From ae131aa2afe7a2b3029ad825fb2120ea9edbfc69 Mon Sep 17 00:00:00 2001 From: Yangwook Kang Date: Wed, 22 Jul 2026 17:46:54 -0700 Subject: [PATCH] report(S5.2): explain why dense mapping wins throughput (per-PE efficiency) Both 1-kv (2 users x 64 PE) and 8-kv (16 users x 8 PE) fill all 128 PEs of the SIP; the throughput gap is per-PE HBM efficiency. Splitting a head over 8 PEs leaves each a single tile, so pipeline-fill and the 8-PE softmax reduce dominate -> 46% HBM util; one head per PE streams contiguously with no reduce -> 76%. Throughput ratio (1.6x) tracks the utilization ratio (1.7x): decode is bandwidth-bound, so throughput follows total HBM efficiency, not PE count. 1-kv spends hardware on latency (8 PE/head for a 4.8x speedup = 60% strong-scaling efficiency). Co-Authored-By: Claude Opus 4.8 (1M context) --- docs/report/1H-codesign-paper/build/main.pdf | Bin 3064176 -> 3066018 bytes .../1H-codesign-paper/sections/05-gqa.tex | 23 ++++++++++++++++-- 2 files changed, 21 insertions(+), 2 deletions(-) diff --git a/docs/report/1H-codesign-paper/build/main.pdf b/docs/report/1H-codesign-paper/build/main.pdf index 35a5490e6c1d324d3677d6ebd8e6f87e978cce1a..9d341a0df2152f1d8dcf7b7ad4be99c23a76541b 100644 GIT binary patch delta 80508 zcmY(qV{k4E45nS%wsmUT)~RjV?Wx@5|g~z+36B=Nstk!{BM??sudk;(3Bd9TTs&Z!*VVl2GUs z0e|n;eMJ(yG#98qZZZtRU2nkd^kShh0sr%HVpY=n#}i>L0P*p|P|Kd4o7{ZAoij7i zAvPqJfwM4heTqfbiy3+(NGB&x#RYKuS}Jk**8}D2I6>)yW=Bz+^JCm?;f%S0x9}&Z zlZ2KB%RA9vf@)K|pRdg+FIA_*b-y605qkA!e^=muF_qwpeDur^aspQ`V0;G@n1Ru8 z>E9K=*PHcX@VABMN;1*q9)f=SKKE%9|GBAK1OAu$xo2(yHkx``MfbaqsSXf_`%QZh=@Xv7;7;s5qJGTv1Rc9%Co_3vR$lwU%9eg46K$gf*|c=~Un&r9 z{AOP7=512DJK|{>e8m1?Ap&53Fjl~ZW3ng`_Kyol(sh0O4ya}|ORLGtK+-kP()1FS zepST8OeS{p+LSBB^`$boRd&}L_b|K1T`Q{41;Z~gO&`yz# 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+256,27 @@ Aggregate throughput at $8$K therefore rises almost linearly with $B$, from \SI{0.86}{}~requests\,/\,\si{\micro\second} (\textsf{1-kv-per-cube}, capped at two users) to \SI{1.41}{} (\textsf{8-kv-per-cube}, sixteen users): the \emph{dense} mapping wins on throughput even though it is the \emph{slowest} -per request, because it packs $8\times$ more concurrent users onto the SIP. -The design conclusion is regime-dependent: \textsf{1-kv-per-cube} is the +per request. + +The reason is worth spelling out, because both mappings fill the same +hardware: at capacity \textsf{1-kv-per-cube} runs two users +$\times$ 64 PEs and \textsf{8-kv-per-cube} sixteen users $\times$ 8 PEs, so +each puts all 128 PEs of the SIP to work. What differs is per-PE +\emph{efficiency}. Splitting one head across eight PEs +(\textsf{1-kv-per-cube}) leaves each PE only $S_{kv}/8$ tokens---a single +tile---so its fixed overheads (pipeline fill on the lazy KV load, and the +eight-PE online-softmax chain-reduce) dominate the little streaming work it +does, and it sustains just \SI{46}{\percent} of the per-PE HBM ceiling. +Giving each head a whole PE (\textsf{8-kv-per-cube}) streams the full +$S_{kv}$-token cache contiguously with no reduce, amortizing those overheads +over $8\times$ more work and reaching \SI{76}{\percent}. The measured +throughput ratio ($1.41/0.86 = 1.6\times$) matches the utilization ratio +($0.76/0.46 = 1.7\times$) almost exactly: because decode is +bandwidth-bound, aggregate throughput is set by total HBM efficiency, not +by PE count. \textsf{1-kv-per-cube} is really spending hardware on +\emph{latency}---eight PEs per head buy only a $4.8\times$ speedup, a $60\%$ +strong-scaling efficiency---and that same $40\%$ overhead is what caps its +throughput once the SIP is full. The design conclusion is regime-dependent: \textsf{1-kv-per-cube} is the latency-optimal choice for single-stream or low-batch decode, while denser mappings are the throughput-optimal choice for high-batch short-context serving. At long context the per-request latencies already scale as $1/C$,