{ "title": "Supporting results: confidence self-training experiments", "snapshot_date": "2026-09-13", "units": { "tables": "percent", "trajectory_metrics": "fraction" }, "protocol": { "models": [ "Qwen2.5-Math-1.5B", "Qwen2.5-Math-7B", "Qwen2.5-7B", "Qwen2.5-14B" ], "adaptation_pool": "30 AIME 2024 questions, answers and solutions removed", "questions_per_update": 1, "responses_per_update": 16, "temperature": 0.5, "response_token_cap": 3072, "context_limit": 4096, "updates": [ 10, 20 ], "learning_rates": [ 1e-05, 1e-06 ], "objective": "negative sum of response-token log probabilities weighted by detached exp(sum log probabilities) + 0.1, averaged over 16 responses globally", "control_weight": 0.1, "standard_evaluation": "greedy final-answer accuracy using frozen project prompts and scorers", "trajectory_prompts": 64, "trajectory_samples_per_prompt": 16, "bootstrap": "pointwise percentile 95% prompt-bootstrap intervals; 2000 repetitions; seed 20270912" }, "tables": [ { "id": "math", "caption": "Math-specialized models: MATH-500 accuracy (%)", "columns": [ "Model / updates / seed", "Original", "Nominal RLSC", "Uniform" ], "rows": [ [ "1.5B 10 2027", "52.60", "46.20", "54.80" ], [ "1.5B 20 2027", "51.80", "43.80", "53.40" ], [ "7B 10 2027", "58.60", "69.80", "62.40" ], [ "7B 20 2027", "58.00", "66.60", "75.00" ], [ "7B 10 2028", "58.20", "60.80", "69.60" ] ], "comparison_note": "Compare within each row. Each 20-update run starts from base. The original scores are the recorded baseline for each condition.", "highlighted_cells_zero_based_including_header": [ [ 1, 3 ], [ 2, 3 ], [ 3, 2 ], [ 4, 3 ], [ 5, 3 ] ] }, { "id": "general", "caption": "General models: MATH-500 accuracy after 10 updates (%)", "columns": [ "Model / seed", "Original", "Nominal RLSC", "Control" ], "rows": [ [ "7B / 2027", "51.20", "14.80", "0.00" ], [ "7B / 2028", "52.00", "0.00", "68.60" ], [ "7B / 2029", "50.80", "17.20", "3.40" ], [ "14B / 2027", "63.80", "51.60", "14.20" ], [ "14B / 2028", "63.40", "0.80", "52.20" ], [ "14B / 2029", "64.40", "19.40", "28.60" ] ], "comparison_note": "Green includes the untouched model. Seeds vary adaptation-question selection/order and generation randomness.", "highlighted_cells_zero_based_including_header": [ [ 1, 1 ], [ 2, 3 ], [ 3, 1 ], [ 4, 1 ], [ 5, 1 ], [ 6, 1 ] ] }, { "id": "lr", "caption": "General 14B: learning rate, sampling seed, and accuracy (%)", "columns": [ "LR / seed", "MATH base", "MATH RLSC", "MATH uniform", "GSM RLSC", "GSM uniform" ], "rows": [ [ "1e-5 / 2027", "64.20", "3.20", "41.20", "4.09", "82.71" ], [ "1e-5 / 2028", "64.00", "1.60", "Missing", "Missing", "Missing" ], [ "1e-6 / 2027", "63.20", "52.00", "78.00", "78.54", "91.43" ], [ "1e-6 / 2028", "64.60", "61.80", "58.60", "91.36", "90.60" ] ], "comparison_note": "All settings target 20 updates with fixed data seed 2027. Green compares MATH including base, and GSM8K between trained arms. Original GSM8K: 89.39%. Missing cells are incomplete measurements.", "highlighted_cells_zero_based_including_header": [ [ 1, 1 ], [ 1, 5 ], [ 2, 1 ], [ 3, 3 ], [ 3, 5 ], [ 4, 1 ], [ 4, 4 ] ] }, { "id": "ranking", "caption": "Choosing one of 16 answers on unchanged models: accuracy (%)", "columns": [ "Selection rule", "7B", "14B" ], "rows": [ [ "Uniform random sample (expected)", "47.86", "60.19" ], [ "Highest summed log probability", "55.40", "61.00" ], [ "Highest mean-token log probability", "49.40", "65.60" ], [ "Shortest response", "48.00", "47.60" ], [ "Any correct among 16 (oracle)", "85.40", "90.60" ] ], "comparison_note": "Green marks the highest value, the oracle upper bound. The oracle uses correctness labels and is not an available confidence selector. All rules compare the same candidate groups.", "highlighted_cells_zero_based_including_header": [ [ 5, 1 ], [ 5, 2 ] ] }, { "id": "cohort", "caption": "Fresh pass@16 at the original and update-20 checkpoints (%)", "columns": [ "Prompt cohort", "N", "Fresh base", "RLSC u20", "Uniform u20" ], "rows": [ [ "All prompts", "64", "90.62", "43.75", "79.69" ], [ "Greedy wrong, sampling rescues", "21", "90.48", "14.29", "66.67" ], [ "Minority-correct", "11", "81.82", "0.00", "45.45" ], [ "First draw wrong, later rescue", "23", "91.30", "30.43", "65.22" ] ], "comparison_note": "N is the number of prompts. Green compares fresh base with both trained arms. Minority-correct: historical greedy failure and 1\u20134 correct samples out of 16. Cohorts overlap.", "highlighted_cells_zero_based_including_header": [ [ 1, 2 ], [ 2, 2 ], [ 3, 2 ], [ 4, 2 ] ] } ], "trajectory": [ { "arm": "base", "update": 0, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.5986328125, "ci95": [ 0.5068359375, 0.6845703125 ] }, "pass@4": { "n": 64, "mean": 0.8171875, "ci95": [ 0.7314646291208792, 0.886461195054945 ] }, "pass@8": { "n": 64, "mean": 0.8828683469308469, "ci95": [ 0.799980574980575, 0.9476252913752914 ] }, "pass@16": { "n": 64, "mean": 0.90625, "ci95": [ 0.828125, 0.96875 ] }, "greedy_accuracy": { "n": 64, "mean": 0.609375, "ci95": [ 0.484375, 0.71875 ] } } }, { "arm": "rlsc", "update": 5, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.359375, "ci95": [ 0.2919921875, 0.427734375 ] }, "pass@4": { "n": 64, "mean": 0.6610233516483517, "ci95": [ 0.5674536401098901, 0.7493389423076923 ] }, "pass@8": { "n": 64, "mean": 0.7686953671328671, "ci95": [ 0.6743747571872571, 0.8521306818181819 ] }, "pass@16": { "n": 64, "mean": 0.84375, "ci95": [ 0.75, 0.921875 ] }, "greedy_accuracy": { "n": 64, "mean": 0.421875, "ci95": [ 0.296875, 0.546875 ] } } }, { "arm": "rlsc", "update": 10, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.3017578125, "ci95": [ 0.2421875, 0.3603515625 ] }, "pass@4": { "n": 64, "mean": 0.6112036401098901, "ci95": [ 0.5149381868131868, 0.7012276785714285 ] }, "pass@8": { "n": 64, "mean": 0.7211599164724165, "ci95": [ 0.6236183954933955, 0.8087303321678322 ] }, "pass@16": { "n": 64, "mean": 0.8125, "ci95": [ 0.703125, 0.90625 ] }, "greedy_accuracy": { "n": 64, "mean": 0.296875, "ci95": [ 0.1875, 0.40625 ] } } }, { "arm": "rlsc", "update": 15, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.123046875, "ci95": [ 0.0888671875, 0.16015625 ] }, "pass@4": { "n": 64, "mean": 0.34423076923076923, "ci95": [ 0.2691191620879121, 0.42112809065934065 ] }, "pass@8": { "n": 64, "mean": 0.4997741841491842, "ci95": [ 0.4004152097902098, 0.5941615675990676 ] }, "pass@16": { "n": 64, "mean": 0.640625, "ci95": [ 0.515625, 0.75 ] }, "greedy_accuracy": { "n": 64, "mean": 0.265625, "ci95": [ 0.171875, 0.375 ] } } }, { "arm": "rlsc", "update": 20, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.044921875, "ci95": [ 0.0302734375, 