Action tokenization — FAST and its learned successor

Papers: FAST (2501.09747, Physical Intelligence) and FASTer (2512.04952, v2). FAST is local canon (#15) — our AR baseline is the π0-FAST recipe — and FASTer was banked from the 2026-08-06 lit radar as the natural rung after the tokenizer-v3 quantile refit. Both re-read at full-text depth for this page. Fed: #5 (tokenization-v3; the learned-VQ follow-on rung now has concrete falsifiers) and #8/#12 (the decode- latency picture).

The theme

Discrete action tokens are what let a language-model trunk treat control as next-token prediction — our whole AR arm rests on them. FAST is the field’s answer to why naive discretization fails and the fix that made AR VLAs competitive; FASTer is the December-2025 bid to replace FAST’s fixed transform with a learned one. Read together they say something specific to us: the tokenizer’s reconstruction quality stopped being the frontier (ours round-trips near-losslessly — owner measurement, 2026-08-05); the live axes are token distribution quality and decode latency.

1. FAST — why binning fails, and the DCT fix (2501.09747)

The problem it solves. Per-dimension binning — the OpenVLA/RT-2 default — collapses at high control frequency. Their account is information-theoretic: as frequency rises, smooth action signals change little per step, so each token’s marginal information approaches zero; highly correlated consecutive tokens make next-token prediction a near-copy task and convergence crawls. Their didactic study makes it vivid: at 800-timestep sampling a binning policy “simply copies the first action.” At 20–50 Hz on real tasks, binning policies were “unable to make progress” at all.

What it contributes. A four-step invertible pipeline: per-dimension quantile normalization (1st→99th percentile mapped to [−1,1] — outlier-robust; exactly the spec our v3 refit targets), DCT per action dimension over the chunk, scale-and-round quantization (scale γ=10 trades lossiness against compression), then BPE over the flattened low-frequency-first sequence (vocab 1024). Compression is strongest exactly where binning dies: 700 naive tokens → 53 for a 50 Hz bimanual shirt-fold chunk (13.2×); rule of thumb ~30 tokens per second per arm. FAST+ is the same pipeline BPE-trained on ~1M cross-embodiment chunks and “closely matches” dataset-specific fits.

What they ran. Five real tasks up to bimanual 50 Hz laundry folding, LIBERO, zero-shot DROID. Against a learned FSQ baseline, FAST is “as good or at times better … despite requiring no separate neural network training.” The headline everyone quotes — π0-FAST matches diffusion π0 with up to 5× fewer GPU-hours — holds (“5x fewer GPU hours” for their flagship comparison; convergence in 3× fewer steps on table bussing), and on zero-shot DROID π0-FAST actually beats diffusion π0 at language following. The counterweight they print themselves: inference is ~7.5× slower — ~750 ms per 1-second chunk on a 4090 vs ~100 ms for the diffusion expert, because 30–60 tokens decode sequentially through the full 2B trunk. Caveats: main results live in bar charts (no numeric task tables), the 5× is a single-run “up to,” and there is no representation-quality measurement anywhere — the evidence that token quality shapes the backbone (KI’s ~95% vs ~85% with FAST-CE vs binning-CE) comes from the KI paper, not this one.

Transfers. Confirmed as our recipe’s foundation. Two working numbers: our v3 refit should reproduce the quantile-normalization spec exactly, and deployment budgeting should treat AR decode latency, not training cost, as the binding constraint — which is precisely the #12 SnapFlow result’s other half (our 1-NFE flow student sidesteps the 750 ms class of cost entirely).

2. FASTer — a learned VQ that internalizes the DCT (2512.04952)

What it contributes. Two separable things. (a) FASTerVQ: the action chunk is reshaped into a 2-D time×dimension grid (their “single-channel image” — not literal pixels), patchified, and passed through a transformer autoencoder into a 3-level residual VQ (codebook 4096). The training loss keeps a DCT-domain L1 term alongside raw-action L1 — the learned tokenizer doesn’t discard FAST’s frequency insight, it internalizes it. Fixed-length output: 21 tokens per single-arm chunk, 84 bimanual, 126 whole-body. (b) Block-wise AR decoding in the VLA: fixed-length tokens make block prediction possible — 21 tokens decode in 3 forward passes instead of 21, plus a small action-expert head.

What they ran. LIBERO average 97.9% vs their π0-FAST reproduction’s 94.2; Simpler-Bridge 87.9% vs 76.5; real XArm 97.5%, whole-body R1Lite 81.0%. Latency (RTX 5090): 112 ms vs π0-FAST’s 197–556 ms single-arm; 237 ms vs 1,100–3,000 ms at 21-DoF whole-body. Tokenizer analytics are the strongest part: FAST’s BPE on Bridge uses only 48% of its vocab with one token eating 9.6% of all occurrences (normalized entropy 0.69); FASTerVQ hits 100% utilization, max frequency 1.35%, entropy 0.91, and reconstructs ~95% of chunks within tolerance where FAST+ manages ~70%.

The scope corrections. Three things the abstract won’t tell you. The speed win is vs AR-FAST, not vs diffusion — diffusion π0 is still faster than FASTer at whole-body (225 vs 237 ms). Their own ablation attributes 2.2 points of the LIBERO headline to the decoding scheme + action expert, not the tokenizer (token-wise 95.5 → block-wise 96.7 → +expert 97.7) — and the expert head collapses to 23.6% if not pretrained. And the cross-backbone table shows the tokenizer gain shrinks as the FAST baseline gets stronger: +17.3 points on their weak InternVL FAST fit, +1.3 on well-tuned Paligemma. All π0-family numbers are author reproductions; no limitations section; no training wall-clock.

Transfers — and the cheap falsifiers. Our FAST fit round-trips near-losslessly, so FASTerVQ’s headline reconstruction advantage is not the operative axis for us. The +1.3-on-a-good-fit row is the honest prior for what a learned-VQ arm would buy. Before any such arm, two CPU-cheap measurements decide (banked into #5): compute our v3 fit’s Table-8-style stats — vocab utilization, max token frequency, normalized unigram entropy — and compare against FAST-on-Bridge’s pathology (48% / 9.6% / 0.69). Near 0.9 entropy → no headroom, the arm dies before it’s born; near the pathology → a real rung. Separately, block-wise decoding of fixed-length tokens is the latency mechanism worth copying independent of the VQ — though for us the 1-NFE flow student (#12) already occupies that deployment slot.