PaperlayerView exact revision on arXiv

Source-linked AI summary

Regret-Based Max-Pressure Traffic Signal Control under Uncertain Traffic States

Chaopeng Tan, Qiongdan Hu, Meng Wang

arXiv:2610.32290v1eess.SYcs.SYmath.OC

TL;DR

Max-pressure traffic control relies on estimated states, but the consequences of state errors depend on whether they change phase rankings and how much pressure is lost. The paper develops stochastic and distributionally robust regret-based controllers; experiments report lower pressure regret, queues, and delay than comparison methods. The authors note that transferability beyond the present simulation settings still needs validation.

  • Problem

    Existing max-pressure studies address individual information components but do not jointly characterize how heterogeneous state errors affect candidate-phase rankings and associated pressure loss.

    Source: §1 · p. 2

  • Method

    The framework evaluates candidate phases by pressure loss relative to the latent-state maximizing phase, then minimizes expected regret or worst-case expected regret over a Wasserstein ambiguity set.

    Source: §1 · p. 2

  • Results

    Normalized true-state pressure regret fell about 70% versus Point-estimate MP, and in the 28-intersection SUMO network mean delay fell 18.9% with Stochastic Regret MP and 14.9% with DR Regret MP.

    Source: §5.2 · p. 14, §7 · p. 19

  • Takeaways & Limitations

    Microscopic experiments report that regret-based controllers retain benefits under detector undercount and detector failure.

    Source: §1 · p. 2

    Transferability remains to be assessed beyond the present simulation settings, including on larger heterogeneous networks, field data, and additional sensing modalities.

    Source: §7 · p. 20

Abstract

from arXiv · show

Max-pressure (MP) traffic signal control selects phases using pressure scores computed from estimated traffic states, but these states are uncertain and their consequences depend on whether they alter phase ranking and how much pressure is lost. This paper develops a decision-oriented Regret-Based Max-Pressure framework that represents uncertainty through phase-selection regret, defined as the pressure loss of a candidate phase relative to the pressure-maximizing phase under each plausible latent state. We propose Stochastic Regret MP, which minimizes expected regret under a calibrated residual distribution, and Distributionally Robust (DR) Regret MP, which minimizes worst-case expected regret over a Wasserstein ambiguity set. Robustifying regret rather than absolute pressure makes the formulation invariant to state-dependent pressure components shared by all candidate phases, thereby focusing robustness on relative phase quality. We further derive an explicit queue-linear pressure-loss bound and establish strong queue stability when the net-service reserve exceeds the resulting loss coefficient. Numerical experiments show that both methods reduce normalized true-state pressure regret by about 70% relative to Point-estimate MP, while DR Regret MP yields lower network queues than direct robustification of absolute pressure. In microscopic SUMO experiments on a 28-intersection real-world network, Stochastic Regret MP and DR Regret MP reduce mean delay by 18.9% and 14.9%, respectively, and retain benefits under detector undercount and failure. The results show that representing uncertainty through its decision consequences can improve reliable traffic control under imperfect state information.

1 Introduction

Max-pressure control depends on traffic-state estimates, but existing treatments do not jointly characterize how heterogeneous errors affect phase rankings and pressure loss. The paper represents uncertainty through phase-selection regret and develops stochastic and distributionally robust controllers.

1 Introduction

  • Existing treatments address individual information components but do not jointly characterize how heterogeneous state errors change phase rankings and associated pressure loss.A large perturbation can leave phase ordering unchanged, while a smaller one can reverse the preferred phase when competing pressures are close.Source: §1 · p. 2
  • Phase-selection regret measures a candidate phase’s pressure loss relative to the pressure-maximizing phase under each plausible latent state.Stochastic Regret MP minimizes expected regret under a calibrated residual distribution; DR Regret MP minimizes worst-case expected regret over a Wasserstein ambiguity set.Source: §1 · p. 2
  • Numerical experiments show lower true-state phase-selection loss and network queues than Point-estimate MP, while microscopic tests reduce delay and retain benefits under detector undercount and failure.The paper also reports lower network queues for regret-based robustification than for direct robustification of absolute pressure.Source: §1 · p. 2
  • The stability analysis bounds uncertainty-induced pressure loss with a queue-linear component and establishes strong stability when net-service reserve exceeds the loss coefficient.It also gives sufficient conditions for comparing the provable stability regions of Regret MP and Point-estimate MP.Source: §1 · p. 2

2 Max-Pressure Control under Uncertain Traffic States

The section sets out the movement-level queue model and max-pressure phase score, then distinguishes the estimated state used by the controller from the latent state that determines actual pressure.

