{
  "config": {
    "n_train": 1000,
    "n_eval": 1200,
    "batch_size": 32,
    "sft_steps": 200,
    "reward_steps": 200,
    "lr": 0.15,
    "seeds": [
      11,
      22,
      33,
      44,
      55
    ],
    "initial_weights": [
      1.6,
      0.0,
      0.0
    ],
    "target_weights": [
      0.8,
      1.2,
      -0.7
    ],
    "frontier_center": -0.6,
    "frontier_width": 0.6,
    "floor": 0.1,
    "monotone_temperature": 0.5
  },
  "examples": {
    "provenance": "HAND-SPECIFIED SEMANTIC ACTION PROBABILITIES, NOT LLM SCORES",
    "question": "计算 1+2+…+100，三个方法均可得到 5050。",
    "actions": [
      {
        "action": "逐项累加 1 到 100",
        "p": 0.8,
        "entropy": 0.6128694524619495,
        "cll": 0.38972590114773975,
        "frontier_raw_weight": 0.1170623766779906,
        "ce_chosen_logit_magnitude": 0.19999999999999996,
        "weighted_chosen_logit_magnitude": 0.023412475335598117
      },
      {
        "action": "首尾配对，50 对，每对 101",
        "p": 0.15,
        "entropy": 0.6128694524619495,
        "cll": -1.2842505324239317,
        "frontier_raw_weight": 0.9996899962412849,
        "ce_chosen_logit_magnitude": 0.85,
        "weighted_chosen_logit_magnitude": 0.8497364968050921
      },
      {
        "action": "用数学归纳法证明求和公式",
        "p": 0.05,
        "entropy": 0.6128694524619495,
        "cll": -2.3828628210920413,
        "frontier_raw_weight": 0.2765836880170781,
        "ce_chosen_logit_magnitude": 0.95,
        "weighted_chosen_logit_magnitude": 0.2627545036162242
      }
    ],
    "paths": [
      {
        "name": "常见前缀",
        "prefix_conditionals": [
          0.8,
          0.8
        ],
        "prefix_probability": 0.6400000000000001,
        "final_action_probability": 0.9,
        "final_action_cll": 0.21972245773362192,
        "full_trajectory_probability": 0.5760000000000002
      },
      {
        "name": "罕见前缀",
        "prefix_conditionals": [
          0.02,
          0.8
        ],
        "prefix_probability": 0.016,
        "final_action_probability": 0.9,
        "final_action_cll": 0.21972245773362192,
        "full_trajectory_probability": 0.014400000000000001
      }
    ],
    "off_policy_counterexample": {
      "p": [
        0.8,
        0.15,
        0.05
      ],
      "q": [
        1.0,
        0.0,
        0.0
      ],
      "expected_cll": 0.38972590114773975,
      "entropy_identity": 0.3897259011477397
    },
    "uniform_counterexample": {
      "p": [
        0.3333333333333333,
        0.3333333333333333,
        0.3333333333333333
      ],
      "cll": [
        -2.220446049250313e-16,
        -2.220446049250313e-16,
        -2.220446049250313e-16
      ]
    }
  },
  "initial": [
    {
      "seed": 11,
      "id": {
        "accuracy": 0.68,
        "expected_reward": 0.6148900810834099,
        "mean_entropy": 0.5066059130979225
      },
      "ood": {
        "accuracy": 0.595,
        "expected_reward": 0.5432322474644323,
        "mean_entropy": 0.6296638587724667
      }
    },
    {
      "seed": 22,
      "id": {
        "accuracy": 0.6666666666666666,
        "expected_reward": 0.6184184769089539,
        "mean_entropy": 0.5048014746358678
      },
      "ood": {
        "accuracy": 0.5625,
        "expected_reward": 0.5211224570310002,
        "mean_entropy": 0.6312084690435511
      }
    },
    {
      "seed": 33,
      "id": {
        "accuracy": 0.6533333333333333,
        "expected_reward": 0.6122147461174526,
        "mean_entropy": 0.5102639638813532
      },
      "ood": {
        "accuracy": 0.5741666666666667,
        "expected_reward": 0.5338784319704429,
        "mean_entropy": 0.6257380350038315
      }
    },
    {
      "seed": 44,
      "id": {
        "accuracy": 0.6575,
        "expected_reward": 0.6056464675365375,
        "mean_entropy": 0.5222711742303352
      },
      "ood": {
        "accuracy": 0.5583333333333333,
        "expected_reward": 0.5272699914064999,
        "mean_entropy": 0.6316693928228275
      }
    },
    {
      "seed": 55,
      "id": {
        "accuracy": 0.6683333333333333,
        "expected_reward": 0.6138456252121405,
        "mean_entropy": 0.5100090277917146
      },
      "ood": {
        "accuracy": 0.555,
        "expected_reward": 0.5179540034219987,
        "mean_entropy": 0.6284325234646073
      }
    }
  ],
  "bucket_diagnostics": [
    {
      "seed": 11,
      "bucket": 1,
      "n": 200,
      "p_mean": 0.21448914708404782,
      "cll_min": -4.327791279459607,
      "cll_max": -0.3736363464033655,
      "x_abs_mean": [
        0.893371731936876,
        1.0573897009458555,
        0.8385979476685445
      ]
    },
    {
      "seed": 11,
      "bucket": 2,
      "n": 200,
      "p_mean": 0.5166293664396652,
      "cll_min": -0.37318327684595753,
      "cll_max": 0.08665727469238438,
      "x_abs_mean": [
        0.4216558158596539,
        0.8531051855655232,
        0.6881880092153422
      ]
    },
    {
      "seed": 11,
      "bucket": 3,
      "n": 200,
      "p_mean": 0.7696373390291661,
      "cll_min": 0.08751248967243562,
      "cll_max": 0.206021076111866,
      "x_abs_mean": [
        1.0276206497401996,
        0.7553231588835092,
        0.7430290990534041
      ]
    },
    {
      "seed": 11,
      "bucket": 4,
      "n": 200,
      "p_mean": 0.7867074121463207,
      "cll_min": 0.20703778302004017,
      "cll_max": 0.2621779072450331,
      "x_abs_mean": [
        0.8876178880708577,
        0.6855281037258734,
        0.8442110539111374
      ]
    },
    {
      "seed": 11,
      "bucket": 5,
      "n": 200,
      "p_mean": 0.7844637160358922,
      "cll_min": 0.26221522591630886,
      "cll_max": 0.27846386935198997,
      "x_abs_mean": [
        0.8170291713523146,
        0.69100381967429,
        0.8364678282987501
      ]
    },
    {
      "seed": 22,
      "bucket": 1,
      "n": 200,
      "p_mean": 0.1954146382665046,
      "cll_min": -3.7535055007028357,
      "cll_max": -0.47739448023266795,
      "x_abs_mean": [
        0.9572627513944753,
        1.0543693776664738,
        0.8570340234668037
      ]
    },
    {
      "seed": 22,
      "bucket": 2,
      "n": 200,
      "p_mean": 0.46180140809681836,
      "cll_min": -0.47635478535167763,
      "cll_max": 0.06550626103600166,
      "x_abs_mean": [
        0.28119566967506177,
        0.8051395606655869,
        0.7874452374001876
      ]
    },
    {
      "seed": 22,
      "bucket": 3,
      "n": 200,
      "p_mean": 0.7554715030766753,
      "cll_min": 0.06631577116715581,
      "cll_max": 0.19139368249625524,
      "x_abs_mean": [
        0.9995984989775057,
        0.7145710162569555,
        0.7570257083800401
      ]
    },
    {
      "seed": 22,
      "bucket": 4,
      "n": 200,
      "p_mean": 0.7764468400526807,
      "cll_min": 0.1916718328872305,
      "cll_max": 0.25788847140143323,
      "x_abs_mean": [
        0.8650150255653964,
        0.6994044648879508,
        0.7430991908913547
      ]
    },
    {
      "seed": 22,
      "bucket": 5,
      "n": 200,
      "p_mean": 0.7802961884444113,
      "cll_min": 0.25790443917116374,
      "cll_max": 0.27845788172599395,
      "x_abs_mean": [
        0.8039754526481075,
        0.7120069789837302,
        0.7901562185887867
      ]
    },
    {
      "seed": 33,
      "bucket": 1,
      "n": 200,
      "p_mean": 0.21954341109689612,
      "cll_min": -3.5392924013423412,
      "cll_max": -0.36146100346631727,
      "x_abs_mean": [
        0.871117534621194,
        1.0603794071907982,
        0.8588941114952818
      ]
    },
    {
      "seed": 33,
      "bucket": 2,
      "n": 200,
      "p_mean": 0.48473234056314196,
