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arXiv · 2610.40858

FARO: Function-Space Adaptive Return-to-Risk Optimization via Cross-Block Equal-Risk-Contribution Budget Allocation

Xiaojun Wu·Cehao Yang·Xuhui Jiang·Chengjin Xu·Jia Li·Jian Guo
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Abstract

Training a large language model is compute-bound, and current recipes divide the work between optimizers by hand: orthogonalized momentum for attention and MLP matrices, Adam-style updates for the rest. This division rests on human priors about which layer type deserves which geometry, and it fixes a single global update budget that every parameter block shares. We introduce FARO, a function-space adaptive optimizer that makes the partition of parameters into blocks the unit of decision. FARO gives each block a geometry by its realized counterfactual return, then freezes the routes and reallocates the frozen per-block budget by cross-block equal risk contribution (ERC), the risk-parity principle from portfolio theory. On a GPT-style model trained on FineWeb, the framework gives two results. First, its FARO-Adaptive arm reaches 3.2780 nats at the full schedule, matching NorMuonAndAdam (3.2794), the best hand-designed recipe: it recovers the human prior without hand-setting per-layer geometry. Second, FARO-ERC improves on that arm: it lowers the clean-data loss at every one of 6 seeds, by 0.0136 nats on average, a modest but robust gain. Its sign test, paired test, and confidence interval all agree, and it exceeds the measured run-to-run noise at no added time or memory. It works by trimming the riskiest blocks rather than expanding the safest, and it holds up under a corrupted stream, still lowering the mean loss and winning the seed majority. The source code is available at https://github.com/wxj630/FARO.

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Xiaojun Wu, Cehao Yang, Xuhui Jiang, Chengjin Xu, Jia Li, Jian Guo. 2026-10-29. FARO: Function-Space Adaptive Return-to-Risk Optimization via Cross-Block Equal-Risk-Contribution Budget Allocation. https://arxiv.org/abs/2610.40858

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