Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints

arXiv:2610.12153 · cs.LG · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints".

Jane: The gist: MA-OPS,

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

Title and authors: Tom: Let's talk about the paper "Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints." We see the authors listed as Qian Zuo and Francesco Emanuele Stradi from the University of Edinburgh and Politecnico di Milano.

Jane: It’s good to know who is behind this work, especially since they are tackling something quite complex like episodic constrained Markov decision processes with adversarial losses.

Lu: The paper is building on previous ideas in online CMDPs, which have been studied in stochastic, adversarial, and non-stationary settings by people like Efroni et al., two thousand twenty and Qiu et al <ref:2610.12153#pg1>., two thousand twenty <ref:2610.12153#pg1>.

Meng: So, they aren't working in a vacuum; they are clearly engaging with the existing body of work on constrained decision processes.

Tom: That’s right, and this paper specifically builds on the approach Stradi et al., 2025b used to achieve a dependence on T through policy optimization <ref:2610.12153#pg3,dependence on T through policy optimization>.

Jane: What's new here is that they are focusing specifically on adversarial losses combined with stochastic hard constraints, which adds another layer of difficulty to the problem.

Lu: They are moving beyond just having a known strictly feasible baseline to situations where no feasible policy is initially known when starting off.

Meng: That’s a big step for real-world applicability because in many constrained systems, you might not have an initial perfect solution ready to go.

Tom: Exactly, and that’s why the focus on learning a baseline when none is known is so important for making this more relevant to practical scenarios.

Jane: It suggests they are addressing a real gap in how we model decision-making under these kinds of hard limitations when things aren't fully specified.

The paper's summary: Tom: So, let’s summarize the core contribution of "Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints." They propose MA-OPS, which is a new algorithm that combines an optimistic search for the Slater margin with a pessimistic evaluation of selected policies.

Jane: The goal here is to safely learn a policy that has a large feasibility margin, and then use this policy as the baseline for minimizing regret while ensuring constraints are met at every step.

Lu: Essentially, they are trying to find that sweet spot between being optimistic about the safety margin and pessimistic about the actual costs or transitions.

Meng: It’s an interesting trade-off, because you want a policy that doesn't just look good on paper but actually performs well under uncertainty.

Tom: They claim this approach lets them obtain optimal regret dependence on both the input margin d and the unknown Slater parameter rho.

Jane: That means their result isn't just tied to one margin; it’s tied to how much safety you start with and how large the theoretical maximum margin is.

Lu: The paper shows that this combination of optimistic search and pessimistic evaluation leads directly to the Oe(√T /ρ + one/(dρ)) bound for regret, which they claim is optimal up to logarithmic factors <ref:2610.12153#pg2,Oe(√T /ρ + 1/(dρ>.

Meng: So, in plain terms, they’re showing a method that systematically balances exploration for safety with evaluation for performance minimization under hard constraints.

Tom: That’s the essence of it—balancing the search and the evaluation phases to get that specific regret bound while guaranteeing constraint satisfaction at every episode.

Jane: It's a very concrete result because it gives us a clear mathematical target for what an algorithm should be achieving in this kind of setting.

The paper's improvements: Tom: Now let’s look at the specific improvements they suggest over previous work, especially Stradi et al., 2025a <ref:2610.12153#pg1>. They focus heavily on the dependence on d when d is small.

Jane: They claim a big improvement in how the dependence on d scales when it's less than or equal to one, moving from quadratic dependence to linear dependence in terms of one/d <ref:2610.12153#pg2>.

Lu: That’s a key technical detail because it means the regret grows much slower as you refine your initial margin.

Meng: Slower growth is always better for deployment because it means that small changes in our initial setup don't lead to huge performance drops.

Tom: Plus, they also noted that this analysis reduces the dependence on L in the leading term when d is less than or equal to one.

Jane: That’s good news because it simplifies the complexity of their analysis when you start with a smaller margin, making things more tractable.

Lu: They are comparing their findings against other bounds, like the DOPE+ regret bound from Yu et al., two thousand twenty-five which shows a leading term linear in one/d and an additive term quadratic in that inverse margin <ref:2610.12153#pg2>.

Meng: So they are showing their result aligns with other advanced results, which gives confidence that this is a well-supported way to improve the performance structure.

Tom: The paper also confirms that both regret terms depend only linearly on one/d for adversarial losses when d is less than or equal to one, according to Theorem one <ref:2610.12153#pg2>.

Jane: That’s a very strong statement because it solidifies the linear relationship in this important regime, confirming their analysis of the dependence on one/d <ref:2610.12153#pg2>.

Conclusion: Tom: So we’ve covered a lot about "Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints," focusing on MA-OPS and its performance bounds. To wrap up, the paper shows how this method systematically balances the search and evaluation phases to get Oe(√T /ρ + one/(dρ)) while guaranteeing constraint satisfaction at every episode <ref:2610.12153#pg2,Oe(√T /ρ + 1/(dρ>.

Jane: The main implication is that we have a much better understanding of how the input margin d influences performance in these constrained settings.

Lu: This work opens up new avenues for more robust reinforcement learning where we can handle uncertainty and hard limits better than before, especially when starting with small safety margins.

