What every section and number below means: reading-reports.md.
Cost model scaled to this device from one probe point: block x0.14, cpu x0.10, whole-matrix x1.00 (1.00 is an M1).
Machine state: load 3.1/18 at the start, load 5.1/18 at the end; power mains.
Probe point after the sweep relative to before it: block x1.01, cpu x0.83, tg x1.00 (stable).
Generated by tuning/tune_eigh.py from 205 (N, batch) points, N in [2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096], batch in [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096], four backends, two or more passes, min-of-repeats.
Row for kTuned[] in src/eigh.mm:
// device, GPU cores, simd_max_n, block_min_n, block_min_n_batched, block_min_batch, gpu_max_n, gpu_min_batch_times_n, gpu_min_batch, values_gpu_max_n, values_gpu_min_batch_times_n, values_gpu_min_batch, tridiag_min_n, values_tridiag_min_n, tridiag_max_batch, values_tridiag_max_batch, ql_min_n, ql_max_n, share_min_batch, gpu_big_batch_max_n, gpu_big_batch_min, values_band_min_n
{"Apple M5 Pro", 20, 0, 96, 0, 0, 48, 16384, 1, 48, 16384, 1, 1024, 1536, 4, 2, 12, 64, 1024, 64, 1024, 4096},
To try it without rebuilding:
EIGH_SIMD_MAX_N=0 EIGH_BLOCK_MIN_N=96 EIGH_BLOCK_MIN_N_BATCHED=0 EIGH_BLOCK_MIN_BATCH=0 EIGH_GPU_MAX_N=48 EIGH_GPU_MIN_BATCH_TIMES_N=16384 EIGH_GPU_MIN_BATCH=1 EIGH_TRIDIAG_MIN_N=1024 EIGH_VALUES_TRIDIAG_MIN_N=1536 EIGH_TRIDIAG_MAX_BATCH=4 EIGH_VALUES_TRIDIAG_MAX_BATCH=2 EIGH_QL_MIN_N=12 EIGH_QL_MAX_N=64 EIGH_SHARE_MIN_BATCH=1024 EIGH_GPU_BIG_BATCH_MAX_N=64 EIGH_GPU_BIG_BATCH_MIN=1024 EIGH_VALUES_BAND_MIN_N=4096 EIGH_VALUES_GPU_MAX_N=48 EIGH_VALUES_GPU_MIN_BATCH_TIMES_N=16384 EIGH_VALUES_GPU_MIN_BATCH=1
The policy in effect on this device came from tuned:Apple M5 Pro. It differs from the fitted one; see the warnings.
Against the best measured backend at every point the whole rule scores 1.0095 geometric-mean regret, worst 1.35x, 8 of 205 points losing more than 10%, and 1.000x the oracle’s total time. The decision is fitted in two stages, below, because the CPU routing would otherwise hide the GPU backend crossover.
Scored against the best GPU backend at each of the 205 points, as if there were no CPU: this is the rule a forced-GPU call (EIGH_DEVICE=gpu) and the detail entry points follow, and it is what a GPU with more cores will lean on.
| rule | geomean regret | worst | >10% | total time / oracle | est. picks |
|---|---|---|---|---|---|
| policy in effect (‘8’, ‘96’, ‘none’, ‘none’) | 1.0277 | 1.76x | 17 | 1.003 | 0 |
| fitted (‘0’, ‘96’, ‘none’, ‘none’) | 1.0186 | 1.57x | 10 | 1.003 | 0 |
3 of 128 (simd_max_n, block_min_n) pairs are within 0.5% of the best geomean: simd_max_n 0 .. 4, block_min_n 96 .. 96.
xychart-beta
title "Regret by block_min_n"
x-axis "block_min_n" [32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, none]
y-axis "geometric-mean regret" 1.0 --> 2.54
line [1.1951, 1.1013, 1.0369, 1.0186, 1.0432, 1.1130, 1.2388, 1.3805, 1.5499, 1.7232, 1.9359, 2.1199, 2.2580, 2.4049, 2.4633, 2.5231]
| block_min_n | 32 | 48 | 64 | 96 | 128 | 192 | 256 | 384 | 512 | 768 | 1024 | 1536 | 2048 | 3072 | 4096 | none |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| geomean | 1.1951 | 1.1013 | 1.0369 | 1.0186 | 1.0432 | 1.1130 | 1.2388 | 1.3805 | 1.5499 | 1.7232 | 1.9359 | 2.1199 | 2.2580 | 2.4049 | 2.4633 | 2.5231 |
| worst | 8.60x | 4.71x | 2.64x | 1.57x | 1.93x | 2.24x | 4.49x | 6.04x | 10.01x | 15.74x | 15.74x | 15.74x | 15.74x | 15.74x | 15.74x | 15.74x |
xychart-beta
title "Regret by simd_max_n"
x-axis "simd_max_n" [0, 2, 4, 8, 12, 16, 24, 32]
y-axis "geometric-mean regret" 1.0 --> 1.18
line [1.0186, 1.0207, 1.0223, 1.0277, 1.0417, 1.0601, 1.1021, 1.1683]
| simd_max_n | 0 | 2 | 4 | 8 | 12 | 16 | 24 | 32 |
|---|---|---|---|---|---|---|---|---|
| geomean | 1.0186 | 1.0207 | 1.0223 | 1.0277 | 1.0417 | 1.0601 | 1.1021 | 1.1683 |
| worst | 1.57x | 1.57x | 1.57x | 1.76x | 2.63x | 2.63x | 3.06x | 4.85x |
Held-out check of a batch-dependent crossover (block from a lower N once the batch is large enough). Fitted on 112 points, scored on the other 93; the verdict is a bootstrap over the test points.
