com-junkawasaki/sqrt-space-kv-paper

Square-Root Space KV Residency for LLM Decode

TL;DR: Williams (STOC 2025) √(t log t) space simulation → set KV checkpoint block (b\approx\sqrt{S\log S}). On real models (135M→14B dense, plus Qwen3.6-35B-A3B MoE), ~50–97% less cache storage depending on architecture and S. Not a TM proof for Transformers—an engineering transfer with measurements.

Headline numbers

Dense (Modal A100, Qwen2.5 bf16)

Model	S=16 384 full → √S	Save
7B	940 MB → 25 MB	97.3%
14B	3.22 GB → 87 MB	97.3%

Hybrid MoE (Modal A100-80GB, Qwen3.6-35B-A3B bf16)

S	full cache	√S resident	Save
4 096	117 MB	38 MB	67.9%
8 192	201 MB	38 MB	81.3%
16 384	369 MB	42 MB	88.5%

Hybrid linear-attn state is O(1) (does not shrink); full-attn K/V shrinks with √S → save grows with S.

Attention fidelity

Block-stream online-softmax vs full: max|Δ| within bf16 noise (~0.016 @ S=2048). Fidelity maintained under compression.

What fails

• Delete cache + full re-forward every token: peak GPU/Metal memory goes up.
• Process RSS alone: often weight-dominated.

Links

• Results (public): https://github.com/com-junkawasaki/sqrt-space-kv
• Code: https://github.com/gftdcojp/cloud-murakumo (scripts/modal_sqrt_kv_bench.py)
• Theory: Williams arXiv:2502.17779 (STOC 2025 Best Paper)
• arXiv draft: 7807366 (cs.CL, submitted 2026-07-10; public id pending)

Cite

@misc{sqrt-space-kv-2026,
  title  = {Simulating Autoregressive Memory with Square-Root Space},
  author = {Kawasaki, Jun},
  year   = {2026},
  note   = {Code: github.com/gftdcojp/cloud-murakumo; Results: github.com/com-junkawasaki/sqrt-space-kv},
  howpublished = {\url{https://github.com/com-junkawasaki/sqrt-space-kv}}
}