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LIT-1069: Scaling Scaling Laws with Board Games

2 min read

Properties4
StatusActive — read in full 2026-10-15 from the author's arXiv preprint, version 2 of 2021-04-15, the latest (arxiv.org/pdf/2104.03113v2; v1 diffed against it, same methods and numbers). Worth reading as a clean small-scale scaling law in problem size as well as compute: AlphaZero on Hex, boards 3×3 to 9×9, one change-point frontier of about 500 Elo per tenfold of training compute, with the compute for perfect play rising about 7× per added board row and fits on small boards predicting larger ones. For QUESTION-095 it shows that, on 9×9 at fixed Elo, training and tree-search compute substitute along a straight log–log line, about 15× less search per 10× more training (fit: log test = −1.2·log train + 0.004·Elo + 29), over roughly 1e14–1e16 FLOPS-seconds of training and down to a one-node search; it does not count the games or moves the trained agent will play, or define the unit of test-time compute, so it prices the iso-performance substitution but not amortization, and has no terms for query shift, staleness or erosion of search.
Tagslearning-theory, anthology-candidate
Cited as “paper” byNOTE-884 — Jones — Scaling scaling laws with board games
Cited as “source” byTHEORY-342 — In AlphaZero on 9×9 Hex, at fixed playing strength, training compute and tree-search compute substitute along a straight line in log–log space, about 15× less search for each 10× more training over about two decades of training compute, down to a one-node search; the trade is measured for one search and counts no games played

Graph View

Jones, A. L. (2021). Scaling scaling laws with board games. ARXIV-2104.03113

Key takeaways

Read in full (NOTE-884). Training AlphaZero on Hex at board sizes 3×3 to 9×9, the best Elo reachable for a given training compute follows one change-point curve across sizes: an incline of about 500 Elo per tenfold of training compute, with the compute for perfect play rising about 7× and the compute to leave random play about 4× per added board row, and fits on small boards predicting larger ones. On 9×9 only, train-time and test-time (tree-search) compute substitute at a fixed rate: at equal Elo, each 10× of train-time compute removes about 15× of test-time compute (fit: log test = −1.2·log train + 0.004·Elo + 29), down to a one-node search, over roughly 1e14–1e16 FLOPS-seconds of training. The trade is an iso-performance curve for a single search; the paper never counts how many games or moves the trained agent will play, so it does not price amortization, and it does not state the unit of test-time compute. One game, small boards, Elo anchored to a perfect player, and no uncertainty on the fitted trade-off coefficients.

Why this is here

The owner, 2026-10-15: “there might also be something useful in the scaling law literature, regarding valuation of pre-training investment (amortized compute) vs inference time search/compute.” Read for QUESTION-095: compute spent once in training is amortized over every later query, against compute spent at inference per query. Machine learning, filed here for that question with the anthology flag (ADR-005, ADR-013); the anthology may hold its own reading.

The cleanest measured exchange rate between amortized (training) and per-query (search) compute.

The reading (NOTE-884) narrows the line above. The paper measures the exchange rate within one system (about 15× less search per 10× more training, at fixed Elo, 9×9 Hex) but prices neither the number of games played nor a per-query unit; the expected-reuse term must be added from outside.

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