      "cll_min": -0.35936675712267896,
      "cll_max": 0.08256102644604636,
      "x_abs_mean": [
        0.3034155894569217,
        0.8305364878633029,
        0.8265755500704522
      ]
    },
    {
      "seed": 33,
      "bucket": 3,
      "n": 200,
      "p_mean": 0.7810370329200484,
      "cll_min": 0.08332134043784933,
      "cll_max": 0.2117097730392517,
      "x_abs_mean": [
        1.0599283348741586,
        0.7968778421176921,
        0.7703786778643034
      ]
    },
    {
      "seed": 33,
      "bucket": 4,
      "n": 200,
      "p_mean": 0.7928548979518968,
      "cll_min": 0.21195813510410175,
      "cll_max": 0.2634519190947375,
      "x_abs_mean": [
        0.9085487058719921,
        0.6804209590051138,
        0.8143107155531348
      ]
    },
    {
      "seed": 33,
      "bucket": 5,
      "n": 200,
      "p_mean": 0.7835383483593754,
      "cll_min": 0.26351560236979255,
      "cll_max": 0.27846443357468864,
      "x_abs_mean": [
        0.8128312165830613,
        0.6864575135436287,
        0.8218442268579919
      ]
    },
    {
      "seed": 44,
      "bucket": 1,
      "n": 200,
      "p_mean": 0.21221551193827218,
      "cll_min": -6.0968291358594895,
      "cll_max": -0.3704749772700049,
      "x_abs_mean": [
        0.9182077985098663,
        1.0693500281738024,
        0.9486369010204694
      ]
    },
    {
      "seed": 44,
      "bucket": 2,
      "n": 200,
      "p_mean": 0.5017378976429108,
      "cll_min": -0.3655431024627027,
      "cll_max": 0.0864811253847948,
      "x_abs_mean": [
        0.35353340015649587,
        0.7674861277520311,
        0.8088357165018475
      ]
    },
    {
      "seed": 44,
      "bucket": 3,
      "n": 200,
      "p_mean": 0.7626474843928653,
      "cll_min": 0.08687766889359905,
      "cll_max": 0.20435151519035788,
      "x_abs_mean": [
        0.9934298062204163,
        0.7052888562871854,
        0.7262056929245941
      ]
    },
    {
      "seed": 44,
      "bucket": 4,
      "n": 200,
      "p_mean": 0.787180428329774,
      "cll_min": 0.20515359583669907,
      "cll_max": 0.2622998979608675,
      "x_abs_mean": [
        0.8923827462195719,
        0.7128927315940401,
        0.8375982509812474
      ]
    },
    {
      "seed": 44,
      "bucket": 5,
      "n": 200,
      "p_mean": 0.779780257218162,
      "cll_min": 0.26237271613417645,
      "cll_max": 0.2784643424488077,
      "x_abs_mean": [
        0.7994532496506234,
        0.661068443953332,
        0.7451260028694559
      ]
    },
    {
      "seed": 55,
      "bucket": 1,
      "n": 200,
      "p_mean": 0.2268037811131374,
      "cll_min": -3.708531877016537,
      "cll_max": -0.3664599870255847,
      "x_abs_mean": [
        0.836162010383519,
        1.0284584943306245,
        0.836006729233984
      ]
    },
    {
      "seed": 55,
      "bucket": 2,
      "n": 200,
      "p_mean": 0.5246898234454026,
      "cll_min": -0.36580015234252317,
      "cll_max": 0.09706595856014233,
      "x_abs_mean": [
        0.4600054442396103,
        0.866482187703236,
        0.7679377090912788
      ]
    },
    {
      "seed": 55,
      "bucket": 3,
      "n": 200,
      "p_mean": 0.7608575049578677,
      "cll_min": 0.09716475792595647,
      "cll_max": 0.20050443909520227,
      "x_abs_mean": [
        0.9925277202715918,
        0.7428359146496455,
        0.7231680149529388
      ]
    },
    {
      "seed": 55,
      "bucket": 4,
      "n": 200,
      "p_mean": 0.7796653238399072,
      "cll_min": 0.20109362176902634,
      "cll_max": 0.2584408539825238,
      "x_abs_mean": [
        0.8709687045976047,
        0.7297299617880622,
        0.7314805052943968
      ]
    },
    {
      "seed": 55,
      "bucket": 5,
      "n": 200,
      "p_mean": 0.7917088633898774,
      "cll_min": 0.2585040193165573,
      "cll_max": 0.27846435844798356,
      "x_abs_mean": [
        0.845135228072802,
        0.696991631791487,
        0.8552321537440702
      ]
    }
  ],
  "runs": [
    {
      "seed": 11,
      "method": "uniform",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.6843559740005944,
        2.6312317268711722,
        -1.3910688677338034
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9891666666666666,
        "expected_reward": 0.8568757410177995,
        "mean_entropy": 0.3222509872473528
      },
      "sft_ood": {
        "accuracy": 0.9833333333333333,
        "expected_reward": 0.8913974018356875,
        "mean_entropy": 0.2478093933603921
      },
      "reward_weights": [
        2.1008585242130766,
        3.3113837643024495,
        -1.8177973397265395
      ],
      "reward_id": {
        "accuracy": 0.9933333333333333,
        "expected_reward": 0.8841422917176829,
        "mean_entropy": 0.2664449668730544
      },
      "reward_ood": {
        "accuracy": 0.9891666666666666,
        "expected_reward": 0.9133365807612589,
        "mean_entropy": 0.20030072582039213
      }
    },
    {
      "seed": 11,
      "method": "probability",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.351806746044427,
        2.074458594728907,
        -1.0515316649516382
      ],
      "mean_batch_ess": 27.17306404868311,
      "sft_id": {
        "accuracy": 0.9066666666666666,
        "expected_reward": 0.8272393313472325,
        "mean_entropy": 0.33028725962954564
      },
      "sft_ood": {
        "accuracy": 0.9591666666666666,
        "expected_reward": 0.8645534808693706,
        "mean_entropy": 0.2923843156459526
      },
      "reward_weights": [
        2.2661852308875283,
        3.098194256917822,
        -1.6738657563411994
      ],
      "reward_id": {
        "accuracy": 0.985,
        "expected_reward": 0.8787102267298627,
        "mean_entropy": 0.27439122913073055
      },
      "reward_ood": {
        "accuracy": 0.9875,
        "expected_reward": 0.9078802871828101,
        "mean_entropy": 0.2118612251377102
      }
    },
    {
      "seed": 11,
      "method": "cll_monotone",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.2301095076603765,
        2.2513691091177663,
        -1.152220555418252
      ],
      "mean_batch_ess": 28.9272255829761,
      "sft_id": {
        "accuracy": 0.9291666666666667,
        "expected_reward": 0.8393539088740261,
        "mean_entropy": 0.3261755690508908
      },
      "sft_ood": {
        "accuracy": 0.9725,
        "expected_reward": 0.8747016739763872,
        "mean_entropy": 0.2765374176159449
      },
      "reward_weights": [
        2.239568562387469,
        3.176457638065309,
        -1.7264554753406012
      ],
      "reward_id": {
        "accuracy": 0.9883333333333333,
        "expected_reward": 0.8812664014513406,
        "mean_entropy": 0.27068157487088734
      },
      "reward_ood": {
        "accuracy": 0.99,
        "expected_reward": 0.9100977540112751,
        "mean_entropy": 0.20729856226473098
      }
    },
    {
      "seed": 11,
      "method": "cll_frontier",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.5988778753555266,
        2.7087898721918746,
        -1.3962912701640746
      ],
      "mean_batch_ess": 28.429469384495036,
      "sft_id": {
        "accuracy": 0.9883333333333333,
        "expected_reward": 0.8573763698299919,
        "mean_entropy": 0.32051703942344156
      },
      "sft_ood": {
        "accuracy": 0.9783333333333334,
        "expected_reward": 0.8926074054578683,
        "mean_entropy": 0.24386276711296703
      },
      "reward_weights": [
        2.0827452211997226,
        3.3385230185185555,
        -1.8227782349291468
      ],
      "reward_id": {
        "accuracy": 0.9916666666666667,
        "expected_reward": 0.8844679387995139,
        "mean_entropy": 0.2656266444805608
