ResearchBenchmarksRobustness Evaluation on Robust LQR adversarial Normalized performance (test)Robustness Evaluation on Robust LQR adversarial Normalized performance (test)Follow1.603Normalized PerformancePathwise Robust ZO Both0.362280.684391.00651.32861Jun 3, 2026Evaluation resultsMethodMethodLinksNormalized PerformancePathwise Robust ZO BothΔt=0.05Δt=0.052026.061.603Pathwise Robust ZO BothΔt=0.01Δt=0.012026.061.579Pathwise Robust ZO BothΔt=0.005Δt=0.0052026.061.576Pathwise Robust ZO BothΔt=0.0005Δt=0.00052026.061.574Pathwise Robust ZO BothΔt=0.001Δt=0.0012026.061.574Discrete Non-robustΔt=0.05Δt=0.052026.061.337Discrete Non-robustΔt=0.01Δt=0.012026.061.308Discrete Non-robustΔt=0.005Δt=0.0052026.061.305Discrete Non-robustΔt=0.0005Δt=0.00052026.061.303Discrete Non-robustΔt=0.001Δt=0.0012026.061.303Stochastic Hamiltonian Robust ZO BothΔt=0.05Δt=0.052026.061.144Stochastic Hamiltonian Robust ZO BothΔt=0.01Δt=0.012026.061.127Stochastic Hamiltonian Robust ZO BothΔt=0.005Δt=0.0052026.061.125Stochastic Hamiltonian Robust ZO BothΔt=0.001Δt=0.0012026.061.124Stochastic Hamiltonian Robust ZO BothΔt=0.0005Δt=0.00052026.061.123Discrete RobustΔt=0.05Δt=0.052026.061.013Discrete RobustΔt=0.01Δt=0.012026.060.981Discrete RobustΔt=0.005Δt=0.0052026.060.977Discrete RobustΔt=0.0005Δt=0.00052026.060.974Discrete RobustΔt=0.001Δt=0.0012026.060.974Stochastic Hamiltonian Robust ZO InnerΔt=0.05Δt=0.052026.060.592Stochastic Hamiltonian Robust ZO InnerΔt=0.01Δt=0.012026.060.576Stochastic Hamiltonian Robust ZO InnerΔt=0.005Δt=0.0052026.060.574Deterministic Hamiltonian (explore) Non-robustΔt=0.05Δt=0.052026.060.574Stochastic Hamiltonian Robust ZO InnerΔt=0.0005Δt=0.00052026.060.572Stochastic Hamiltonian Robust ZO InnerΔt=0.001Δt=0.0012026.060.572Deterministic Hamiltonian Non-robustΔt=0.05Δt=0.052026.060.565Pathwise Non-robustΔt=0.05Δt=0.052026.060.565Deterministic Hamiltonian (explore) Non-robustΔt=0.01Δt=0.012026.060.561Deterministic Hamiltonian (explore) Non-robustΔt=0.005Δt=0.0052026.060.559Deterministic Hamiltonian (explore) Non-robustΔt=0.0005Δt=0.00052026.060.558Deterministic Hamiltonian (explore) Non-robustΔt=0.001Δt=0.0012026.060.558Deterministic Hamiltonian Non-robustΔt=0.01Δt=0.012026.060.553Pathwise Non-robustΔt=0.01Δt=0.012026.060.553Deterministic Hamiltonian Non-robustΔt=0.005Δt=0.0052026.060.551Pathwise Non-robustΔt=0.005Δt=0.0052026.060.551Deterministic Hamiltonian Non-robustΔt=0.0005Δt=0.00052026.060.55Deterministic Hamiltonian Non-robustΔt=0.001Δt=0.0012026.060.55Pathwise Non-robustΔt=0.0005Δt=0.00052026.060.55Pathwise Non-robustΔt=0.001Δt=0.0012026.060.55Deterministic Hamiltonian (explore) RobustΔt=0.05Δt=0.052026.060.535Pathwise Robust ZO InnerΔt=0.05Δt=0.052026.060.535Deterministic Hamiltonian RobustΔt=0.05Δt=0.052026.060.533Stochastic Hamiltonian Non-robustΔt=0.05Δt=0.052026.060.528Deterministic Hamiltonian (explore) RobustΔt=0.01Δt=0.012026.060.525Deterministic Hamiltonian (explore) RobustΔt=0.005Δt=0.0052026.060.524Pathwise Robust ZO InnerΔt=0.01Δt=0.012026.060.524Deterministic Hamiltonian (explore) RobustΔt=0.001Δt=0.0012026.060.523Deterministic Hamiltonian (explore) RobustΔt=0.0005Δt=0.00052026.060.522Pathwise Robust ZO InnerΔt=0.005Δt=0.0052026.060.522Deterministic Hamiltonian RobustΔt=0.01Δt=0.012026.060.521Pathwise Robust ZO InnerΔt=0.0005Δt=0.00052026.060.521Pathwise Robust ZO InnerΔt=0.001Δt=0.0012026.060.521Deterministic Hamiltonian RobustΔt=0.005Δt=0.0052026.060.52Deterministic Hamiltonian RobustΔt=0.0005Δt=0.00052026.060.518Deterministic Hamiltonian RobustΔt=0.001Δt=0.0012026.060.518Stochastic Hamiltonian Non-robustΔt=0.01Δt=0.012026.060.513Stochastic Hamiltonian Non-robustΔt=0.005Δt=0.0052026.060.511Stochastic Hamiltonian Non-robustΔt=0.0005Δt=0.00052026.060.51Stochastic Hamiltonian Non-robustΔt=0.001Δt=0.0012026.060.51Pathwise RobustΔt=0.05Δt=0.052026.060.445Pathwise RobustΔt=0.01Δt=0.012026.060.436Pathwise RobustΔt=0.005Δt=0.0052026.060.435Pathwise RobustΔt=0.0005Δt=0.00052026.060.434Pathwise RobustΔt=0.001Δt=0.0012026.060.434Stochastic Hamiltonian RobustΔt=0.05Δt=0.052026.060.42Stochastic Hamiltonian RobustΔt=0.005Δt=0.0052026.060.411Stochastic Hamiltonian RobustΔt=0.01Δt=0.012026.060.411Stochastic Hamiltonian RobustΔt=0.0005Δt=0.00052026.060.41Stochastic Hamiltonian RobustΔt=0.001Δt=0.0012026.060.41