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Model-Adaptive and Risk-Constrained Frequency Hopping Against Predictive Jammers

Yanbo Chen, Xinjing Zhou

arXiv:2609.06514v1cs.LGeess.SP

TL;DR

Predictive-jamming-resistant frequency hopping must handle uncertainty about whether global or local contextual models fit, alongside exposure to attacked channels. D-PACT-AFH combines those models and adds risk-aware selection and policy projection; experiments report adaptation to mismatch and a controllable goodput-risk frontier. The guarantees rely on a non-anticipating attacker and accurate channel-wise risk marginals.

  • Problem

    Global models can be misspecified across observable regimes, local models incur greater finite-horizon estimation cost, and nominal utility alone does not control exposure to predictably attacked channels.

    Source: §I · p. 1

  • Method

    A Tsallis-FTRL master combines global and partitioned local learners, with marginal-risk-aware selection and a minimum-KL projection to enforce a per-slot risk budget.

    Source: §I · p. 1, §VII · p. 8

  • Results

    Experiments showed adaptation to observable model mismatch, reduced predictive exposure with negligible communication loss in Hit mode, and a controllable goodput-risk frontier across channel regimes and jammer types in Safe mode.

    Source: §VII · p. 9

  • Takeaways & Limitations

    The reported results support adapting between global and local models while using separate Hit and Safe modes to manage predictive exposure and per-slot risk.

    Source: §I · p. 1, §VII · p. 9

    The guarantees rely on a non-anticipating attacker and accurate channel-wise risk marginals; misspecification is covered only through a bounded-error extension.

    Source: §VII · p. 9

Abstract

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Adaptive frequency hopping against predictive jamming must address both model uncertainty and policy exposure: the context-loss relationship may vary across operating regimes, while persistent hopping patterns may expose high-probability channels to attack. We propose D-PACT-AFH, a model-adaptive and risk-constrained adversarial contextual-bandit framework in which a Tsallis-FTRL master combines a global linear learner with a partitioned local learner and selects the model class online. D-PACT-Hit incorporates channel-wise marginal hit risk into model selection, while D-PACT-Safe applies a minimum-Kullback-Leibler projection to enforce a per-slot risk budget. We establish estimator validity under non-anticipating attacks, an oracle decomposition relative to the better fixed base, and an exact conditional-risk guarantee for the Safe projection. Experiments across diverse channel regimes and jammer types demonstrate effective model adaptation and a controllable goodput-risk tradeoff: D-PACT-AFH recovers 95.5% of the local learner's gain under observable switching while avoiding 77.7% of its degradation in a negative-control regime.

I. INTRODUCTION

The introduction frames predictive-jamming-resistant hopping as requiring both online choice between global and local models and control of channel exposure. D-PACT-AFH combines these goals, while the system model specifies what the learner knows and observes.

I. INTRODUCTION

  • D-PACT-AFH combines global and partitioned local learners through a Tsallis-FTRL master, with hit-aware model selection and a minimum-KL projection for per-slot risk control.The paper reports conditional estimator validity, an oracle decomposition, and an exact conditional hit-risk guarantee for feasible Safe projections.Source: §I · p. 1

II. RELATED WORK

  • Prior work addresses bandit hopping, contextual model adaptation, predictive jamming, and risk constraints, but these concerns have largely been treated separately.The paper distinguishes its global learner, which pools observations across regimes, from its local learner, which specializes over partitioned context regions.Source: §II · p. 2

A. Contextual Frequency-Hopping System

B. Public Attack Prediction and Hidden Attack Realization

  • The public predictor is converted into channel-wise marginal hit risks under a fixed attack budget, and a hidden set of exactly that size is then sampled.The marginals are the exact conditional hit-risk vector under the fixed-budget attack model.Source: B. Public Attack Prediction and Hidden… · p. 2

C. Communication Reward and Bandit Feedback

  • The learner receives only the selected channel’s reward and hit indicator, not full feedback across channels.Observed loss is one minus the selected-channel reward; reward is zero under occupancy or outage and otherwise follows a capped spectral-efficiency model.Source: C. Communication Reward and Bandit… · p. 2

D. Learning Objectives

The learning objectives balance model adaptation against estimation uncertainty and attack exposure. D-PACT-AFH combines global and local contextual bases, then adjusts model selection using predictable risk.

