further into more complex environments.

University of Michigan

eess.SY, cs.RO, cs.SY

Submitted: 2026-04-07

Updated: 2026-10-09

Comments: Accepted to CDC 2026

Code: https://github.com/dcherenson/staggered-integral-ocp

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 81/100

The gist: The gist: Safety-critical control of uncertain, adaptive systems often relies on conservative, worst-case uncertainty bounds that limit closed-loop performance, and this research proposes Staggered

Key concepts

Online Conformal Prediction (OCP): A data-driven technique used to estimate uncertainty in predictions as new data arrives. It works without needing prior knowledge about the true distribution of errors, providing coverage guarantees for prediction intervals.
Integral Non-Conformity Score ($S_k$): This is a novel score that quantifies the cumulative error over a rolling time window. It measures how much the actual system trajectory deviates from the nominal model's prediction over that period, effectively capturing both disturbance and learning errors.
Staggered Integral Online Conformal Prediction (SI-OCP): This proposed algorithm uses multiple independent thresholds, each associated with a different 'thread' or time window. It computes an integral score for each window to provide a more robust and accurate uncertainty estimate than standard OCP alone.
Robust Tube Model Predictive Control (MPC): A safety-critical control strategy that designs a controller to keep the system within a predefined safe set, even when facing uncertainties. SI-OCP provides the necessary disturbance bounds to correctly size the 'tube' around the nominal trajectory for this MPC.

Terminology

Adaptation — 28× in this paper · explained in 1 episode
Conformal Prediction — 22× in this paper · explained in 10 episodes
Uncertainty Quantification — 9× in this paper · explained in 9 episodes
Robustness — 8× in this paper · explained in 5 episodes
Score Function — 8× in this paper · explained in 4 episodes
American Control Conference — 3× in this paper
Distribution Shift — 3× in this paper · explained in 9 episodes
Empirical Coverage — 2× in this paper · explained in 2 episodes
Exchangeability — 2× in this paper · explained in 2 episodes
IEEE Conference on Decision and Control — 2× in this paper
Quantifying Uncertainty — 2× in this paper · explained in 5 episodes
Reinforcement Learning — 2× in this paper · explained in 4 episodes
Activation Function — 1× in this paper · explained in 1 episode
Active Exploration — 1× in this paper · explained in 2 episodes
Adaptive Safety — 1× in this paper · explained in 1 episode
Control Barrier Functions (CBFs) — 1× in this paper · explained in 6 episodes
Decoupling — 1× in this paper · explained in 2 episodes
IEEE International Conference on Robotics and Automation — 1× in this paper
Meta-learning — 1× in this paper · explained in 3 episodes
Model-Agnostic — 1× in this paper · explained in 2 episodes
Regularization — 1× in this paper · explained in 3 episodes
Stochastic Dynamics — 1× in this paper · explained in 3 episodes
Verification — 1× in this paper · explained in 2 episodes

Summary

The gist: Safety-critical control of uncertain, adaptive systems often relies on conservative, worst-case uncertainty bounds that limit closed-loop performance, and this research proposes Staggered Integral Online Conformal Prediction (SI-OCP), an algorithm utilizing an integral score function to quantify the lumped effect of disturbance and learning error.

Problem Formulation

The central obstacle is that standard online conformal prediction would require a ground-truth label for the residual after each prediction, which is not directly available online because evaluating the instantaneous residual requires the state derivative, whereas only state measurements are available <ref:2604.06058#pg2> The second challenge is that safety objective is inherently multi-step because the controller reasons over a finite prediction horizon <ref:2604.06058#pg2>

System Dynamics and Adaptation

The system dynamics are considered as a continuous-time nonlinear dynamical system where the term ∆ denotes the unknown unmodeled dynamics <ref:2604.06058#pg2> The unknown dynamics are approximated with a known parametric function F parameterized by ˆθ, which is updated using an adaptive law of the form ˙ˆθ(t) = Γ(x(t), u(t), ˆθ(t)) <ref:2604.06058#pg2> The continuous-time dynamics are rewritten as x˙ = fnom + ∆, where d denotes the unknown residual disturbance <ref:2604.06058#pg2>