2.1 Network and traffic dynamics

  • The network uses movement queues, routing proportions, and saturation service; vehicle conservation removes realized discharges from current queues and routes their fractions downstream.The control problem is to allocate service across feasible phases to create sufficient net service relative to external arrivals.Source: §2.1 · p. 3

2.2 Max-pressure phase selection

  • Each served movement contributes its saturation service times the upstream-minus-routed-downstream queue imbalance, and network pressure decomposes across intersections.This separability permits local phase decisions while acting on a network-level Lyapunov drift.Source: §2.2 · p. 3
  • Fixed positive diagonal queue weighting is supported by the same pressure and regret construction, including link-length normalization used in the microscopic experiments.The complete weighted formulation and stability result are given in Appendix C.4.1.Source: §2.2 · pp. 3–4

2.3 Estimated traffic state and point-estimate MP

  • The controller observes estimates rather than the latent state, and a joint residual vector represents queue, routing, and service errors while retaining their dependence.Queue and service errors are multiplicative, routing errors additive, and the stated residual constraints ensure unique reconstruction of latent queue and service states.Source: §2.3 · p. 4
  • Point-estimate MP applies the phase rule directly to the available estimate without aggregating residual uncertainty.The estimated pressure is evaluated at the current estimate; historical paired data are used to construct a joint empirical residual distribution.Source: §2.3 · p. 4
  • The retained local scenarios form a discrete distribution over joint error patterns, keeping queue, routing, and service errors together rather than treating them as independent marginal perturbations.Scenario construction, scaling, reduction, and projection are described as steps for producing local scenarios used to reconstruct plausible latent states.Source: §2.3 · p. 4

3 Regret-Based Max-Pressure Control

The framework represents traffic-state uncertainty by the pressure lost through phase selection, then aggregates that regret under calibrated or distributionally robust scenario probabilities. This focuses robustness on relative phase quality and connects the stochastic, DR, and support-wise formulations through nested probability sets.

3.1 Decision-relative representation of uncertain traffic states

  • Phase-selection regret is the pressure gap between a candidate phase and the scenario-best phase under each reconstructed latent traffic state.Under a fixed probability distribution, minimizing expected regret preserves the probability-weighted pressure decision because the expected scenario-best score is action-independent.Source: §3.1 · pp. 4–5

3.2 Stochastic and distributionally robust regret control

  • Stochastic Regret MP aggregates scenario regrets using calibrated probabilities, while DR Regret MP minimizes worst-case expected regret over nearby distributions in a Wasserstein ambiguity set.The ambiguity radius is a transport budget: zero recovers the calibrated distribution, and larger values allow more probability redistribution.Source: §3.2 · pp. 5, 7

3.3 Structural Advantage of Decision-Relative Robustification

  • Regret is unchanged by any state-dependent score component shared by all phases, because that component cancels between the scenario-best and candidate-phase scores.Regret and absolute-score robustification have the same optimizer when the expected state-specific best score is constant across the ambiguity set; otherwise their rankings can differ.Source: §3.3 · p. 8

3.4 Relationships among regret formulations

  • The stochastic, DR, and support-wise regret scores are ordered from lowest to highest as their admissible probability sets expand from calibrated probabilities to the full simplex.At zero ambiguity radius, DR equals the stochastic score; when the transport budget reaches the support-wise limit, DR reaches maximum regret over the retained scenarios.Source: §3.4 · p. 9

4 Network Stability

The stability analysis links uncertain phase selection to queue drift through achieved pressure, then bounds the resulting pressure loss and compares its stability conditions with Point-estimate MP.