      },
      "reward_ood": {
        "accuracy": 0.9875,
        "expected_reward": 0.9136957028801004,
        "mean_entropy": 0.1992439085116901
      }
    },
    {
      "seed": 11,
      "method": "bucket_1",
      "unique_pool_size": 200,
      "sft_weights": [
        -1.1151294025831404,
        2.6345149206609118,
        -1.333801216501272
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7216666666666667,
        "expected_reward": 0.6889709334206523,
        "mean_entropy": 0.3498025634386594
      },
      "sft_ood": {
        "accuracy": 0.9058333333333334,
        "expected_reward": 0.8491865607577858,
        "mean_entropy": 0.26498754809185526
      },
      "reward_weights": [
        0.8244572571935036,
        3.2768251696201043,
        -1.753059929925077
      ],
      "reward_id": {
        "accuracy": 0.9141666666666667,
        "expected_reward": 0.8419499176951284,
        "mean_entropy": 0.29809368048147644
      },
      "reward_ood": {
        "accuracy": 0.9666666666666667,
        "expected_reward": 0.9023614166893598,
        "mean_entropy": 0.2098328034738164
      }
    },
    {
      "seed": 11,
      "method": "bucket_2",
      "unique_pool_size": 200,
      "sft_weights": [
        1.2659522699471943,
        2.7842846613459615,
        -1.5133808876570254
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9591666666666666,
        "expected_reward": 0.8515866216795432,
        "mean_entropy": 0.3226476975444785
      },
      "sft_ood": {
        "accuracy": 0.9825,
        "expected_reward": 0.8948567661408883,
        "mean_entropy": 0.23783184177610042
      },
      "reward_weights": [
        1.9857819658885656,
        3.347428851956346,
        -1.8458294263490571
      ],
      "reward_id": {
        "accuracy": 0.9866666666666667,
        "expected_reward": 0.8835110534742803,
        "mean_entropy": 0.26692930106194435
      },
      "reward_ood": {
        "accuracy": 0.9875,
        "expected_reward": 0.9138700791474564,
        "mean_entropy": 0.19880214761579415
      }
    },
    {
      "seed": 11,
      "method": "bucket_3",
      "unique_pool_size": 200,
      "sft_weights": [
        2.89982845090997,
        1.5458052310094312,
        -0.7584271108870773
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.8316666666666667,
        "expected_reward": 0.7823543369804102,
        "mean_entropy": 0.3216741410334489
      },
      "sft_ood": {
        "accuracy": 0.9041666666666667,
        "expected_reward": 0.8216249064557418,
        "mean_entropy": 0.3426351818454519
      },
      "reward_weights": [
        2.5181525357916152,
        2.8944066808874043,
        -1.5044290345422926
      ],
      "reward_id": {
        "accuracy": 0.9541666666666667,
        "expected_reward": 0.8690568516521099,
        "mean_entropy": 0.2786733619580241
      },
      "reward_ood": {
        "accuracy": 0.98,
        "expected_reward": 0.9002721357977619,
        "mean_entropy": 0.22446878216376587
      }
    },
    {
      "seed": 11,
      "method": "bucket_4",
      "unique_pool_size": 200,
      "sft_weights": [
        3.395139066507279,
        1.001013929329562,
        -0.5226056069481917
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7741666666666667,
        "expected_reward": 0.7367045284769882,
        "mean_entropy": 0.30639338154448215
      },
      "sft_ood": {
        "accuracy": 0.8283333333333334,
        "expected_reward": 0.7590639184697607,
        "mean_entropy": 0.3969787299319419
      },
      "reward_weights": [
        2.9176737948328415,
        2.6269943118368198,
        -1.300922481462687
      ],
      "reward_id": {
        "accuracy": 0.9075,
        "expected_reward": 0.8496114665472583,
        "mean_entropy": 0.27695941298594723
      },
      "reward_ood": {
        "accuracy": 0.9591666666666666,
        "expected_reward": 0.8870059487632352,
        "mean_entropy": 0.24131091225473386
      }
    },
    {
      "seed": 11,
      "method": "bucket_5",
      "unique_pool_size": 200,
      "sft_weights": [
        3.536764459822444,
        0.7777345871371185,
        -0.4802917996704758
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.755,
        "expected_reward": 0.7207999348389469,
        "mean_entropy": 0.3013033311986531
      },
      "sft_ood": {
        "accuracy": 0.7933333333333333,
        "expected_reward": 0.73076797813993,
        "mean_entropy": 0.41729737455474203
      },
      "reward_weights": [
        3.0745689774437364,
        2.4913379817212475,
        -1.244653403951204
      ],
      "reward_id": {
        "accuracy": 0.895,
        "expected_reward": 0.8400978973365079,
        "mean_entropy": 0.2753780234086577
      },
      "reward_ood": {
        "accuracy": 0.9516666666666667,
        "expected_reward": 0.8806917203658048,
        "mean_entropy": 0.24914166576032246
      }
    },
    {
      "seed": 22,
      "method": "uniform",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.5957286179845072,
        2.598697623269953,
        -1.4798203822856109
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9866666666666667,
        "expected_reward": 0.8498579174781822,
        "mean_entropy": 0.3310186298722128
      },
      "sft_ood": {
        "accuracy": 0.9975,
        "expected_reward": 0.8793100698585585,
        "mean_entropy": 0.27204782075665657
      },
      "reward_weights": [
        1.8986145408284607,
        3.353090224479443,
        -1.836700745588429
      ],
      "reward_id": {
        "accuracy": 0.9775,
        "expected_reward": 0.8746446070227928,
        "mean_entropy": 0.2776312153102246
      },
      "reward_ood": {
        "accuracy": 0.99,
        "expected_reward": 0.9022696461542807,
        "mean_entropy": 0.2224707394035221
      }
    },
    {
      "seed": 22,
      "method": "probability",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.268923518523838,
        2.0659826760660196,
        -1.1732728038692974
      ],
      "mean_batch_ess": 26.837357685413753,
      "sft_id": {
        "accuracy": 0.9141666666666667,
        "expected_reward": 0.8330135428731894,
        "mean_entropy": 0.33072659574728336
      },
      "sft_ood": {
        "accuracy": 0.9666666666666667,
        "expected_reward": 0.8542463225451422,
        "mean_entropy": 0.31776147941366956
      },
      "reward_weights": [
        2.061481037885832,
        3.173961236319966,
        -1.7379048298224726
      ],
      "reward_id": {
        "accuracy": 0.9925,
        "expected_reward": 0.8742957129587458,
        "mean_entropy": 0.2819065433546612
      },
      "reward_ood": {
        "accuracy": 0.9941666666666666,
        "expected_reward": 0.8988341752077917,
        "mean_entropy": 0.2318082571361435
      }
    },
    {
      "seed": 22,
      "method": "cll_monotone",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.150400308884912,
        2.23771903717703,
        -1.2687650537672324
      ],
      "mean_batch_ess": 28.6777778356647,
      "sft_id": {
        "accuracy": 0.9333333333333333,
        "expected_reward": 0.8423571349909804,
        "mean_entropy": 0.3272601860522873
      },
      "sft_ood": {
        "accuracy": 0.9766666666666667,
        "expected_reward": 0.8644313190174864,
        "mean_entropy": 0.3012585163956384
      },
      "reward_weights": [
        2.04515912156482,
        3.2443799629119106,
        -1.776332126161305
      ],
      "reward_id": {
        "accuracy": 0.9908333333333333,
        "expected_reward": 0.8756009080068862,
        "mean_entropy": 0.27893553200645166
      },
      "reward_ood": {
        "accuracy": 0.9925,