| rule | fitted on train | train geomean | test geomean | test worst | verdict |
|---|---|---|---|---|---|
| two constants | [0, 96, 1000000000, 1000000000] | 1.0229 | 1.0134 | 1.53x | baseline |
| batch-dependent block crossover | {“block_lo”: 64, “batch_hi”: 128} | 1.0026 | 1.0071 | 1.49x | rejected (better in 78% of resamples, median gain 0.6%) |
Best GPU backend per point (s simd, t threadgroup, B block, q ql, Q ql shared with the CPU), then what the split picks, the ql window of stage 1b included:
N \ batch 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
2 s q q t t q q t q q q q s
4 q q s q s t q q q s q q q
8 t s q q t t q t s t s s s
12 t t t t t t q t q q q q q
16 t t t t t t t t q q q q q
24 t t t t t t t q q q q q q
32 t t t t t t q q q q q q q
48 t t t t t q q q q q q q q
64 t t t t t q q q q q q q q
96 t t t t t B B B B B B B B
128 B B B B B B B B B B B B B
192 B B B B B B B B B B B B B
256 B B B B B B B B B B B . .
384 B B B B B B B B B B . . .
512 B B B B B B B B . . . . .
768 B B B B B B B . . . . . .
1024 B B B B B . . . . . . . .
1536 B B B . . . . . . . . . .
2048 B B B . . . . . . . . . .
3072 B . . . . . . . . . . . .
4096 B . . . . . . . . . . . .
N \ batch 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
2 t t t t t t t t t t t t t
4 t t t t t t t t t t t t t
8 t t t t t t t t t t t t t
12 q q q q q q q q q q Q Q Q
16 q q q q q q q q q q Q Q Q
24 q q q q q q q q q q Q Q Q
32 q q q q q q q q q q Q Q Q
48 q q q q q q q q q q Q Q Q
64 q q q q q q q q q q Q Q Q
96 B B B B B B B B B B B B B
128 B B B B B B B B B B B B B
192 B B B B B B B B B B B B B
256 B B B B B B B B B B B . .
384 B B B B B B B B B B . . .
512 B B B B B B B B . . . . .
768 B B B B B B B . . . . . .
1024 B B B B B . . . . . . . .
1536 B B B . . . . . . . . . .
2048 B B B . . . . . . . . . .
3072 B . . . . . . . . . . . .
4096 B . . . . . . . . . . . .
Inside a window of N, ql (tridiagonalization and implicit QL, one threadgroup per matrix, N <= 87 on this device) instead of the Jacobi backend the split picks, fitted over the 205 points against the best GPU backend, ql included. Chosen: N = 12 .. 64.
| rule | geomean regret | worst | >10% | total time / oracle | est. picks |
|---|---|---|---|---|---|
| without ql (the split alone) | 1.1580 | 3.65x | 48 | 1.010 | 0 |
| with ql for N in (12, 64) | 1.0474 | 1.62x | 34 | 1.000 | 0 |
3 windows are within 0.5% of the best geomean: ql_min_n 2 .. 12, ql_max_n 64 .. 64.