4.1 Stability framework

  • Under bounded network primitives, quadratic Lyapunov drift is bounded by a queue-independent constant plus arrivals minus the pressure achieved by the selected phase.With perfect information, all regret-based controllers recover the true-state MP phase; under the stated reserve and i.i.d. conditions, this benchmark strongly stabilizes the network.Source: §4.1 · pp. 9–10

4.2 Conditional pressure performance under uncertainty

  • Finite residual and probability-coverage errors yield a bound on conditional pressure loss because minimizing conditional expected regret is equivalent to maximizing conditional expected pressure.The resulting controller-specific loss coefficient is carried into the network stability analysis.Source: §4.2 · pp. 10–11

4.3 Queue-linear pressure loss and strong stability

  • A queue-linear pressure-loss bound leads to strong stability when the comparison policy’s net-service reserve exceeds the queue-proportional uncertainty-loss coefficient.The bound follows from affine pressure sensitivity to queue magnitude under the stated reconstruction-sensitivity assumption.Source: §4.3 · p. 12

4.4 Comparison with Point-estimate MP

  • Under the stated coefficient conditions, Regret MP has a no-larger queue-linear loss coefficient than Point-estimate MP, yielding an inclusion between their provable stability regions.The comparison depends on local uncertainty-approximation conditions and uses the same comparison-policy reserve criterion.Source: §4.4 · p. 13

5 Numerical Experiments

Numerical experiments compare regret-based controllers with point-estimate and pressure-robust alternatives under traffic-state uncertainty. The regret-based methods reduce normalized true-state pressure regret by about 70%, while decision disagreement remains limited even as uncertainty varies.

5.1 Experimental setting

  • The numerical study uses an eight-intersection corridor with alternating off-peak and peak demand, stochastic service and routing, and residual scenarios calibrated from historical errors.Runs use 5,000 control steps, discard the first 500 as burn-in, and average over 20 error environments with two common traffic seeds each.Source: §5.1 · p. 13, Figure 2 · p. 14

5.2 Results and analysis

  • DR Regret MP yields lower queues than DR-Pressure using the same latent-state scenarios, nominal probabilities, scenario metric, and ambiguity radius.The comparison supports focusing robustness on phase-score differences rather than pressure components shared across candidate phases.Source: §5.2 · p. 14
  • At the largest residual magnitude, plausible-state best phases vary in about 26% of cases while Stochastic and DR Regret MP decisions differ in less than 1%.When online probabilities shift from calibration, DR Regret MP degrades slightly less than Stochastic Regret MP.Source: §5.2 · pp. 14–15
  • About 70% lower normalized true-state pressure regret and about 15% lower mean network queue are reported for Stochastic Regret MP and DR Regret MP relative to Point-estimate MP.Their baseline performance is nearly identical when online scenario probabilities remain representative.Source: §5.2 · p. 14, Figure 3 · p. 15
  • As the Wasserstein-radius multiplier increases, DR Regret MP moves toward the support-wise benchmark while mean network queue changes by only about 1%.Stochastic–DR disagreement is concentrated in the lowest regret-margin decisions and becomes negligible when the leading phase has a clear score advantage.Source: §5.2 · pp. 15–16

6 Microscopic Simulation Evaluation

The microscopic evaluation tests the regret-based controllers on a 28-intersection SUMO network, measuring baseline performance, intervention timing, and robustness to sensing degradation. The experiments report lower delay and queues than Point-estimate MP, with benefits shaped by where and when improved phase decisions occur.

6 Microscopic Simulation Evaluation

  • In the 28-intersection SUMO evaluation, demand varies over 7200 s and phase decisions occur every 15 s; uncertainty calibration and baseline evaluation use disjoint runs.Calibration uses three independent runs, and baseline evaluation uses three disjoint runs.Source: §6 · p. 16

6.1 Baseline closed-loop performance

  • 18.9% and 14.9%: Stochastic Regret MP and DR Regret MP reduce mean delay relative to Point-estimate MP with complete unbiased detectors.Both also reduce travel time and time-mean network total movement queue.Source: §6.1 · p. 16, Figure 5 · p. 17
  • Queue benefits accumulate through repeated phase choices that alter movement discharge and downstream arrivals, progressively separating network trajectories from Point-estimate MP.Five intersections account for 80% of positive Stochastic Regret MP benefit and six for 80% of positive DR Regret MP benefit; high-impact locations tend to have larger queues and estimation errors.Source: §6.1 · p. 17

