Weak convergence properties of constrained emphatic temporal-difference learning with constant and slowly diminishing stepsize
Bibliographic record
Abstract
We consider the emphatic temporal-difference (TD) algorithm, ETD(λ), for learning the value functions of stationary policies in a discounted, finite state and action Markov decision process. The ETD(λ) algorithm was recently proposed by Sutton, Mahmood, and White (2016) to solve a long-standing divergence problem of the standard TD algorithm when it is applied to off-policy training, where data from an exploratory policy are used to evaluate other policies of interest. The almost sure convergence of ETD(λ) has been proved in our recent work under general off-policy training conditions, but for a narrow range of diminishing stepsize. In this paper we present convergence results for constrained versions of ETD(λ) with constant stepsize and with diminishing stepsize from a broad range. Our results characterize the asymptotic behavior of the trajectory of iterates produced by those algorithms, and are derived by combining key properties of ETD(λ) with powerful convergence theorems from the weak convergence methods in stochastic approximation theory. For the case of constant stepsize, in addition to analyzing the behavior of the algorithms in the limit as the stepsize parameter approaches zero, we also analyze their behavior for a fixed stepsize and bound the deviations of their averaged iterates from the desired solution. These results are obtained by exploiting the weak Feller property of the Markov chains associated with the algorithms, and by using ergodic theorems for weak Feller Markov chains, in conjunction with the convergence results we get from the weak convergence methods. Besides ETD(λ), our analysis also applies to the off-policy TD(λ) algorithm, when the divergence issue is avoided by setting λ sufficiently large. It yields, for that case, new results on the asymptotic convergence properties of constrained off-policy TD(λ) with constant or slowly diminishing stepsize.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.039 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
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