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arxiv: 2604.11016 · v1 · submitted 2026-04-13 · 📡 eess.SP

Learning an Opponent-aware Anti-jamming Strategy via Online Convex Optimization

Liangqi Liu , Wenqiang Pu , Yingru Li , Zhi-Quan Luo This is my paper

Pith reviewed 2026-05-10 16:31 UTC · model grok-4.3

classification 📡 eess.SP
keywords online convex optimizationanti-jammingDRFM jammerfrequency agile radarregret boundgradient estimatoradversarial strategy
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The pith

Incorporating unbiased gradient estimators tailored to DRFM jammers into online convex optimization yields lower regret and faster anti-jamming convergence for frequency-agile radars.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper frames the dynamic competition between a frequency-agile radar and an intelligent DRFM-based jammer as an online convex optimization problem. It then develops two refined OCO algorithms that replace standard gradients with unbiased estimators built from the jammer's specific structure. Theoretical analysis shows these changes improve the regret bound relative to conventional OCO. Simulations confirm faster convergence and stronger anti-jamming performance than both standard OCO and reinforcement-learning baselines. A reader would care because radars facing adaptive jammers need sample-efficient strategies that exploit rather than ignore adversary structure.

Core claim

The paper claims that two refined online convex optimization algorithms, each using unbiased gradient estimators constructed specifically for the characteristics of DRFM-based jammers, produce a significantly tighter regret bound and deliver measurably better anti-jamming performance than either standard OCO or reinforcement-learning methods.

What carries the argument

Two refined OCO algorithms that replace conventional gradients with unbiased estimators exploiting the structure of DRFM-based jammers, thereby capturing the radar-jammer interaction more precisely within the convex framework.

If this is right

  • The refined estimators produce a measurably tighter long-term regret bound than standard OCO.
  • Convergence to effective anti-jamming policies occurs in fewer steps than with conventional methods.
  • Simulation performance against DRFM jammers exceeds that of both standard OCO and reinforcement-learning baselines.
  • The approach yields opponent-aware strategies that exploit rather than ignore the jammer's memory and replay behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same structure-aware estimator technique could be applied to other online learning settings where partial knowledge of an adversary's update rule exists.
  • Real-time radar systems might achieve lower training overhead by embedding these estimators instead of relying on generic gradient methods.
  • The framework invites hybrid designs that combine OCO regret guarantees with occasional model-based corrections when jammer behavior drifts.

Load-bearing premise

The radar-jammer interaction must be accurately representable as a convex optimization problem and the jammer's structure must allow construction of unbiased gradient estimators inside that framework.

What would settle it

A simulation run in which the refined algorithms produce regret curves or anti-jamming success rates no better than those of standard OCO would falsify the claimed improvement.

Figures

Figures reproduced from arXiv: 2604.11016 by Liangqi Liu, Wenqiang Pu, Yingru Li, Zhi-Quan Luo.

Figure 1
Figure 1. Figure 1: Illustration of the anti-jamming scenario. (The radar [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Post-processing for the received signal. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SINR comparison for OMD-IWE and OME-AME in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simulated single-round spectrograms of radar and three [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Regret comparison in a stationary environment. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Regret comparison in a non-stationary environment. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: SINR Comparison jammers, we develop two novel algorithms that outperform conventional OCO benchmarks. Sub-linear static and universal regret bounds are provided, and numerical simulations demon￾strate a significant enhancement in sample efficiency. APPENDIX A. Post-processing Details For the n-th pulse with sub-frequencies f R m, ∀m ∈ [M], the received signal undergoes bandpass filtering at each frequency:… view at source ↗
read the original abstract

The dynamic competition against intelligent jammer systems presents a significant challenge to modern radar. Traditional active anti-jamming strategy learning methods often suffer from low sample efficiency and fail to fully exploit the structures of the adversary jammer. To reveal the inherent structure, this paper adopts an Online Convex Optimization (OCO) framework to capture the competition between a frequency agile radar and a digital radio frequency memory (DRFM)-based intelligent jammer. Recognizing that conventional OCO algorithms also suffer from suboptimal sample efficiency, two refined algorithms are developed that incorporate unbiased gradient estimators specifically tailored to the unique characteristics of DRFM-based jammers. Our theoretical analysis of the regret bound indicates significant improvements in long-term performance compared to standard OCO. The simulation results consistently show that our algorithms outperform traditional OCO and reinforcement learning baselines, achieving faster convergence and better anti-jamming performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper models the dynamic competition between a frequency-agile radar and a DRFM-based intelligent jammer as an online convex optimization (OCO) problem. It develops two refined OCO algorithms incorporating unbiased gradient estimators that exploit the jammer's memory and replay structure, claiming improved regret bounds over standard OCO and superior empirical performance (faster convergence, better anti-jamming) versus both OCO and reinforcement-learning baselines.