        "expected_reward": 0.9005462802172169,
        "mean_entropy": 0.2278269103199804
      }
    },
    {
      "seed": 22,
      "method": "cll_frontier",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.4986608452465497,
        2.6628458143329117,
        -1.4931591564932776
      ],
      "mean_batch_ess": 28.09940760198522,
      "sft_id": {
        "accuracy": 0.9783333333333334,
        "expected_reward": 0.8486553508188335,
        "mean_entropy": 0.3309377876290776
      },
      "sft_ood": {
        "accuracy": 0.9941666666666666,
        "expected_reward": 0.8806863842956502,
        "mean_entropy": 0.2679860152110082
      },
      "reward_weights": [
        1.871508589613833,
        3.3730599306714906,
        -1.8407204186986026
      ],
      "reward_id": {
        "accuracy": 0.9758333333333333,
        "expected_reward": 0.8742486143336975,
        "mean_entropy": 0.2774751851491088
      },
      "reward_ood": {
        "accuracy": 0.99,
        "expected_reward": 0.9024633199493199,
        "mean_entropy": 0.22162407070171639
      }
    },
    {
      "seed": 22,
      "method": "bucket_1",
      "unique_pool_size": 200,
      "sft_weights": [
        -1.0968172170764778,
        2.4574850066791947,
        -1.4646543470446187
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7016666666666667,
        "expected_reward": 0.6717810606791174,
        "mean_entropy": 0.35750585180682104
      },
      "sft_ood": {
        "accuracy": 0.8841666666666667,
        "expected_reward": 0.8325708187068629,
        "mean_entropy": 0.27987734198810127
      },
      "reward_weights": [
        0.7053666397801699,
        3.193444600845128,
        -1.8203107026986425
      ],
      "reward_id": {
        "accuracy": 0.8933333333333333,
        "expected_reward": 0.8235992547622906,
        "mean_entropy": 0.3090375118588544
      },
      "reward_ood": {
        "accuracy": 0.9525,
        "expected_reward": 0.8865230503779936,
        "mean_entropy": 0.23362387300167575
      }
    },
    {
      "seed": 22,
      "method": "bucket_2",
      "unique_pool_size": 200,
      "sft_weights": [
        0.8315220353152591,
        3.0217812566940365,
        -1.5716115937887158
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9075,
        "expected_reward": 0.8258498638046406,
        "mean_entropy": 0.325234192591246
      },
      "sft_ood": {
        "accuracy": 0.9583333333333334,
        "expected_reward": 0.8827835035643107,
        "mean_entropy": 0.24673176021997717
      },
      "reward_weights": [
        1.654557302416486,
        3.5021018056568893,
        -1.8730710895232847
      ],
      "reward_id": {
        "accuracy": 0.9583333333333334,
        "expected_reward": 0.8694370529535103,
        "mean_entropy": 0.2765867485511538
      },
      "reward_ood": {
        "accuracy": 0.9808333333333333,
        "expected_reward": 0.9030533441310439,
        "mean_entropy": 0.21633615249601404
      }
    },
    {
      "seed": 22,
      "method": "bucket_3",
      "unique_pool_size": 200,
      "sft_weights": [
        2.7618900854488233,
        1.5989455586369306,
        -0.9304714812016815
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.8533333333333334,
        "expected_reward": 0.7975381292966351,
        "mean_entropy": 0.3268835907075591
      },
      "sft_ood": {
        "accuracy": 0.9233333333333333,
        "expected_reward": 0.8146799805127573,
        "mean_entropy": 0.36507121563369377
      },
      "reward_weights": [
        2.2445778723142973,
        3.0025058998691585,
        -1.6545165994579107
      ],
      "reward_id": {
        "accuracy": 0.9808333333333333,
        "expected_reward": 0.8720637114527472,
        "mean_entropy": 0.2833683368962834
      },
      "reward_ood": {
        "accuracy": 0.99,
        "expected_reward": 0.8947265142993752,
        "mean_entropy": 0.24126212765134147
      }
    },
    {
      "seed": 22,
      "method": "bucket_4",
      "unique_pool_size": 200,
      "sft_weights": [
        3.363996157107638,
        1.0455182320696703,
        -0.6088041012368085
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.775,
        "expected_reward": 0.7445880025018294,
        "mean_entropy": 0.3090669121466218
      },
      "sft_ood": {
        "accuracy": 0.8166666666666667,
        "expected_reward": 0.7403697957023165,
        "mean_entropy": 0.4209741051723918
      },
      "reward_weights": [
        2.723367007759875,
        2.6960590305373824,
        -1.4927920896220492
      ],
      "reward_id": {
        "accuracy": 0.9225,
        "expected_reward": 0.8605814606022099,
        "mean_entropy": 0.27631163122314434
      },
      "reward_ood": {
        "accuracy": 0.9716666666666667,
        "expected_reward": 0.8834382735395198,
        "mean_entropy": 0.25978818384883934
      }
    },
    {
      "seed": 22,
      "method": "bucket_5",
      "unique_pool_size": 200,
      "sft_weights": [
        3.5269499810338987,
        0.8205076691628088,
        -0.509432675537263
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7533333333333333,
        "expected_reward": 0.7254005940858096,
        "mean_entropy": 0.3028245670466256
      },
      "sft_ood": {
        "accuracy": 0.7758333333333334,
        "expected_reward": 0.7050357676088792,
        "mean_entropy": 0.44081365842408216
      },
      "reward_weights": [
        2.930396499603685,
        2.5272286596073523,
        -1.4173222198998567
      ],
      "reward_id": {
        "accuracy": 0.905,
        "expected_reward": 0.8512022103794753,
        "mean_entropy": 0.27392877453347153
      },
      "reward_ood": {
        "accuracy": 0.9641666666666666,
        "expected_reward": 0.8750832139628236,
        "mean_entropy": 0.270937226323021
      }
    },
    {
      "seed": 33,
      "method": "uniform",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.720930646136228,
        2.5352569269651,
        -1.4474867157308386
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9966666666666667,
        "expected_reward": 0.8586398800822003,
        "mean_entropy": 0.3170792865328325
      },
      "sft_ood": {
        "accuracy": 0.9966666666666667,
        "expected_reward": 0.8854948126065479,
        "mean_entropy": 0.26157491767740204
      },
      "reward_weights": [
        2.1837390054549415,
        3.2000917192864353,
        -1.8383006802307926
      ],
      "reward_id": {
        "accuracy": 0.9966666666666667,
        "expected_reward": 0.8850502332391471,
        "mean_entropy": 0.26202825295737536
      },
      "reward_ood": {
        "accuracy": 0.9958333333333333,
        "expected_reward": 0.9077099129018128,
        "mean_entropy": 0.21349793370311335
      }
    },
    {
      "seed": 33,
      "method": "probability",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.392487156264376,
        1.9787911029082998,
        -1.1624481733058125
      ],
      "mean_batch_ess": 27.26626302434232,
      "sft_id": {
        "accuracy": 0.9175,
        "expected_reward": 0.8294254723847236,
        "mean_entropy": 0.3265111113550604
      },
      "sft_ood": {
        "accuracy": 0.9583333333333334,
        "expected_reward": 0.8584345015279615,
        "mean_entropy": 0.30626130894961034
      },
      "reward_weights": [
        2.3538474318110296,
        3.007879955559885,
        -1.726571874938536
      ],
      "reward_id": {
        "accuracy": 0.9775,
        "expected_reward": 0.8797562299999108,
        "mean_entropy": 0.26772709506810005
      },
      "reward_ood": {
        "accuracy": 0.995,
        "expected_reward": 0.9028592324001847,