0.060546875 ] }, "pass@4": { "n": 64, "mean": 0.16171016483516484, "ci95": [ 0.1123798076923077, 0.21492960164835165 ] }, "pass@8": { "n": 64, "mean": 0.28221153846153846, "ci95": [ 0.20208333333333334, 0.36790865384615384 ] }, "pass@16": { "n": 64, "mean": 0.4375, "ci95": [ 0.3125, 0.5625 ] }, "greedy_accuracy": { "n": 64, "mean": 0.0625, "ci95": [ 0.015625, 0.125 ] } } }, { "arm": "alpha_only", "update": 5, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.380859375, "ci95": [ 0.3056640625, 0.4580078125 ] }, "pass@4": { "n": 64, "mean": 0.6911658653846154, "ci95": [ 0.5978794642857143, 0.7720037774725275 ] }, "pass@8": { "n": 64, "mean": 0.7993395493395493, "ci95": [ 0.7058275058275059, 0.8794386169386169 ] }, "pass@16": { "n": 64, "mean": 0.859375, "ci95": [ 0.765625, 0.9375 ] }, "greedy_accuracy": { "n": 64, "mean": 0.421875, "ci95": [ 0.296875, 0.546875 ] } } }, { "arm": "alpha_only", "update": 10, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.49609375, "ci95": [ 0.3994140625, 0.5849609375 ] }, "pass@4": { "n": 64, "mean": 0.7092548076923076, "ci95": [ 0.6026442307692308, 0.8013307005494505 ] }, "pass@8": { "n": 64, "mean": 0.7717863733488733, "ci95": [ 0.6660766317016317, 0.8576061091686091 ] }, "pass@16": { "n": 64, "mean": 0.8125, "ci95": [ 0.703125, 0.90625 ] }, "greedy_accuracy": { "n": 64, "mean": 0.453125, "ci95": [ 0.328125, 0.578125 ] } } }, { "arm": "alpha_only", "update": 15, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.458984375, "ci95": [ 0.3671875, 0.5478515625 ] }, "pass@4": { "n": 64, "mean": 0.6842376373626373, "ci95": [ 0.5795587225274725, 0.7760044642857142 ] }, "pass@8": { "n": 64, "mean": 0.7604251651126651, "ci95": [ 0.6553491647241647, 0.8459365287490288 ] }, "pass@16": { "n": 64, "mean": 0.8125, "ci95": [ 0.703125, 0.90625 ] }, "greedy_accuracy": { "n": 64, "mean": 0.4375, "ci95": [ 0.3125, 0.5625 ] } } }, { "arm": "alpha_only", "update": 20, "prompts": 64, "metrics": { "pass@1": { "n": 64, "mean": 0.3984375, "ci95": [ 0.314453125, 0.4833984375 ] }, "pass@4": { "n": 64, "mean": 0.6498540521978022, "ci95": [ 0.5481541895604396, 0.7402300824175825 ] }, "pass@8": { "n": 64, "mean": 0.7400762432012432, "ci95": [ 0.6350135975135975, 0.8328865578865579 ] }, "pass@16": { "n": 64, "mean": 0.796875, "ci95": [ 0.6875, 0.890625 ] }, "greedy_accuracy": { "n": 64, "mean": 0.421875, "ci95": [ 0.296875, 0.546875 ] } } } ], "applied_weight_audit": { "model": "general Qwen2.5-14B", "seed": 2027, "updates": 20, "responses": 320, "underflowed_fp32": 277, "largest_nonzero_confidence_approx": 8.75e-10, "all_applied_weights": 0.10000000149011612, "same_scalar_weights_as_control": true }, "caveats": [ "This is a recorded study snapshot, not live scheduler state.", "The audited trajectory pair had identical applied scalar weights; its curves cannot isolate active confidence weighting.", "AIME24 overlaps adaptation.", "Per-row recorded original baselines are preserved.", "Cohorts overlap and use independent historical discovery samples.", "Zero correct among 16 samples is not zero underlying success probability.", "Truncation and the generation budget affect answer coverage.", "Confidence selection uses unchanged models and does not establish a training benefit.", "Bootstrap intervals do not quantify independent training-seed variation." ] }