Meng: From an engineering view, it confirms that systematic approaches to learning are effective even when the problem is highly uncertain and constrained.

Lalam: And I see this framework allowing us to build models that are inherently more reliable by incorporating safety margin optimization directly into the learning process.

Tom: To summarize, "Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints" presents MA-OPS as a way to get optimal regret bounds involving the Slater margin rho and the input margin d.

Jane: It’s a very concrete contribution that gives us a clear mathematical benchmark for what good performance looks like under these difficult conditions.

Lu: We are now ready to look at what this means for future research in constrained AI, where safety is paramount.

Meng: It definitely provides a solid foundation for tackling those harder problems in the long run.

Lalam: And this paper suggests that building models that are inherently more reliable by incorporating safety margin optimization directly into the learning process is a very promising direction.

Qian Zuo Francesco Emanuele Stradi

University of Edinburgh · Politecnico di Milano

cs.LG

Submitted: 2026-10-08

Updated: 2026-10-08

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist: MA-OPS, an algorithm that combines an optimistic search for the Slater margin with a pessimistic evaluation of selected policies to safely learn a policy with a large feasibility margin,

Key concepts

Optimistic Margin Search: This phase aims to find a policy that can maintain a large safety margin, denoted by ρ. It involves selecting policies and estimating the largest attainable margin while strictly enforcing safety constraints throughout this search period.
Pessimistic Evaluation: This phase uses the policy found in the first step as a baseline and evaluates its expected costs under uncertainty. This provides a lower bound on the policy's actual margin, which is then used to guide loss minimization.
Safety Constraint Enforcement: Safety is maintained during exploration through a randomized policy selection rule based on a probability $\lambda n$. This rule ensures that the occupancy measure satisfies specific constraints in every single episode of the search process.

Terminology

Regret — 1× in this paper · explained in 3 episodes

Summary

The gist: MA-OPS, an algorithm that combines an optimistic search for the Slater margin with a pessimistic evaluation of selected policies to safely learn a policy with a large feasibility margin, attains regret Oe(√T /ρ + 1/(dρ)) while satisfying constraints at every episode

How it works

The algorithm MA-OPS first searches for a policy with a large safety margin while ensuring safety throughout this phase, and then uses this policy as the baseline for loss minimization. This approach allows the algorithm to attain a regret bound of order Oe(√T /ρ + 1/(dρ)), which is shown to be optimal in T, ρ, d up to logarithmic factors.

The core mechanism involves two phases: an optimistic margin search and a pessimistic policy evaluation.

Optimistic margin search: This phase selects a policy and estimates the largest attainable margin ρ while ensuring safety throughout this phase.
Pessimistic evaluation: This phase accounts for cost and transition uncertainty to obtain a lower bound on the selected policy’s margin, which is used as the baseline for loss minimization.

The algorithm's safety during exploration is maintained by a randomized policy selection rule where the probability of executing the baseline policy π⋄ is determined by λn = max i: wt,i>αi (wt,i − αi)/(wt,i − βi). This randomization ensures that the resulting occupancy measure satisfies G⊤qt ≤ λnβ + (1 − λn)wt ≤ α in every episode of the search.

Key Contributions and Guarantees

The paper establishes several key results regarding performance and optimality.

**: Linear 1/d dependence for S-OPS: The work establishes a Oe(√T /d) regret bound for S-OPS with high probability, which improves the dependence from 1/d2 to 1/d for d < 1 with respect to the bound provided in Stradi et al. (2025a). This is shown in Theorem 1. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al., 2025). Theorem 1 shows that both regret terms depend only linearly on 1/d for adversarial losses. The original S-OPS analysis assumes that the baseline policy’s expected costs are known exactly and gives regret Oe(L 3XpAT / min[d, d 2]) (Stradi et al., 2025a). Our analysis improves the dependence on 1/d from quadratic to linear when d < 1, and reduces the dependence on L in the leading term. The DOPE+ regret bound has a leading term linear in 1/d and an additive term quadratic in this inverse margin (Yu et al.

Improvements for AI systems

Improve policy selection using Margin-Adaptive Optimistic Policy Search (MA-OPS). This algorithm combines an optimistic search for the Slater margin with a pessimistic evaluation of the selected policies to safely learn a policy with a large feasibility margin, allowing it to attain regret Oe(√T /ρ + 1/(dρ)).
Implement safe learning of unknown constraints using MA-OPS. This system is the first algorithm capable of learning the Slater margin and the associated policy in constant time, while satisfying the constraints at each episode, guaranteeing safety throughout with a learned margin at least ρ/2 within Oe(1/(dρ)) episodes.
Optimize regret minimization by leveraging joint margin dependence. The system is designed to attain optimal regret dependence on the initial input margin d and the unknown Slater parameter ρ, while satisfying the constraints at every episode, achieving a bound of Oe(√T /ρ + 1/(dρ)).
Establish optimal lower bounds for constraint satisfaction. The system provides a matching lower bound showing that the dependence on T, d, ρ in the regret bound is optimal up to logarithmic factors, specifically attaining an omega(√T /ρ + 1/(dρ)) regret lower bound.