        "mean_entropy": 0.22409427638288007
      }
    },
    {
      "seed": 33,
      "method": "cll_monotone",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.2773257800970805,
        2.146623778822605,
        -1.263813675947117
      ],
      "mean_batch_ess": 29.0239726499571,
      "sft_id": {
        "accuracy": 0.9375,
        "expected_reward": 0.8412589696467315,
        "mean_entropy": 0.3208152004021651
      },
      "sft_ood": {
        "accuracy": 0.9683333333333334,
        "expected_reward": 0.8688336098269177,
        "mean_entropy": 0.2903161554044507
      },
      "reward_weights": [
        2.33184808493327,
        3.082416036166017,
        -1.770037589453774
      ],
      "reward_id": {
        "accuracy": 0.9825,
        "expected_reward": 0.8822204815109028,
        "mean_entropy": 0.2644093647660217
      },
      "reward_ood": {
        "accuracy": 0.995,
        "expected_reward": 0.9049617885974627,
        "mean_entropy": 0.21969585843641828
      }
    },
    {
      "seed": 33,
      "method": "cll_frontier",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.623096126463441,
        2.6017932367792787,
        -1.5228222422130526
      ],
      "mean_batch_ess": 28.319023248581633,
      "sft_id": {
        "accuracy": 0.9933333333333333,
        "expected_reward": 0.860095337336415,
        "mean_entropy": 0.3145248094764783
      },
      "sft_ood": {
        "accuracy": 0.9941666666666666,
        "expected_reward": 0.8881002141100888,
        "mean_entropy": 0.25522377973335725
      },
      "reward_weights": [
        2.1636467669389154,
        3.227829310538742,
        -1.8657407483378727
      ],
      "reward_id": {
        "accuracy": 0.9991666666666666,
        "expected_reward": 0.8857722760610377,
        "mean_entropy": 0.2608275768275966
      },
      "reward_ood": {
        "accuracy": 0.9991666666666666,
        "expected_reward": 0.9084509200130994,
        "mean_entropy": 0.21169784702306588
      }
    },
    {
      "seed": 33,
      "method": "bucket_1",
      "unique_pool_size": 200,
      "sft_weights": [
        -1.1187322690227108,
        2.6039301529987497,
        -1.3993061911844453
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7191666666666666,
        "expected_reward": 0.6948706872135262,
        "mean_entropy": 0.34192323882796394
      },
      "sft_ood": {
        "accuracy": 0.8891666666666667,
        "expected_reward": 0.8429620197874951,
        "mean_entropy": 0.2602541283832285
      },
      "reward_weights": [
        0.8746176148378729,
        3.163325156857973,
        -1.87341294839647
      ],
      "reward_id": {
        "accuracy": 0.9141666666666667,
        "expected_reward": 0.846297696908867,
        "mean_entropy": 0.2935475189400374
      },
      "reward_ood": {
        "accuracy": 0.95,
        "expected_reward": 0.8968773187029979,
        "mean_entropy": 0.21629514390346366
      }
    },
    {
      "seed": 33,
      "method": "bucket_2",
      "unique_pool_size": 200,
      "sft_weights": [
        1.0639057783794292,
        2.89721281649728,
        -1.7536347802662395
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.935,
        "expected_reward": 0.8506833926706162,
        "mean_entropy": 0.3073137488999156
      },
      "sft_ood": {
        "accuracy": 0.9608333333333333,
        "expected_reward": 0.8931298113249639,
        "mean_entropy": 0.2325851307302936
      },
      "reward_weights": [
        2.0090557263623627,
        3.3371459234008007,
        -1.9590322941237743
      ],
      "reward_id": {
        "accuracy": 0.9908333333333333,
        "expected_reward": 0.8868289303836848,
        "mean_entropy": 0.25810343043135786
      },
      "reward_ood": {
        "accuracy": 0.9908333333333333,
        "expected_reward": 0.910565939643977,
        "mean_entropy": 0.2053983405860314
      }
    },
    {
      "seed": 33,
      "method": "bucket_3",
      "unique_pool_size": 200,
      "sft_weights": [
        2.8709011436021523,
        1.4008383603382812,
        -0.9210011104562793
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.8375,
        "expected_reward": 0.7813630384069287,
        "mean_entropy": 0.3293773813215838
      },
      "sft_ood": {
        "accuracy": 0.9041666666666667,
        "expected_reward": 0.810544873652727,
        "mean_entropy": 0.3622831097932306
      },
      "reward_weights": [
        2.5598666636228815,
        2.7487494265624126,
        -1.6264285171761759
      ],
      "reward_id": {
        "accuracy": 0.955,
        "expected_reward": 0.8696942533500419,
        "mean_entropy": 0.27433857839987097
      },
      "reward_ood": {
        "accuracy": 0.9825,
        "expected_reward": 0.8950311353854747,
        "mean_entropy": 0.23789003451177973
      }
    },
    {
      "seed": 33,
      "method": "bucket_4",
      "unique_pool_size": 200,
      "sft_weights": [
        3.332907953661111,
        0.8756438502469723,
        -0.725393244065203
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7516666666666667,
        "expected_reward": 0.7343512858888971,
        "mean_entropy": 0.3155685117299776
      },
      "sft_ood": {
        "accuracy": 0.825,
        "expected_reward": 0.7433809681535675,
        "mean_entropy": 0.41256237323568196
      },
      "reward_weights": [
        2.9321343441324235,
        2.435385501212131,
        -1.5391487630765206
      ],
      "reward_id": {
        "accuracy": 0.9183333333333333,
        "expected_reward": 0.8515140628968577,
        "mean_entropy": 0.27623065249028983
      },
      "reward_ood": {
        "accuracy": 0.9566666666666667,
        "expected_reward": 0.8821487760719834,
        "mean_entropy": 0.2535705961900335
      }
    },
    {
      "seed": 33,
      "method": "bucket_5",
      "unique_pool_size": 200,
      "sft_weights": [
        3.581532456894627,
        0.7226891338250315,
        -0.4190627754091752
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.73,
        "expected_reward": 0.7111891301870776,
        "mean_entropy": 0.30685311798796794
      },
      "sft_ood": {
        "accuracy": 0.7691666666666667,
        "expected_reward": 0.7015286181964224,
        "mean_entropy": 0.4333705058480822
      },
      "reward_weights": [
        3.194793148034798,
        2.2899451800793416,
        -1.3776842667835179
      ],
      "reward_id": {
        "accuracy": 0.8975,
        "expected_reward": 0.8369394395349984,
        "mean_entropy": 0.2755452528914945
      },
      "reward_ood": {
        "accuracy": 0.9525,
        "expected_reward": 0.8718383796110206,
        "mean_entropy": 0.26521372994171577
      }
    },
    {
      "seed": 44,
      "method": "uniform",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.6711110447079243,
        2.5790784768095114,
        -1.5211134661044816
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9966666666666667,
        "expected_reward": 0.8540283424190096,
        "mean_entropy": 0.32708191397020664
      },
      "sft_ood": {
        "accuracy": 0.9991666666666666,
        "expected_reward": 0.8837711033425742,
        "mean_entropy": 0.2622706707060319
      },
      "reward_weights": [
        2.1608766729247315,
        3.2368885925816397,
        -1.903988724532783
      ],
      "reward_id": {
        "accuracy": 0.9983333333333333,
        "expected_reward": 0.881264949544745,
        "mean_entropy": 0.27176263279263346
      },
      "reward_ood": {
        "accuracy": 0.9991666666666666,
        "expected_reward": 0.9049702260487547,
        "mean_entropy": 0.21753032083471588
      }
