ResearchBenchmarksRobustness Evaluation on Robust LQRRobustness Evaluation on Robust LQRFollow0.57Mean Normalized PerformancePathwise Robust ZO Both0.074960.203480.3320.46052Jun 3, 2026Evaluation resultsMethodMethodLinksMean Normalized PerformanceMaximum Normalized PerformancePathwise Robust ZO Both∆t=0.05∆t=0.052026.060.571.53Pathwise Robust ZO Both∆t=0.01∆t=0.012026.060.5611.514Pathwise Robust ZO Both∆t=0.005∆t=0.0052026.060.561.513Pathwise Robust ZO Both∆t=0.0005∆t=0.00052026.060.5591.511Pathwise Robust ZO Both∆t=0.001∆t=0.0012026.060.5591.511Discrete Non-robust∆t=0.05∆t=0.052026.060.4311.21Discrete Non-robust∆t=0.01∆t=0.012026.060.4231.208Discrete Non-robust∆t=0.005∆t=0.0052026.060.4221.207Discrete Non-robust∆t=0.0005∆t=0.00052026.060.4211.207Discrete Non-robust∆t=0.001∆t=0.0012026.060.4211.207Discrete Robust∆t=0.05∆t=0.052026.060.3330.907Stochastic Hamiltonian Robust ZO Both∆t=0.05∆t=0.052026.060.3291.02Stochastic Hamiltonian Robust ZO Both∆t=0.005∆t=0.0052026.060.3231.001Stochastic Hamiltonian Robust ZO Both∆t=0.01∆t=0.012026.060.3231.003Stochastic Hamiltonian Robust ZO Both∆t=0.0005∆t=0.00052026.060.3220.999Stochastic Hamiltonian Robust ZO Both∆t=0.001∆t=0.0012026.060.3220.999Discrete Robust∆t=0.01∆t=0.012026.060.3190.887Discrete Robust∆t=0.005∆t=0.0052026.060.3170.884Discrete Robust∆t=0.0005∆t=0.00052026.060.3160.882Discrete Robust∆t=0.001∆t=0.0012026.060.3160.882Stochastic Hamiltonian Robust ZO Inner∆t=0.05∆t=0.052026.060.1360.391Stochastic Hamiltonian Robust∆t=0.05∆t=0.052026.060.1340.46Stochastic Hamiltonian Robust ZO Inner∆t=0.005∆t=0.0052026.060.1290.399Stochastic Hamiltonian Robust ZO Inner∆t=0.01∆t=0.012026.060.1290.398Stochastic Hamiltonian Robust ZO Inner∆t=0.0005∆t=0.00052026.060.1280.4Stochastic Hamiltonian Robust ZO Inner∆t=0.001∆t=0.0012026.060.1280.4Stochastic Hamiltonian Robust∆t=0.01∆t=0.012026.060.1250.42Stochastic Hamiltonian Robust∆t=0.005∆t=0.0052026.060.1240.421Pathwise Robust∆t=0.05∆t=0.052026.060.1240.471Stochastic Hamiltonian Robust∆t=0.0005∆t=0.00052026.060.1230.423Stochastic Hamiltonian Robust∆t=0.001∆t=0.0012026.060.1230.423Deterministic Hamiltonian Robust∆t=0.05∆t=0.052026.060.1210.484Pathwise Robust∆t=0.01∆t=0.012026.060.1210.466Pathwise Robust∆t=0.0005∆t=0.00052026.060.120.465Pathwise Robust∆t=0.001∆t=0.0012026.060.120.465Pathwise Robust∆t=0.005∆t=0.0052026.060.120.466Deterministic Hamiltonian (explore) Robust∆t=0.05∆t=0.052026.060.1190.464Deterministic Hamiltonian Robust∆t=0.01∆t=0.012026.060.1170.478Deterministic Hamiltonian Robust∆t=0.0005∆t=0.00052026.060.1160.477Deterministic Hamiltonian Robust∆t=0.001∆t=0.0012026.060.1160.477Deterministic Hamiltonian Robust∆t=0.005∆t=0.0052026.060.1160.478Deterministic Hamiltonian (explore) Robust∆t=0.0005∆t=0.00052026.060.1150.459Deterministic Hamiltonian (explore) Robust∆t=0.001∆t=0.0012026.060.1150.459Deterministic Hamiltonian (explore) Robust∆t=0.005∆t=0.0052026.060.1150.459Deterministic Hamiltonian (explore) Robust∆t=0.01∆t=0.012026.060.1150.46Pathwise Robust ZO Inner∆t=0.05∆t=0.052026.060.1090.437Pathwise Robust ZO Inner∆t=0.01∆t=0.012026.060.1060.435Pathwise Robust ZO Inner∆t=0.0005∆t=0.00052026.060.1050.434Pathwise Robust ZO Inner∆t=0.001∆t=0.0012026.060.1050.434Pathwise Robust ZO Inner∆t=0.005∆t=0.0052026.060.1050.434Stochastic Hamiltonian Non-robust∆t=0.05∆t=0.052026.060.1030.36Deterministic Hamiltonian (explore) Non-robust∆t=0.05∆t=0.052026.060.1010.362Stochastic Hamiltonian Non-robust∆t=0.01∆t=0.012026.060.10.359Stochastic Hamiltonian Non-robust∆t=0.0005∆t=0.00052026.060.0990.358Stochastic Hamiltonian Non-robust∆t=0.001∆t=0.0012026.060.0990.358Stochastic Hamiltonian Non-robust∆t=0.005∆t=0.0052026.060.0990.358Pathwise Non-robust∆t=0.05∆t=0.052026.060.0990.356Deterministic Hamiltonian Non-robust∆t=0.05∆t=0.052026.060.0980.356Deterministic Hamiltonian (explore) Non-robust∆t=0.01∆t=0.012026.060.0980.355Deterministic Hamiltonian (explore) Non-robust∆t=0.0005∆t=0.00052026.060.0970.353Deterministic Hamiltonian (explore) Non-robust∆t=0.001∆t=0.0012026.060.0970.353Deterministic Hamiltonian (explore) Non-robust∆t=0.005∆t=0.0052026.060.0970.354Deterministic Hamiltonian Non-robust∆t=0.01∆t=0.012026.060.0950.354Pathwise Non-robust∆t=0.01∆t=0.012026.060.0950.354Deterministic Hamiltonian Non-robust∆t=0.0005∆t=0.00052026.060.0940.353Deterministic Hamiltonian Non-robust∆t=0.001∆t=0.0012026.060.0940.353Deterministic Hamiltonian Non-robust∆t=0.005∆t=0.0052026.060.0940.353Pathwise Non-robust∆t=0.0005∆t=0.00052026.060.0940.353Pathwise Non-robust∆t=0.001∆t=0.0012026.060.0940.353Pathwise Non-robust∆t=0.005∆t=0.0052026.060.0940.353