6.2 Timing of regret-control intervention

  • Activating either regret-aware controller at 2400 s recovers roughly 170–190 veh·h of subsequent queue burden, while activation at 6000 s leaves too little time to offset accumulated congestion.The benefit is path dependent: earlier local improvements alter downstream arrivals and later decisions.Source: Figure 6 · p. 18, §6.2 · p. 18

6.3 Performance under sensing degradation

  • Stochastic Regret MP and DR Regret MP remain better than Point-estimate MP across tested detector-undercount settings and retain benefits under detector failure.At 15% detector failure, they reduce mean delay by 6.0% and 14.8%, respectively, relative to Point-estimate MP; performance dispersion across layouts also widens.Source: §6.3 · p. 18, Figure 7 · p. 19

7 Conclusion

The conclusion presents regret-based max-pressure as a decision-oriented treatment of traffic-state uncertainty, reports benefits in numerical and microscopic experiments, and identifies extensions and validation still needed. The appendix passages describe residual-scenario construction and preserve dependence among error types.

7 Conclusion

  • Phase-selection regret measures pressure loss against the latent-state pressure-maximizing phase; Stochastic Regret MP minimizes expected regret, while DR Regret MP minimizes worst-case expected regret over probability ambiguity.Regret removes state-dependent score components shared by all actions before uncertainty aggregation.Source: §7 · p. 19
  • Strong queue stability is established when conditional net-service reserve exceeds the queue-linear uncertainty-loss coefficient.The regret-margin result identifies when uncertainty perturbations cannot change the selected phase.Source: §7 · p. 19
  • In the 28-intersection SUMO network, Stochastic Regret MP and DR Regret MP reduce mean delay by 18.9% and 14.9%, respectively, relative to Point-estimate MP.The numerical experiments also reduce normalized true-state pressure regret by about 70% and mean network queues by about 15%.Source: §7 · p. 19
  • The authors identify online residual-distribution and Wasserstein-radius updates, stability analysis for time-varying uncertainty, and validation beyond the present simulation settings as future work.The proposed validation scope includes larger heterogeneous networks, field data, and additional sensing modalities.Source: §7 · pp. 19–20

A Residual Calibration and Scenario Construction

  • Retaining complete residual vectors preserves dependence among queue, routing, and service errors; scenario reduction selects representative residuals and assigns them aggregated probabilities.Nonuniform empirical weights can support regime balancing or time-of-day reweighting.Source: A Residual Calibration and Scenario… · p. 22

B Proofs and Derivations for Regret-Based Control

The derivations establish when regret minimization matches expected-score selection, explain invariance to action-shared score terms, and show that robust regret and robust absolute pressure can choose different actions. They also give a dual form for the finite transport problem.

B.1 Score-gap equivalence and robustification structure

  • For a fixed distribution, minimizing expected regret and maximizing expected score have exactly the same optimizer set because the statewise maximum-score term is action-independent.The identity follows from regret being the maximum score minus the candidate action's score.Source: §B.1 · p. 22
  • Adding the same state-dependent term to every action's score leaves regret unchanged, including under fixed-distribution expectation, ambiguity-set supremum, or support-wise maximization.The action-independent term cancels when score gaps are formed.Source: §B.1 · p. 22
  • For ε > 1/6, robust regret selects a2 while robust absolute-score maximization selects a1 in the stated two-action, two-state Wasserstein example.The robust-regret scores are 2 + 4ε for a1 and 1 + 2ε for a2.Source: §B.1 · p. 23
  • The finite transport linear-program dual uses a nonnegative multiplier for the Wasserstein budget, with each smallest feasible row multiplier equal to the maximum scenario regret minus its transport penalty.Substitution yields the stated dual form, and strong duality applies because the primal is feasible and bounded.Source: §B.1 · p. 23

C Proofs and Extensions for Network Stability

The appendix outlines technical results for the network-stability analysis, from its common drift framework through a fixed-positive-weight extension.

C Proofs and Extensions for Network Stability

C.1 Drift framework and perfect-information benchmark

The drift analysis establishes the perfect-information phase-selection benchmark and relates regret minimization to conditional-pressure maximization. It also bounds the approximation between aggregate and conditional regret.