    },
    {
      "seed": 44,
      "method": "probability",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.4024909496651308,
        2.008619435127794,
        -1.1782227176370152
      ],
      "mean_batch_ess": 27.258509277744125,
      "sft_id": {
        "accuracy": 0.9133333333333333,
        "expected_reward": 0.8267407605238559,
        "mean_entropy": 0.3336616741863516
      },
      "sft_ood": {
        "accuracy": 0.9558333333333333,
        "expected_reward": 0.8526933980861513,
        "mean_entropy": 0.30931726886601796
      },
      "reward_weights": [
        2.3838116199486166,
        3.01536079604564,
        -1.7763937123445444
      ],
      "reward_id": {
        "accuracy": 0.9758333333333333,
        "expected_reward": 0.8762091213653226,
        "mean_entropy": 0.2771260021229066
      },
      "reward_ood": {
        "accuracy": 0.9891666666666666,
        "expected_reward": 0.8983503738260793,
        "mean_entropy": 0.22958565380886772
      }
    },
    {
      "seed": 44,
      "method": "cll_monotone",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.2659462544993603,
        2.1913011278505743,
        -1.279258040835921
      ],
      "mean_batch_ess": 29.017417004723836,
      "sft_id": {
        "accuracy": 0.9308333333333333,
        "expected_reward": 0.8388575280600296,
        "mean_entropy": 0.32935779778102897
      },
      "sft_ood": {
        "accuracy": 0.9666666666666667,
        "expected_reward": 0.8642195605329056,
        "mean_entropy": 0.29285775720430285
      },
      "reward_weights": [
        2.3430106897517016,
        3.0974049359616282,
        -1.8220682752695252
      ],
      "reward_id": {
        "accuracy": 0.9808333333333333,
        "expected_reward": 0.8787726287843967,
        "mean_entropy": 0.27413337302885193
      },
      "reward_ood": {
        "accuracy": 0.9925,
        "expected_reward": 0.900907556241121,
        "mean_entropy": 0.22493048682045372
      }
    },
    {
      "seed": 44,
      "method": "cll_frontier",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.6453335998958114,
        2.6235701080061236,
        -1.561267348120157
      ],
      "mean_batch_ess": 28.322118917614894,
      "sft_id": {
        "accuracy": 0.99,
        "expected_reward": 0.8552334521858371,
        "mean_entropy": 0.3242206126631463
      },
      "sft_ood": {
        "accuracy": 0.9983333333333333,
        "expected_reward": 0.8856696501845879,
        "mean_entropy": 0.25839968083658005
      },
      "reward_weights": [
        2.161695098466436,
        3.2611409743674136,
        -1.9223710242552736
      ],
      "reward_id": {
        "accuracy": 0.9975,
        "expected_reward": 0.8819178241656042,
        "mean_entropy": 0.2703725372091804
      },
      "reward_ood": {
        "accuracy": 0.9991666666666666,
        "expected_reward": 0.9056654390029998,
        "mean_entropy": 0.2160937473859507
      }
    },
    {
      "seed": 44,
      "method": "bucket_1",
      "unique_pool_size": 200,
      "sft_weights": [
        -0.9862715349629805,
        2.4507726002150103,
        -1.6275389259572608
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7441666666666666,
        "expected_reward": 0.7061636873678454,
        "mean_entropy": 0.35170540627454827
      },
      "sft_ood": {
        "accuracy": 0.9116666666666666,
        "expected_reward": 0.852835665957333,
        "mean_entropy": 0.2643777568014033
      },
      "reward_weights": [
        1.1234882457781603,
        3.0807817881917563,
        -1.8470005613953828
      ],
      "reward_id": {
        "accuracy": 0.9341666666666667,
        "expected_reward": 0.8496692002278505,
        "mean_entropy": 0.3020735956634137
      },
      "reward_ood": {
        "accuracy": 0.9783333333333334,
        "expected_reward": 0.8976101871154145,
        "mean_entropy": 0.22717741544255363
      }
    },
    {
      "seed": 44,
      "method": "bucket_2",
      "unique_pool_size": 200,
      "sft_weights": [
        1.3573719408485296,
        2.8813640866404637,
        -1.5244460999255542
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.96,
        "expected_reward": 0.8521272807018622,
        "mean_entropy": 0.3189097394316518
      },
      "sft_ood": {
        "accuracy": 0.9825,
        "expected_reward": 0.8894353799811086,
        "mean_entropy": 0.24606109257390057
      },
      "reward_weights": [
        2.094012066782739,
        3.346712651159979,
        -1.9215444874362244
      ],
      "reward_id": {
        "accuracy": 0.9925,
        "expected_reward": 0.8824556589439114,
        "mean_entropy": 0.26865917188302285
      },
      "reward_ood": {
        "accuracy": 0.9966666666666667,
        "expected_reward": 0.9070328332863553,
        "mean_entropy": 0.2129038772604162
      }
    },
    {
      "seed": 44,
      "method": "bucket_3",
      "unique_pool_size": 200,
      "sft_weights": [
        2.868553672692428,
        1.4153977889796747,
        -1.0094815191135382
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.845,
        "expected_reward": 0.7827247799015468,
        "mean_entropy": 0.33597410753321655
      },
      "sft_ood": {
        "accuracy": 0.8908333333333334,
        "expected_reward": 0.8060798063978153,
        "mean_entropy": 0.3611682708784148
      },
      "reward_weights": [
        2.606338132948791,
        2.7586456667317494,
        -1.6895237450621308
      ],
      "reward_id": {
        "accuracy": 0.9491666666666667,
        "expected_reward": 0.8663621901513557,
        "mean_entropy": 0.28163490220890447
      },
      "reward_ood": {
        "accuracy": 0.9783333333333334,
        "expected_reward": 0.8896694954360798,
        "mean_entropy": 0.24296576399307404
      }
    },
    {
      "seed": 44,
      "method": "bucket_4",
      "unique_pool_size": 200,
      "sft_weights": [
        3.3541352257499732,
        0.982044616987561,
        -0.5951259588749166
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7725,
        "expected_reward": 0.7347355997782882,
        "mean_entropy": 0.32805788716779927
      },
      "sft_ood": {
        "accuracy": 0.8091666666666667,
        "expected_reward": 0.7386246753874456,
        "mean_entropy": 0.415729479036803
      },
      "reward_weights": [
        2.986542102506506,
        2.4943662594659926,
        -1.496663334299371
      ],
      "reward_id": {
        "accuracy": 0.9141666666666667,
        "expected_reward": 0.8483682809770232,
        "mean_entropy": 0.28092997497012395
      },
      "reward_ood": {
        "accuracy": 0.955,
        "expected_reward": 0.8750295269738415,
        "mean_entropy": 0.26056447233796975
      }
    },
    {
      "seed": 44,
      "method": "bucket_5",
      "unique_pool_size": 200,
      "sft_weights": [
        3.638763835835621,
        0.6435599147893447,
        -0.419319638012859
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.73,
        "expected_reward": 0.7030992765149102,
        "mean_entropy": 0.31957449622941997
      },
      "sft_ood": {
        "accuracy": 0.7316666666666667,
        "expected_reward": 0.6815606463171817,
        "mean_entropy": 0.44538324329192297
      },
      "reward_weights": [
        3.30705128207713,
        2.2346958239763444,
        -1.3790327345123763
      ],
      "reward_id": {
        "accuracy": 0.8858333333333334,
        "expected_reward": 0.8293348351216979,
        "mean_entropy": 0.2789467428702412
      },
      "reward_ood": {
        "accuracy": 0.9333333333333333,
        "expected_reward": 0.8588282041954174,
        "mean_entropy": 0.27692704394766016
