Source
Jones (2021), LIT-1069 (arXiv 2104.03113v2), read in NOTE-884.
What was actually shown
The frontier. About 200 AlphaZero models and 2,800 snapshots on Hex boards 3×3 to 9×9, rated by Elo with perfect play (MoHex) at zero, fitted by a five-parameter change-point model. “The slope of the incline is 500 Elo per order of magnitude increase in compute”; “The minimum compute needed for perfect play increases 7× for each increment in board size.” Fits on small boards predict larger ones, with residual variance falling as boards are added. Training compute is varied by depth, width and run length only, with a 64-node search in training and in evaluation.
The exchange rate. For 9×9 snapshots, test-time tree size was varied from 1 to 512 nodes, and for each Elo level the minimum training and test-time compute reaching it were traced. “the trade-off is linear in log-compute: for each additional 10× of train-time compute, about 15× of test-time compute can be eliminated, down to a floor of a single-node tree search.” The fitted line is log(test) = −1.2·log(train) + 0.004·elo + 29; a slope of −1.2 is about 16× per decade. The contours cover training compute of about 1e14 to 1e16 FLOPS-seconds, test-time compute of about 1e6 to 1e9, and Elo from −1500 to −250.
What could have come out the other way. The contours could have curved, or search could have been far cheaper than training at equal Elo; the author expected the second and was “surprised that one could easily substitute for the other”. His account of the near one-for-one rate: “an optimization at test-time needs only optimise over one sample, while train-time compute meanwhile must optimise over the entire distribution of samples.”
What this does not say
- That the trade prices amortization. The paper never says how many games or moves the trained agent will play, and does not state whether test-time compute is per move, per game or per evaluation. Adding a reuse count N to the fit (total cost train + N·k·train^−1.2) is the record’s arithmetic in NOTE-884, not the paper’s.
- That it holds outside the measured range. Below a one-node search no further training replaces search. Beyond 1e16 FLOPS-seconds, at Elo near perfect play, or on other board sizes the trade was not measured.
- That the rate is general. One game, small boards, a modified AlphaZero with small fully connected nets, Elo assuming transitivity (the top-agent check moves fitted parameters by up to about 40%), and no standard errors on −1.2, 0.004 or 29. The author asks twice for the test on other games.
- That the two fits agree. Holding test-time compute fixed, the trade fit implies about 300 Elo per decade of training (1.2/0.004), against the frontier’s 500 at a fixed 64-node search; the paper does not reconcile them (NOTE-884’s arithmetic).
- That a trained policy stays fit. Training and test draw positions from the same game and board, so the later query never leaves the training distribution. Nothing measures a shift, a stale policy or bias carried by training beyond what Elo absorbs.
- Anything about how to allocate compute in practice. This is a measured relation in one system. A rule for spending compute would be a practice and belongs in the Anthology of the SOTA.
Connections
- QUESTION-095’s note of 2026-10-15 places this as precompute-and-leave-a- result (in the weights) against compute at the moment of use, lying on one smooth iso-performance curve, the shape the owner’s conjecture predicts for those two strategies. It supplies the computational part of the trade (CLAIM-295’s second component) and none of expected reuse, retrieval cost, staleness or erosion. A one-node search is the floor of per-use cost: the retrieval cost of the precomputed store, which here is not traded.
- THEORY-331 adds the term this lacks, expected use, in language models.