C.1 Drift framework and perfect-information benchmark

  • With a single latent state, minimizing phase regret is equivalent to maximizing true-state phase pressure for Stochastic, DR, and Set Regret MP.The phase-independent term in regret leaves the true-state pressure ranking unchanged.Source: §C.1 · p. 24
  • The drift proof bounds the time-average queue using a finite quadratic-drift bound and a positive service reserve.The proof conditions on the queue and history before the exogenous i.i.d. routing and service pair is realized.Source: §C.1 · pp. 23–24

C.2 Conditional pressure approximation

  • Conditional-pressure maximization and regret minimization have the same phase ranking because their score difference equals the conditional-pressure gap.The finite-scenario approximation analysis bounds differences between aggregate controller regret and true conditional regret, using Lipschitz regret scores on retained support.Source: §C.2 · pp. 24–25

C.3 Queue-linear pressure loss and strong stability

The analysis bounds pressure sensitivity by a queue-linear term and uses that loss bound in a drift argument. Strong stability follows when the service reserve exceeds the resulting loss coefficient.

C.3 Queue-linear pressure loss and strong stability

  • A finite queue-linear pressure-sensitivity bound yields strong stability when the service reserve ε0 exceeds the pressure-loss coefficient δψ.The proof sums the drift inequality over time under ε0 > δψ.Source: §C.3 · pp. 25–26
  • The pressure-loss analysis bounds regret variation by twice the pressure-sensitivity modulus and then sums the bound across intersections.For additive and clipped reconstruction, the sensitivity bound includes a finite queue-independent term as well as a queue-magnitude term.Source: §C.3 · p. 25

C.4 Point-estimate comparison and fixed positive queue weighting

The point-estimate comparison derives a pressure-loss bound and compares the resulting stability reserve with regret-based controllers. The analysis also extends stability to fixed positive queue weights, including link-length normalization.

C.4 Point-estimate comparison and fixed positive queue weighting

  • The Point-estimate pressure-loss bound supports strong stability for every arrival rate in the stated provable stability region.The proof applies the network-wide loss bound in the drift comparison.Source: §C.4 · p. 26
  • The regret-based loss coefficients satisfy δSto ≤ δP and δSto ≤ δDR, giving the stated inclusions under the common reserve definition.A comparison policy with reserve ε0 > δP also has ε0 > δSto.Source: §C.4 · p. 26

C.4.1 Fixed positive queue weighting and link-length normalization

  • Weighted Regret MP strongly stabilizes the network when the comparison policy's service reserve meets the weighted loss condition.Choosing weights w_m = 1/L_i makes the weighted queue component Q_ij/L_i, so the result applies to the link-length-normalized pressure used in the microscopic experiments.Source: §C.4.1 · p. 27

D Additional Proof for the Numerical Analysis

The appendix details the detector-based estimator and residual calibration used for the numerical analysis. It also specifies how the implementation treats undercount and detector failures.

D.1 Phase-decision invariance

  • The invariance proof concludes that every competing phase retains a strictly larger perturbed score than phase s0.Source: §D.1 · p. 27

E Detector-Based State Estimation and Uncertainty Calibration

E.1 Logical-link conservation and movement composition

  • Movement composition redistributes estimated link mass without creating or removing it, with movement shares summing to the estimated link total.Source: §E.1 · p. 28

E.2 Length-normalized uncertainty state

  • Calibration retains eight representative joint residual vectors per local model and sets the operational DR radius to the 95th percentile of bootstrap Wasserstein distances.Calibration and baseline evaluation use three runs each; the operational ambiguity radius is distinct from the analytical coverage condition.Source: §E.2 · p. 28

E.3 Detector undercount

  • In the undercount experiment, selected detectors receive a fixed undercount fraction drawn once and held for the full simulation.Source: §E.3 · p. 28

E.4 Detector failure

  • For failed detectors, the estimator reconstructs unknown flows with a bounded L1 conservation program and blends reconstructed flows with historical movement-share and service information.Network-entry detectors remain protected in the formal failure experiment so external inflows stay directly observed.Source: §E.4 · p. 29
Loading 2610.32290v1…