D. Learning Objectives

  • The global model shares a common parameter across operating regimes, while the partitioned local model permits region-dependent parameters.Global pooling is efficient when the shared structure is adequate; local specialization reduces approximation bias but fragments data and increases finite-horizon uncertainty.Source: D. Learning Objectives · p. 3

IV. ONLINE MODEL SELECTION FOR ADAPTIVE FREQUENCY HOPPING

  • The online architecture combines structurally distinct contextual learners through a Tsallis-FTRL master and can incorporate predictable hit risk and a per-slot policy constraint.Its objectives include performance relative to the better fixed base on the common execution trajectory.Source: §IV · p. 3, D. Learning Objectives · p. 3

A. Global Contextual Base

  • The global base uses LC-Tsallis-INF with an online covariance estimate and assumes a shared linear representation across channels and operating regimes.Its shared representation is statistically efficient when a common linear structure is adequate.Source: A. Global Contextual Base · pp. 3–4

B. Partitioned Local-Linear Base

  • The local base partitions public features and maintains a ridge model with an intercept for each cell.Partitioning captures observable heterogeneity; regularization, optimism, forgetting, and softmax randomization limit finite-sample instability.Source: B. Partitioned Local-Linear Base · p. 4

C. Predictable Risk Signal and Model-Class Master

  • The master distributes weight over the global and local model classes and can incorporate predicted exposure relative to uniform random hopping.A nonnegative risk weight controls hit awareness, and the current predictable risk term requires no bandit estimation.Source: C. Predictable Risk Signal and… · p. 4

D. Coverage-Aware Exploration

D-PACT-AFH explores actions supported by its active bases, can project the resulting policy onto a risk-feasible set, and updates from selected-action feedback.

D. Coverage-Aware Exploration

  • Coverage-aware exploration assigns every channel probability of at least γ_t μ/K before any Safe projection.The main configuration adds no separate risk-directed exploration tilt, so D-PACT-Hit’s effect comes from model selection.Source: D. Coverage-Aware Exploration · p. 4

E. Safe Policy Projection

  • D-PACT-Safe projects the complete pre-projection policy onto the risk-feasible simplex, leaving feasible policies unchanged.When the budget is below the minimum channel risk, no feasible policy exists; the implementation uses minimum-risk channels and records the unavoidable violation.Source: E. Safe Policy Projection · p. 4

F. Bandit and Off-Policy Updates

  • After sampling an action, the algorithm updates the master from an unclipped importance-weighted estimate and updates the selected local cell’s statistics.Explicit exploration preserves common support, clipping stabilizes base updates, and the master estimator is unbiased when the executed policy has full support.Source: F. Bandit and Off-Policy Updates · p. 5

G. Complete Procedure

The analysis evaluates the procedure against continuously updated bases on the same execution trajectory and establishes conditional estimation validity under non-anticipating attacks. It also connects communication-loss comparisons to goodput.

V. PERFORMANCE ANALYSIS

  • The analysis covers estimator validity, model-selection overhead, and the effects of anchoring, clipping, exploration, Safe projection, and fallback.The complete procedure summarizes three operating modes.Source: §V · p. 5, G. Complete Procedure · p. 5

A. Information Structure and Comparator

  • The primary comparator is the better continuously updated shadow base on the common execution trajectory, not an independently executed counterfactual base.Comparison with independently executed bases requires an additional transfer term.Source: A. Information Structure and Comparator · p. 5, §V · p. 5

B. Conditional Estimation and Goodput

  • The estimator is conditionally unbiased when the executed policy assigns positive probability to the relevant channels.An attacker may depend on past interactions and current public contexts without invalidating the estimator, provided it does not react to the current sampled action.Source: B. Conditional Estimation and Goodput · p. 5
  • Communication-loss and goodput differences between policies are identical up to the positive factor W R_max.This relationship lets the loss analysis apply directly to the reported goodput metric.Source: B. Conditional Estimation and Goodput · p. 5

C. Risk-Aware Tsallis Model Selection

D. Base Updates and Execution Remainders

Clipping stabilizes base updates at the cost of downward bias, while Safe execution projects policies to control conditional hit risk when the budget is feasible. The section also characterizes infeasible budgets and risk-model error.