      }
    },
    {
      "seed": 55,
      "method": "uniform",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.6785669239415417,
        2.6178890004973616,
        -1.2889360450071747
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9725,
        "expected_reward": 0.8544350776145223,
        "mean_entropy": 0.3193729613032715
      },
      "sft_ood": {
        "accuracy": 0.9791666666666666,
        "expected_reward": 0.873363885580317,
        "mean_entropy": 0.2800865532085644
      },
      "reward_weights": [
        2.0663895158192376,
        3.359488114828245,
        -1.6038986843002616
      ],
      "reward_id": {
        "accuracy": 0.9716666666666667,
        "expected_reward": 0.8793456298151808,
        "mean_entropy": 0.2649390619134582
      },
      "reward_ood": {
        "accuracy": 0.9733333333333334,
        "expected_reward": 0.8961964486443985,
        "mean_entropy": 0.2294168702483596
      }
    },
    {
      "seed": 55,
      "method": "probability",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.3184244632426516,
        2.0824525075013995,
        -1.0070116970898526
      ],
      "mean_batch_ess": 27.263774731850134,
      "sft_id": {
        "accuracy": 0.9108333333333334,
        "expected_reward": 0.8294443608033825,
        "mean_entropy": 0.3273186579058476
      },
      "sft_ood": {
        "accuracy": 0.9391666666666667,
        "expected_reward": 0.8407320752016563,
        "mean_entropy": 0.32801635059443174
      },
      "reward_weights": [
        2.228126547394618,
        3.159655241181193,
        -1.4979941542667896
      ],
      "reward_id": {
        "accuracy": 0.9625,
        "expected_reward": 0.8754821205766545,
        "mean_entropy": 0.2705154729658427
      },
      "reward_ood": {
        "accuracy": 0.9691666666666666,
        "expected_reward": 0.8897340925540994,
        "mean_entropy": 0.2416463066539329
      }
    },
    {
      "seed": 55,
      "method": "cll_monotone",
      "unique_pool_size": 1000,
      "sft_weights": [
        2.2112328855577443,
        2.2522319757668026,
        -1.0960937697120106
      ],
      "mean_batch_ess": 29.0641545276901,
      "sft_id": {
        "accuracy": 0.9283333333333333,
        "expected_reward": 0.8402040562646041,
        "mean_entropy": 0.3217967870345584
      },
      "sft_ood": {
        "accuracy": 0.9533333333333334,
        "expected_reward": 0.8523853997197879,
        "mean_entropy": 0.31163053444217464
      },
      "reward_weights": [
        2.2083380671872845,
        3.2354218127121666,
        -1.5391377766172776
      ],
      "reward_id": {
        "accuracy": 0.965,
        "expected_reward": 0.8775839047679552,
        "mean_entropy": 0.2672008061570889
      },
      "reward_ood": {
        "accuracy": 0.9691666666666666,
        "expected_reward": 0.8922554957407931,
        "mean_entropy": 0.236873667675125
      }
    },
    {
      "seed": 55,
      "method": "cll_frontier",
      "unique_pool_size": 1000,
      "sft_weights": [
        1.5671704793243735,
        2.7032602595704907,
        -1.3297142790593097
      ],
      "mean_batch_ess": 28.442675621673946,
      "sft_id": {
        "accuracy": 0.9775,
        "expected_reward": 0.8549712069972346,
        "mean_entropy": 0.3176788707908481
      },
      "sft_ood": {
        "accuracy": 0.9758333333333333,
        "expected_reward": 0.87679680347683,
        "mean_entropy": 0.27321734947915227
      },
      "reward_weights": [
        2.042670808889806,
        3.3925170828243645,
        -1.6198355350075364
      ],
      "reward_id": {
        "accuracy": 0.97,
        "expected_reward": 0.8797380125633046,
        "mean_entropy": 0.2639826442431079
      },
      "reward_ood": {
        "accuracy": 0.9716666666666667,
        "expected_reward": 0.897098117994694,
        "mean_entropy": 0.22750652349825537
      }
    },
    {
      "seed": 55,
      "method": "bucket_1",
      "unique_pool_size": 200,
      "sft_weights": [
        -1.0543526291941359,
        2.6405150943048614,
        -1.4357367234733769
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7441666666666666,
        "expected_reward": 0.7012244019276677,
        "mean_entropy": 0.34597955013599807
      },
      "sft_ood": {
        "accuracy": 0.905,
        "expected_reward": 0.849391931680818,
        "mean_entropy": 0.26755975366259405
      },
      "reward_weights": [
        0.8417560095193886,
        3.337278807149232,
        -1.6400991041837985
      ],
      "reward_id": {
        "accuracy": 0.9133333333333333,
        "expected_reward": 0.8401989272700685,
        "mean_entropy": 0.2966816975452083
      },
      "reward_ood": {
        "accuracy": 0.96,
        "expected_reward": 0.8926037306259961,
        "mean_entropy": 0.22888544972828617
      }
    },
    {
      "seed": 55,
      "method": "bucket_2",
      "unique_pool_size": 200,
      "sft_weights": [
        1.3433140499062246,
        2.862761175427957,
        -1.3738068742917924
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.9558333333333333,
        "expected_reward": 0.8521314374272517,
        "mean_entropy": 0.315639055019244
      },
      "sft_ood": {
        "accuracy": 0.9708333333333333,
        "expected_reward": 0.8811506304340287,
        "mean_entropy": 0.26168373236797504
      },
      "reward_weights": [
        1.9908140027922365,
        3.4520066708203645,
        -1.6385283414627259
      ],
      "reward_id": {
        "accuracy": 0.9708333333333333,
        "expected_reward": 0.8799065331421065,
        "mean_entropy": 0.26274543965629604
      },
      "reward_ood": {
        "accuracy": 0.9708333333333333,
        "expected_reward": 0.8983606886655163,
        "mean_entropy": 0.22428757695883375
      }
    },
    {
      "seed": 55,
      "method": "bucket_3",
      "unique_pool_size": 200,
      "sft_weights": [
        2.84196108331579,
        1.655537827775793,
        -0.7049512113461718
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.835,
        "expected_reward": 0.789081615510342,
        "mean_entropy": 0.3233149725353472
      },
      "sft_ood": {
        "accuracy": 0.8841666666666667,
        "expected_reward": 0.7951668780319431,
        "mean_entropy": 0.37169046912542686
      },
      "reward_weights": [
        2.467524365287279,
        2.984535682005334,
        -1.3714212201351388
      ],
      "reward_id": {
        "accuracy": 0.9433333333333334,
        "expected_reward": 0.8685400980146107,
        "mean_entropy": 0.27300468109319426
      },
      "reward_ood": {
        "accuracy": 0.9625,
        "expected_reward": 0.8812449851309648,
        "mean_entropy": 0.2535717949622236
      }
    },
    {
      "seed": 55,
      "method": "bucket_4",
      "unique_pool_size": 200,
      "sft_weights": [
        3.3330322935961716,
        1.1023981434958674,
        -0.575392766968823
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7633333333333333,
        "expected_reward": 0.7450365232911795,
        "mean_entropy": 0.30949363804054836
      },
      "sft_ood": {
        "accuracy": 0.8033333333333333,
        "expected_reward": 0.7312137821539505,
        "mean_entropy": 0.4177048616651413
      },
      "reward_weights": [
        2.8596374117132783,
        2.685206766766383,
        -1.267213898919674
      ],
      "reward_id": {
        "accuracy": 0.9141666666666667,
        "expected_reward": 0.8535636784704059,
        "mean_entropy": 0.27352259076438273
      },
      "reward_ood": {
        "accuracy": 0.9408333333333333,
        "expected_reward": 0.8670508360316185,
