Comparing Direct and Indirect Temporal-Difference Methods for Estimating the Variance of the Return
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Bibliographic record
Abstract
Temporal-difference (TD) learning methods are widely used in reinforcement learning to estimate the expected return for each state, without a model, because of their significant advantages in computational and data efficiency. For many applications involving risk mitigation it would also be useful to estimate the variance of the return by TD methods. In this paper we describe a way of doing this that is substantially simpler than those proposed by Tamar, Di Castro, and Mannor in 2012, or that proposed by White and White in 2016. We show that expectation and variance estimates can be learned by two TD learners operating in series. The trick is to use the square of the TD error of the expectation learner as the reward of the variance learner, and the square of the expectation learner’s discount rate as the discount rate of the variance learner. With these two modifications, the variance learning problem becomes a conventional TD learning problem to which standard theoretical results can be applied. Our formal results are limited to the table lookup case, for which our method is still novel, but the extension to function approximation is immediate and also given here, along with empirical results. Our experimental results show that our direct method behaves just as well as a comparable indirect method, but is generally more robust.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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