Significance. If the per-round loss is convex in the radar action and the proposed estimators are unbiased, the work would strengthen the applicability of OCO to structured adversarial radar-jammer settings by delivering both tighter theoretical guarantees and practical sample-efficiency gains, as demonstrated in the reported simulations.

major comments (2)
  1. [Problem formulation and loss definition] The modeling of the radar-DRFM interaction as a convex OCO problem is load-bearing for all regret claims, yet the effective loss (e.g., detection probability or post-jammer SINR) is not shown to be convex in the radar's frequency choice. DRFM replay and signal-dependent interference can produce history-dependent, non-convex responses; without an explicit verification or proof that convexity is preserved under the DRFM model, the standard OCO regret machinery does not apply.
  2. [Algorithm development and regret analysis] The unbiasedness of the two tailored gradient estimators is asserted but not derived in detail. Because the estimators rely on estimates of the jammer's internal state, any bias introduced by finite memory, estimation error, or the replay mechanism would invalidate the claimed regret improvement; a full bias analysis (including variance bounds) is required.
minor comments (2)
  1. [Numerical results] The simulation section should report exact jammer parameters (memory length, replay probability, power levels) and radar action discretization to permit reproduction of the reported convergence curves.
  2. [Preliminaries] Notation for the action set, loss function, and gradient estimator should be introduced once and used consistently; several symbols appear to be redefined across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment point by point below, providing clarifications and committing to revisions that strengthen the theoretical foundations without altering the core contributions.

read point-by-point responses

  1. Referee: [Problem formulation and loss definition] The modeling of the radar-DRFM interaction as a convex OCO problem is load-bearing for all regret claims, yet the effective loss (e.g., detection probability or post-jammer SINR) is not shown to be convex in the radar's frequency choice. DRFM replay and signal-dependent interference can produce history-dependent, non-convex responses; without an explicit verification or proof that convexity is preserved under the DRFM model, the standard OCO regret machinery does not apply.

    Authors: We appreciate this observation, as convexity is indeed foundational. In the manuscript, the radar action is the instantaneous frequency choice and the per-round loss is the resulting post-jamming SINR (or detection probability), which is convex in the chosen frequency for a fixed jammer state because interference power is a convex function of frequency mismatch. The DRFM replay introduces history dependence through the jammer's internal state, but the instantaneous loss at each round remains convex in the current action once the state is fixed. To make this rigorous and address the referee's concern directly, we will add an explicit lemma and proof in the revised manuscript (new Section 3.1) showing that the effective loss is convex under the stated DRFM model assumptions. This will confirm the applicability of standard OCO regret bounds as a baseline before presenting our refined estimators. revision: yes

  2. Referee: [Algorithm development and regret analysis] The unbiasedness of the two tailored gradient estimators is asserted but not derived in detail. Because the estimators rely on estimates of the jammer's internal state, any bias introduced by finite memory, estimation error, or the replay mechanism would invalidate the claimed regret improvement; a full bias analysis (including variance bounds) is required.

    Authors: We agree that a detailed derivation is necessary for the claimed regret improvements. The manuscript describes the two DRFM-specific estimators (one exploiting replay memory and one using an unbiased reconstruction of the jammer state) but presents their unbiasedness primarily through high-level arguments rather than full derivations. In the revision we will expand the analysis (new Appendix B) to include: (i) step-by-step proofs that both estimators are unbiased conditional on the jammer's finite memory and replay structure, (ii) explicit bounds on the bias introduced by state estimation error, and (iii) variance bounds that are used to derive the improved regret rates. These additions will directly support the theoretical claims and allow readers to verify the conditions under which the regret improvement holds. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the OCO modeling or regret analysis

full rationale

The paper adopts the established OCO framework to model radar-DRFM interaction (a standard modeling choice requiring per-round convexity), then constructs tailored unbiased gradient estimators that exploit DRFM memory structure and derives improved regret bounds from the standard OCO machinery applied to these estimators. No load-bearing step reduces by construction to its own inputs: the regret improvement follows from the new estimator variance reduction, not from re-labeling a fit or from a self-citation chain. Simulations supply independent empirical checks. Convexity is an explicit modeling assumption rather than a derived claim that loops back to itself. This matches the reader's assessment of low circularity risk.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that the radar-jammer interaction fits an OCO model and that unbiased estimators exist for DRFM characteristics; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption The anti-jamming competition between frequency agile radar and DRFM jammer can be modeled as an online convex optimization problem.
    The paper adopts an OCO framework to capture the competition.
  • domain assumption Unbiased gradient estimators can be constructed that are tailored to the unique characteristics of DRFM-based jammers.
    The refined algorithms rely on this property to improve sample efficiency over standard OCO.

pith-pipeline@v0.9.0 · 5442 in / 1318 out tokens · 56995 ms · 2026-05-10T16:31:20.543743+00:00 · methodology

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