        "mean_entropy": 0.27236889213341964
      }
    },
    {
      "seed": 55,
      "method": "bucket_5",
      "unique_pool_size": 200,
      "sft_weights": [
        3.5292076011181637,
        0.6949475519693623,
        -0.42369081955789606
      ],
      "mean_batch_ess": 32.0,
      "sft_id": {
        "accuracy": 0.7316666666666667,
        "expected_reward": 0.7122696991696136,
        "mean_entropy": 0.3045293700301091
      },
      "sft_ood": {
        "accuracy": 0.7233333333333334,
        "expected_reward": 0.6698272193541825,
        "mean_entropy": 0.4506256777513702
      },
      "reward_weights": [
        3.1116780508954998,
        2.403479373978373,
        -1.1341263991294206
      ],
      "reward_id": {
        "accuracy": 0.8808333333333334,
        "expected_reward": 0.8351406562542255,
        "mean_entropy": 0.27660893865066016
      },
      "reward_ood": {
        "accuracy": 0.9191666666666667,
        "expected_reward": 0.8495841997023145,
        "mean_entropy": 0.2924387395228071
      }
    }
  ],
  "summary": [
    {
      "method": "uniform",
      "sft_id_accuracy_mean": 0.9883333333333334,
      "sft_id_accuracy_sd": 0.0099128143778086,
      "sft_id_expected_reward_mean": 0.8547673917223427,
      "sft_id_expected_reward_sd": 0.003324186620136372,
      "sft_ood_accuracy_mean": 0.9911666666666666,
      "sft_ood_accuracy_sd": 0.0092157775339662,
      "sft_ood_expected_reward_mean": 0.8826674546447371,
      "sft_ood_expected_reward_sd": 0.006768537533842414,
      "reward_ood_accuracy_mean": 0.9895,
      "reward_ood_accuracy_sd": 0.009940796972297744,
      "reward_ood_expected_reward_mean": 0.9048965629021011,
      "reward_ood_expected_reward_sd": 0.006358988631064186,
      "mean_batch_ess": 32.0
    },
    {
      "method": "probability",
      "sft_id_accuracy_mean": 0.9125,
      "sft_id_accuracy_sd": 0.004039733214513615,
      "sft_id_expected_reward_mean": 0.8291726935864768,
      "sft_id_expected_reward_sd": 0.0024769947219377285,
      "sft_ood_accuracy_mean": 0.9558333333333333,
      "sft_ood_accuracy_sd": 0.010155048005794934,
      "sft_ood_expected_reward_mean": 0.8541319556460564,
      "sft_ood_expected_reward_sd": 0.008785775614751603,
      "reward_ood_accuracy_mean": 0.987,
      "reward_ood_accuracy_sd": 0.01046820583162815,
      "reward_ood_expected_reward_mean": 0.8995316322341931,
      "reward_ood_expected_reward_sd": 0.0066829365519433396,
      "mean_batch_ess": 27.15979375360669
    },
    {
      "method": "cll_monotone",
      "sft_id_accuracy_mean": 0.9318333333333333,
      "sft_id_accuracy_sd": 0.003698723503528688,
      "sft_id_expected_reward_mean": 0.8404063195672743,
      "sft_id_expected_reward_sd": 0.0014212606041074786,
      "sft_ood_accuracy_mean": 0.9675,
      "sft_ood_accuracy_sd": 0.008819171036881965,
      "sft_ood_expected_reward_mean": 0.864914312614697,
      "sft_ood_expected_reward_sd": 0.008198215578105203,
      "reward_ood_accuracy_mean": 0.9878333333333333,
      "reward_ood_accuracy_sd": 0.01058366141223771,
      "reward_ood_expected_reward_mean": 0.9017537749615737,
      "reward_ood_expected_reward_sd": 0.006563245685373822,
      "mean_batch_ess": 28.942109520202365
    },
    {
      "method": "cll_frontier",
      "sft_id_accuracy_mean": 0.9855,
      "sft_id_accuracy_sd": 0.0071589105316381355,
      "sft_id_expected_reward_mean": 0.8552663434336624,
      "sft_id_expected_reward_sd": 0.00422979917400466,
      "sft_ood_accuracy_mean": 0.9881666666666666,
      "sft_ood_accuracy_sd": 0.010297653454387852,
      "sft_ood_expected_reward_mean": 0.884772091505005,
      "sft_ood_expected_reward_sd": 0.006196216162518556,
      "reward_ood_accuracy_mean": 0.9895,
      "reward_ood_accuracy_sd": 0.011282360864050855,
      "reward_ood_expected_reward_mean": 0.9054746999680426,
      "reward_ood_expected_reward_sd": 0.006239284638272,
      "mean_batch_ess": 28.322538954870147
    },
    {
      "method": "bucket_1",
      "sft_id_accuracy_mean": 0.7261666666666666,
      "sft_id_accuracy_sd": 0.01814869141288153,
      "sft_id_expected_reward_mean": 0.6926021541217617,
      "sft_id_expected_reward_sd": 0.013323928470303035,
      "sft_ood_accuracy_mean": 0.8991666666666667,
      "sft_ood_accuracy_sd": 0.011829224638815328,
      "sft_ood_expected_reward_mean": 0.8453893993780589,
      "sft_ood_expected_reward_sd": 0.008001464982248224,
      "reward_ood_accuracy_mean": 0.9615,
      "reward_ood_accuracy_sd": 0.011462499242699629,
      "reward_ood_expected_reward_mean": 0.8951951407023524,
      "reward_ood_expected_reward_sd": 0.0059568918133271614,
      "mean_batch_ess": 32.0
    },
    {
      "method": "bucket_2",
      "sft_id_accuracy_mean": 0.9435,
      "sft_id_accuracy_sd": 0.02257241433451212,
      "sft_id_expected_reward_mean": 0.8464757192567828,
      "sft_id_expected_reward_sd": 0.011545335065042517,
      "sft_ood_accuracy_mean": 0.971,
      "sft_ood_accuracy_sd": 0.011492751338706222,
      "sft_ood_expected_reward_mean": 0.8882712182890601,
      "sft_ood_expected_reward_sd": 0.0061063168838493935,
      "reward_ood_accuracy_mean": 0.9853333333333334,
      "reward_ood_accuracy_sd": 0.009923317097736147,
      "reward_ood_expected_reward_mean": 0.9065765769748698,
      "reward_ood_expected_reward_sd": 0.006108034639975192,
      "mean_batch_ess": 32.0
    },
    {
      "method": "bucket_3",
      "sft_id_accuracy_mean": 0.8405,
      "sft_id_accuracy_sd": 0.008692269873603553,
      "sft_id_expected_reward_mean": 0.7866123800191726,
      "sft_id_expected_reward_sd": 0.006824021426940577,
      "sft_ood_accuracy_mean": 0.9013333333333333,
      "sft_ood_accuracy_sd": 0.015041608956491323,
      "sft_ood_expected_reward_mean": 0.8096192890101969,
      "sft_ood_expected_reward_sd": 0.00989952367428668,
      "reward_ood_accuracy_mean": 0.9786666666666667,
      "reward_ood_accuracy_sd": 0.010079544742805702,
      "reward_ood_expected_reward_mean": 0.8921888532099312,
      "reward_ood_expected_reward_sd": 0.0071758825465496265,
      "mean_batch_ess": 32.0
    },
    {
      "method": "bucket_4",
      "sft_id_accuracy_mean": 0.7673333333333333,
      "sft_id_accuracy_sd": 0.009919817426635306,
      "sft_id_expected_reward_mean": 0.7390831879874364,
      "sft_id_expected_reward_sd": 0.005307909801649744,
      "sft_ood_accuracy_mean": 0.8165,
      "sft_ood_accuracy_sd": 0.010481465122353407,
      "sft_ood_expected_reward_mean": 0.7425306279734082,
      "sft_ood_expected_reward_sd": 0.010272164256714974,
      "reward_ood_accuracy_mean": 0.9566666666666667,
      "reward_ood_accuracy_sd": 0.011008204011352437,
      "reward_ood_expected_reward_mean": 0.8789346722760397,
      "reward_ood_expected_reward_sd": 0.007940940048252703,
      "mean_batch_ess": 32.0
    },
    {
      "method": "bucket_5",
      "sft_id_accuracy_mean": 0.74,
      "sft_id_accuracy_sd": 0.012963624321753367,
      "sft_id_expected_reward_mean": 0.7145517269592716,
      "sft_id_expected_reward_sd": 0.00872369400974448,
      "sft_ood_accuracy_mean": 0.7586666666666667,
      "sft_ood_accuracy_sd": 0.029933955078917752,
      "sft_ood_expected_reward_mean": 0.6977440459233192,
      "sft_ood_expected_reward_sd": 0.023448475003614296,
      "reward_ood_accuracy_mean": 0.9441666666666667,
      "reward_ood_accuracy_sd": 0.017804883849351225,
      "reward_ood_expected_reward_mean": 0.8672051435674761,
      "reward_ood_expected_reward_sd": 0.012709144336547538,
      "mean_batch_ess": 32.0
    }
  ],
  "scope": "Synthetic binary contextual bandit. No LLM, no actual GRPO/PPO. Bucket comparisons are exploratory and not covariate-matched."
}