Online Conformal Prediction

To quantify the uncertainty of the learned dynamics, OCP is used because it is a distribution-free method that provides coverage guarantees without requiring any prior knowledge of D or assuming exchangeability <ref:2604.06058#pg2> All versions of conformal prediction make use of a nonconformity score Sk, k ∈ N, as a metric for the error between a model’s prediction and the ground truth <ref:2604.06058#pg2> The threshold is updated recursively using qk+1 = qk + ηk(1(Sk > qk) − α) <ref:2604.06058#pg2>

Staggered Integral Online Conformal Prediction (SI-OCP)

The SI-OCP framework addresses the gaps by proposing a new integral non-conformity score computed from a rolling window of measured state, input, and parameter trajectories <ref:2604.06058#pg2> At time tk, the supremum integral score is defined as Sk = sup τ1,τ2∈[tk−Tp,tk],τ1≤τ2 x(t2) − x(t1) - Z τ2 τ1 fnom(x(s), u(s), ˆθ)(s)ds <ref:2604.06058#pg2> This score serves as the true label for the prediction made by thread j at time tk − Tp <ref:2604.06058#pg2> The algorithm maintains an array of P independent non-conformity thresholds, updated according to the active thread index j = k mod P <ref:2604.06058#pg2>

Safety-Critical Control and Guarantees

The safety objective is to design a controller for the closedloop system (7) that renders the safe set S forward invariant <ref:2604.06058#pg2> The problem statement is to synthesize a dynamic robustness margin dk at each time tk such that it correctly bounds d(t) with a longrun average rate of at least 1 − α <ref:2604.06058#pg2> Theorem 1 shows that the closed-loop system satisfies the long-run probabilistic safety guarantee: lim K→∞ 1/K X K k=1 Pr D x(τ) ∈ S, ∀τ ∈ [tk, tk + Tp] ≥ 1 − α <ref:2604.06058#pg2>

Simulation and Results

The proposed algorithm is evaluated through a case study in involving a 3D quadcopter navigating to a target region while avoiding obstacles under time-varying, complex aerodynamic disturbances <ref:2604.06058#pg2> With adaptation turned on, the DNN reduces the approximation error of the unmodeled dynamics and the resulting estimated disturbance bound is lower, allowing the quadcopter to fit through the narrow gap between obstacles with a smaller tube <ref:2604.06058#pg2> In contrast, in Figure 3, with adaptation disabled and the DNN is frozen at its offline metalearned weights, the lumped disturbance is high because the frozen DNN does not accurately model the unmodeled dynamics <ref:2604.06058#pg2>

Conclusion

The effectiveness of this approach is validated through simulation of an all-layer DNN adaptive quadcopter navigating a constrained environment under unmodeled aerodynamics and time-varying wind disturbances, demonstrating that the algorithm successfully modulates robust tubes using the SI-OCP-derived disturbance bound to maintain safety <ref:2604.06058#pg2> Future work will explore methods to alleviate assumptions on the derivative of the disturbance and will incorporate partial observability of the states <ref:2604.06058#pg2>

References

[1] B. T. Lopez, J.-J. E. Slotine, and J. P. How, “Robust adaptive control barrier functions: An adaptive and data-driven approach to safety,” IEEE Control Systems Letters, vol 5, no 3, pp 1031–1036 <ref:2604.06058#pg2>

[2] D. Fan, A. Agha, and E. Theodorou, “Deep Learning Tubes for Tube MPC,” in Proceedings of Robotics: Science and Systems (RSS), 2020 <ref:2604.06058#pg2>

[3] M. Black, E. Arabi, and D. Panagou, “A fixed-time stable adaptation law for safety-critical control under parametric uncertainty,” in European Control Conference (ECC), 2021, pp 1328–1333 <ref:2604.06058#pg2>

[4] M. H. Cohen, M. Mann, K. Leahy, and C. Belta, “Uncertainty quantification for recursive estimation in adaptive safety-critical control,” in American Control Conference (ACC), 2024, pp 3885–3890 <ref:2604.06058#pg2>

[5] R. Tao, P. Zhao, I. Kolmanovsky, and N. Hovakimyan, “Robust adaptive mpc using uncertainty compensation,” in American Control Conference (ACC), 2024, pp 1873–1878 <ref:2604.06058#pg2>