D. Base Updates and Execution Remainders

E. Safe Execution and Hit-Risk Control

  • The Safe projection is a unique exponential tilt when the risk constraint is active, and at the minimum feasible budget it renormalizes the policy over minimum-risk channels.Source: E. Safe Execution and Hit-Risk Control · p. 6
  • For feasible Safe rounds, the projection controls conditional hit risk; below the minimum achievable risk, every policy violates the budget and a minimum-risk fallback attains the smallest violation.The Safe projection is unique and inactive when the unprojected policy already meets the budget.Source: E. Safe Execution and Hit-Risk Control · p. 6
  • When Safe uses estimated channel risks within ε_t of the true risks in sup norm, conditional hit risk is bounded by τ_t + ε_t.Source: E. Safe Execution and Hit-Risk Control · p. 6

F. Overall Oracle Guarantee

The oracle guarantee compares D-PACT-AFH with the better fixed Global or Local base on a common execution trajectory, with implementation costs accounted for. Under specified schedules and a zero anchor floor, average excess cost relative to the better independent base converges to zero.

F. Overall Oracle Guarantee

  • The oracle decomposition compares against the better Global or Local shadow base on the common execution trajectory and includes prior, anchor, exploration, Safe-distortion, and fallback costs.Source: F. Overall Oracle Guarantee · p. 7
  • With η_t = Θ(t^-1/2), bounded expected variation, a_min = 0, and a_t, γ_t = O(t^-1/2), average excess cost relative to the better independent base converges to zero.A positive anchor floor instead contributes an explicit linear term G_c a_min T that must be retained in finite-horizon interpretation.Source: F. Overall Oracle Guarantee · p. 7

VI. NUMERICAL RESULTS

  • Numerical results use 20 matched evaluation seeds per comparison, with confidence intervals computed as 1.96s/√n; Table I gives the default simulation configuration.Methods within each comparison share channel, occupancy, and attack realizations.Source: §VI · p. 7, Table I · p. 7

A. Setup and Evaluation Protocol

D-PACT-AFH combines global and local contextual learners with risk-aware model selection and Safe projection. Across the reported regimes, it adapts to model mismatch and offers controllable goodput–risk tradeoffs.

A. Setup and Evaluation Protocol

  • The default setup uses K=12 channels over T=10^4 slots, compares six baselines with three D-PACT modes, and calibrates Safe95 at τ=0.11 on independent seeds.Safe95 selects the lowest-hit-risk Pareto-efficient budget retaining at least 95% of unconstrained Hit goodput.Source: A. Setup and Evaluation Protocol · p. 7

B. End-to-End Goodput–Risk Performance

  • D-PACT-Safe95 lowers the hit rate to 0.108 while retaining 95.9% of D-PACT-Hit's goodput, occupying a more favorable goodput–risk region than comparable low-exposure baselines.Risk-aware EXP4 and AUFH-EXP3++ have goodputs approximately 0.17 and 0.67 Mbps lower, respectively.Source: B. End-to-End Goodput–Risk Performance · pp. 7–8, Figure 2 · p. 8

D. Robustness and Safe Risk Control

  • Tightening τ from 0.14 to 0.085 lowers empirical hit rate from 0.124 to 0.0846 while goodput falls from 3.214 to 2.662 Mbps.The maximum numerical budget violation is below 10^-10.Source: D. Robustness and Safe Risk Control · p. 8
  • Under random jamming, all D-PACT variants coincide at 3.580 Mbps and a 0.0834 hit rate; across sweeping and FOLPETTI-inspired attacks, they retain high utility.Additional nominal, nonlinear, observable-switching, contaminated, exogenous-sweep, mixed, and hidden-Markov regimes show the same qualitative behavior.Source: D. Robustness and Safe Risk Control · p. 8

E. Scaling and Computational Cost

  • As K increases from 8 to 32, D-PACT-Hit goodput rises from 2.931 to 3.639 Mbps while its empirical hit rate falls from 0.199 to 0.0467.Across horizons from 10^3 to 2×10^4, Hit maintains 3.22–3.31 Mbps and Safe95 3.08–3.23 Mbps; runtime grows approximately linearly.Source: E. Scaling and Computational Cost · p. 8

C. Mechanism Validation and Model Adaptation

  • Under observable switching, D-PACT recovers 95.5% of Local-only's 0.449 Mbps gain; in the nonlinear negative control, it avoids 77.7% of Local-only's degradation.The master assigns 0.964 mass to Local under switching and 0.258 in the negative control.Source: C. Mechanism Validation and Model… · p. 8, Figure 4 · p. 9

VII. CONCLUSION

  • The analysis establishes conditional estimator validity, an oracle decomposition, and exact conditional hit-risk control; guarantees rely on non-anticipating attacks and accurate channel-wise risk marginals.Risk-marginal misspecification is covered only through a bounded-error extension.Source: §VII · p. 9

APPENDIX A PROOF OF THE RISK-AWARE TSALLIS MASTER BOUND

The proof separates predictable costs from observed-cost residuals, applies an optimistic-FTRL stability–penalty decomposition, and controls regularization changes under a local-stability condition.

