i2_s AVX2 GEMM folds the int16 accumulator 10x more often than its own bound requires (7.5% measured)

ggml/src/ggml-cpu/llamafile/sgemm.cpp の non-VNNI tinyBLAS_I2S_AVX::gemm パスから始め、次に ggml/src/ggml-cpu/ggml-cpu-i2s.c の ggml_gemm_i2_i8_s と比較します。ggml/src/ggml-cpu/arch/x86/quants.c の TQ2_0 の処理を読み、non-VNNI パスの前後でベンチマークを実行します。完了の条件は、両方の箇所で安全な int16 folding を遅延させ、VNNI の動作や perplexity を変更しないことです。

Summary

tinyBLAS_I2S_AVX::gemm folds its int16 accumulator into int32 once per
128-weight block. Its own operands license once per ten. Deferring the fold
is worth a measured 7.5 % on the dispatched path, with perplexity unchanged.

This is the path that actually runs: on a default build the per-row vec_dot is
never called (calls=0 sgemm=52068 from the kernel's own counters), so the
saving is on the hot path rather than a fallback.

The bound, from the kernel's own operands

ggml/src/ggml-cpu/llamafile/sgemm.cpp, non-VNNI branch:

__m256i d0 = _mm256_maddubs_epi16(a_vals[r][0], b0);
__m256i d1 = _mm256_maddubs_epi16(a_vals[r][1], b1);
__m256i d2 = _mm256_maddubs_epi16(a_vals[r][2], b2);
__m256i d3 = _mm256_maddubs_epi16(a_vals[r][3], b3);
__m256i sum = _mm256_add_epi16(_mm256_add_epi16(d0, d1),
                               _mm256_add_epi16(d2, d3));
acc[r][c] = _mm256_add_epi32(acc[r][c],
                             _mm256_madd_epi16(sum, one16));   // every block

Codes are masked to 0..3 (the lut comment a few lines above says so) and
activations are int8, so one vpmaddubsw lane takes two products of at most
3 * 127:

per lane, per plane      2 * 3 * 127 = 762
four planes per block    4 *     762 = 3048
safe fold interval       floor(32767 / 3048) = 10 blocks

With the codes an encoder actually emits (0..2; code 3 measured absent in
2,084,044,800 weights of BitNet-b1.58-2B-4T) the bound is 16. Ten is the
conservative figure and is what I measured against.

Measurement

Ryzen 5 3600 (Zen 2, AVX2, no VNNI), same binary, same session, mean of three:

folding every block   10,254,922,296 cycles
folding every ten      9,488,166,828 cycles
difference                      7.5 %

llama-perplexity over the same corpus is identical between the two builds, so
this is a cost and not a trade.

Scope

• Non-VNNI only. The __AVXVNNI__ / __AVX512VNNI__ branch above uses
  _mm256_dpbusd_epi32, which accumulates straight to int32; there is no fold
  to amortise and nothing to gain there.
• A second site has the same shape and I have not benchmarked it:
  ggml/src/ggml-cpu/ggml-cpu-i2s.c, in ggml_gemm_i2_i8_s, builds the same
  sum16 from four maddubs and folds it per block. A fix should probably
  cover both.

Suggested shape of a fix

Hoist the madd_epi16 out of the block loop and run an int16 accumulator for
up to ten blocks, folding at the boundary and at the tail. Upstream ggml does
exactly this for its own ternary types and documents the reasoning in
ggml/src/ggml-cpu/arch/x86/quants.c — TQ2_0 carries the comment
// 16-bit sums, because 256*127 still fits. That is the idiom this kernel is
missing, and the best reference for the change.

Provenance

Found while measuring this kernel as a baseline for unrelated work. The 7.5 %
figure, the bound derivation and the calls=0 sgemm=52068 dispatch counts are
reproducible from https://github.com/purpleskulll/bitnet-t5b — results/i2s_fold_interval.txt
carries the run. I have no patch to offer that I have tested against your CI;
happy to prepare one if the direction is welcome.

I searched issues and PRs for fold, accumulator, int16, tinyBLAS_I2S,
maddubs and madd_epi16 before opening this and found nothing covering it.
#259 is the nearest neighbour and is a different kernel and a different idea.