Reinforcement Learning With Non-Cumulative Objective
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Bibliographic record
Abstract
In reinforcement learning, the objective is almost always defined as a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cumulative</i> function over the rewards along the process. However, there are many optimal control and reinforcement learning problems in various application fields, especially in communications and networking, where the objectives are not naturally expressed as summations of the rewards. In this paper, we recognize the prevalence of non-cumulative objectives in various problems, and propose a modification to existing algorithms for optimizing such objectives. Specifically, we dive into the fundamental building block for many optimal control and reinforcement learning algorithms: the Bellman optimality equation. To optimize a non-cumulative objective, we replace the original <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">summation operation</i> in the Bellman update rule with a generalized operation corresponding to the objective. Furthermore, we provide sufficient conditions on the form of the generalized operation as well as assumptions on the Markov decision process under which the globally optimal convergence of the generalized Bellman updates can be guaranteed. We demonstrate the idea experimentally with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bottleneck</i> objective, i.e., the objectives determined by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">minimum</i> reward along the process, on classical optimal control and reinforcement learning tasks, as well as on two network routing problems on maximizing the flow rates.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it