  • The proof distinguishes each base's predictable look-ahead point from the residual between observed scalarized cost and its predictable part.The look-ahead point is introduced after the within-round quantity is identified as predictable before action sampling.Source: Appendix A · p. 10
  • For every fixed comparator in Δ2, optimistic-FTRL stability and penalty terms follow from first-order optimality and summed three-point inequalities.The decomposition uses the Bregman divergence generated by Φ_t.Source: Appendix A · p. 10
  • A nonincreasing learning rate controls the changing-regularizer penalty, while the assumed local-stability condition keeps the update segment in a constant-factor local neighborhood.The condition is η_t∥z_t∥*_t ≤ c0 < 1.Source: Appendix A · p. 10

APPENDIX B SAFE PROJECTION AND ORACLE DECOMPOSITION

The appendix characterizes the Safe policy as a unique minimum-KL projection and establishes conditional hit-risk control when the risk budget is feasible.

A. Safe Projection

  • The Safe projection is feasible exactly when τ≥h_min; if the pre-projection policy already meets the budget, it remains the unique optimum.Strict convexity of KL over the convex compact feasible set gives uniqueness.Source: A. Safe Projection · pp. 10–11
  • When the pre-projection policy exceeds the budget and channel risks differ, the optimizer is an exponential tilt with a unique multiplier enforcing risk τ.At τ=h_min, the limiting distribution renormalizes the pre-projection policy over minimum-risk channels.Source: A. Safe Projection · p. 11

B. Conditional Hit-Risk Guarantee

  • On feasible Safe rounds, conditional hit risk is at most τ; when τ<h_min, at least h_min−τ violation is unavoidable.A distribution supported on minimum-risk channels attains that unavoidable lower bound.Source: B. Conditional Hit-Risk Guarantee · p. 11

C. Oracle Decomposition

The proof establishes a common-trajectory oracle decomposition against the better fixed shadow base, then extends it to independently executed references. With a_min = 0 and sublinear Safe, fallback, and transfer terms, average excess cost vanishes.

C. Oracle Decomposition

  • The comparator cost is decomposed, with the first term bounded by Theorem 2 together with anchor and Local-prior perturbations.Source: C. Oracle Decomposition · p. 11
  • Support loss, infeasible budgets, and fallback execution are collected in E_fallback.Source: C. Oracle Decomposition · p. 11
  • Choosing the comparator as the better shadow base yields the common-trajectory oracle decomposition.The proof then extends the bound to independently executed references by adding the shadow/reference discrepancy.Source: C. Oracle Decomposition · pp. 11–12
  • With a_min = 0, sublinear Safe, fallback, and transfer terms imply vanishing average excess cost.Source: C. Oracle Decomposition · p. 12

APPENDIX C SUPPLEMENTARY EXPERIMENTAL RESULTS

The appendix reports saved full-run aggregates rather than rerun experiments, generally using 20 matched evaluation seeds; scaling uses 10 seeds, and Safe calibration uses disjoint seeds. It covers attacker comparisons, communication and attacked regimes, scaling, and Safe calibration.

A. Cross-Attacker Robustness

  • 3.580 Mbps and 0.0834 empirical hit rate are shared by all three D-PACT modes under random jamming; under white-box jamming, Hit is 3.230 Mbps at 0.1352 versus Safe95 at 3.105 Mbps and 0.1080.Source: A. Cross-Attacker Robustness · p. 12

B. Regime Robustness

C. Scaling and Runtime

D. Additional Ablation and Safe Calibration

  • 3.111 Mbps in the independent calibration scan and approximately 3.104 Mbps in the disjoint main evaluation are separate Safe95 values; maximum policy-budget violation across the scan is at most 10^-10.The calibration diagnostics distinguish conditional policy risk before and after projection from empirical jam-hit frequency.Source: D. Additional Ablation and Safe… · pp. 15–16, Table VIII · p. 17
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