[6] L. Lindemann, M. Cleaveland, G. Shim, and G. J. Pappas, “Safe planning in dynamic environments using conformal prediction,” IEEE Robotics and Automation Letters, vol 8, no 8, pp 5116–5123 <ref:2604.06058#pg2>

[7] A. N. Angelopoulos and S. Bates, “Conformal prediction: A gentle introduction,” Foundations and Trends in Machine Learning, vol 16, no 4, pp 494–591 <ref:2604.06058#pg2>

[8] I. Gibbs and E. Candes, “Adaptive conformal inference under distribution shift,” Advances in Neural Information Processing Systems (NeurIPS), pp 1660–1672 <ref:2604.06058#pg2>

[9] A. N. Angelopoulos, R. Barber, and S. Bates, “Online conformal prediction with decaying step sizes,” in International Conference on Machine Learning (ICML), vol 235, 2024, pp 1616–1630 <ref:2604.06058#pg2>

[10] F. Areces, C. Mohri, T. Hashimoto, and J. Duchi, “Online conformal prediction via online optimization,” in International Conference on Machine Learning (ICML), 2025 <ref:2604.06058#pg2>

[11] A. Dixit, L. Lindemann, S. X. Wei, M. Cleaveland, G. J. Pappas, and J. W. Burdick, “Adaptive conformal prediction for motion planning among dynamic agents,” in Learning for Dynamics and Control Conference (L4DC), 2023, pp 300–314 <ref:2604.06058#pg2>

[12] H. Zhou, Y. Zhang, and W. Luo, “Safety-critical control with uncertainty quantification using adaptive conformal prediction,” in American Control Conference (ACC), 2024, pp 574–580 <ref:2604.06058#pg2>

[13] J. Zhang, S. Z. Yong, and D.

Improvements for AI systems

The system can achieve asymptotic long-run safety by solving Problem 1, which is to Synthesize a dynamic robustness margin dk at each time tk such that it correctly bounds d(t) with a longrun average rate of at least 1 − α.
The AI system can quantify the uncertainty of the lumped residual disturbance d(t) by computing an integral non-conformity score Sk, which is defined as sup τ1,τ2∈[tk−Tp,tk],τ1≤τ2 x(τ2) − x(τ1) - Z τ2 τ1 fnom(x(s), u(s), ˆθ(s))ds.
The system can provide a multi-step horizon coverage guarantee by using the staggered approach that maintains an array of P independent non-conformity thresholds, updated according to the active thread index j = k mod P, allowing predictions at each step with multi-step coverage guarantees.
The AI system can integrate these adaptive robustness margins with safety-critical control, yielding asymptotic longrun safety when applied to controllers like robust tube model predictive control (MPC).
The system can operate effectively on complex, learned predictors, as the framework is "agnostic to the underlying adaptation law and therefore applies broadly to online dynamics predictors, ranging from classical linear-in-parameter models [3], [4], [13], [15] to recent all-layer deep neural network (DNN) adaptation schemes [16], [17]."
The system can handle unmodeled dynamics and time-varying disturbances in real time, as the SI-OCP algorithm quantifies parameter adaptation errors and stochastic disturbances in real time.

Abstract

Safety-critical control of uncertain, adaptive systems often relies on conservative, worst-case uncertainty bounds that limit closed-loop performance. Online conformal prediction is a powerful data-driven method for quantifying uncertainty when truth values of predicted outputs are revealed online; however, for systems that adapt the dynamics without measurements of the state derivatives, standard online conformal prediction is insufficient to quantify the model uncertainty. We propose Staggered Integral Online Conformal Prediction (SI-OCP), an algorithm utilizing an integral score function to quantify the lumped effect of disturbance and learning error. This approach provides long-run coverage guarantees, resulting in long-run safety when synthesized with safety-critical controllers, including robust tube model predictive control. Finally, we validate the proposed approach through a numerical simulation of an all-layer deep neural network (DNN) adaptive quadcopter using robust tube (MPC), highlighting the applicability of our method to complex learning parameterizations and control strategies.