Held out: fitted on 112 points (window (2, 64)), scored on the other 93: geomean 1.0468x, worst 1.40x, against 1.1511x, worst 3.32x without ql.
ql over the best Jacobi backend, N x batch: 2x1 0.93x, 2x2 1.11x, 2x4 1.01x, 2x8 0.91x, 2x16 1.00x, 2x32 1.02x, 2x64 1.07x, 2x128 1.00x, 2x256 1.02x, 2x512 1.00x, 2x1024 1.07x, 2x2048 1.04x, 2x4096 1.00x, 4x1 1.06x, 4x2 1.09x, 4x4 0.95x, 4x8 1.03x, 4x16 0.96x, 4x32 0.97x, 4x64 1.22x, 4x128 1.27x, 4x256 1.03x, 4x512 0.94x, 4x1024 1.04x, 4x2048 1.11x, 4x4096 1.03x, 8x1 0.88x, 8x2 0.86x, 8x4 1.06x, 8x8 1.01x, 8x16 0.88x, 8x32 0.91x, 8x64 1.03x, 8x128 0.95x, 8x256 0.99x, 8x512 0.98x, 8x1024 0.96x, 8x2048 0.93x, 8x4096 0.96x, 12x1 0.95x, 12x2 0.92x, 12x4 0.87x, 12x8 0.95x, 12x16 0.95x, 12x32 0.96x, 12x64 1.00x, 12x128 1.00x, 12x256 1.19x, 12x512 1.19x, 12x1024 1.26x, 12x2048 1.40x, 12x4096 1.34x, 16x1 0.89x, 16x2 0.90x, 16x4 0.93x, 16x8 0.89x, 16x16 0.88x, 16x32 0.91x, 16x64 0.91x, 16x128 0.98x, 16x256 1.24x, 16x512 1.28x, 16x1024 1.48x, 16x2048 1.41x, 16x4096 1.46x, 24x1 0.77x, 24x2 0.77x, 24x4 0.69x, 24x8 0.75x, 24x16 0.74x, 24x32 0.76x, 24x64 0.92x, 24x128 1.19x, 24x256 1.93x, 24x512 1.93x, 24x1024 2.02x, 24x2048 2.17x, 24x4096 2.34x, 32x1 0.62x, 32x2 0.68x, 32x4 0.72x, 32x8 0.72x, 32x16 0.75x, 32x32 0.79x, 32x64 1.07x, 32x128 1.71x, 32x256 2.53x, 32x512 2.21x, 32x1024 2.72x, 32x2048 2.98x, 32x4096 3.05x, 48x1 0.80x, 48x2 0.81x, 48x4 0.85x, 48x8 0.81x, 48x16 0.87x, 48x32 1.13x, 48x64 1.76x, 48x128 2.68x, 48x256 2.48x, 48x512 3.11x, 48x1024 3.20x, 48x2048 3.32x, 48x4096 3.65x, 64x1 0.92x, 64x2 0.95x, 64x4 0.94x, 64x8 0.91x, 64x16 0.90x, 64x32 1.30x, 64x64 2.28x, 64x128 2.03x, 64x256 2.07x, 64x512 2.06x, 64x1024 2.17x, 64x2048 2.17x, 64x4096 2.16x
From a batch on, ql_share: ql and the CPU path at once on one batch, the GPU taking chunks from the front and the CPU from the back. Fitted against the best GPU backend, the shared one included, on the 42 points where it was timed and ql is the GPU’s choice. Chosen: from batch 1024.
| rule | geomean regret | worst | >10% | total time / oracle | est. picks |
|---|---|---|---|---|---|
| ql alone | 1.1334 | 1.94x | 15 | 1.501 | 0 |
| shared from batch 1024 | 1.0509 | 1.54x | 7 | 1.043 | 0 |
ql_share over ql alone, N x batch: 12x64 0.70x, 12x128 0.69x, 12x256 0.62x, 12x512 0.74x, 12x1024 0.84x, 12x2048 0.88x, 12x4096 1.10x, 16x64 0.72x, 16x128 0.75x, 16x256 0.67x, 16x512 0.80x, 16x1024 0.78x, 16x2048 1.09x, 16x4096 1.13x, 24x64 0.81x, 24x128 0.82x, 24x256 0.76x, 24x512 0.65x, 24x1024 1.02x, 24x2048 1.08x, 24x4096 1.20x, 32x64 0.83x, 32x128 0.85x, 32x256 0.81x, 32x512 0.83x, 32x1024 1.10x, 32x2048 1.08x, 32x4096 1.19x, 48x64 0.87x, 48x128 0.88x, 48x256 1.54x, 48x512 1.23x, 48x1024 1.26x, 48x2048 1.41x, 48x4096 1.44x, 64x64 0.90x, 64x128 0.95x, 64x256 1.36x, 64x512 1.47x, 64x1024 1.72x, 64x2048 1.90x, 64x4096 1.94x
Given the split above, GPU iff N <= gpu_max_n, batch * N >= gpu_min_batch_times_n and batch >= gpu_min_batch, scored against the best of all four backends. worst is over the points where the chosen backend was timed; a pick the cost model had to guess is listed in the warnings instead.
| rule | geomean regret | worst | >10% | total time / oracle | est. picks |
|---|---|---|---|---|---|
| oracle (best per point) | 1.0000 | 1.00x | 0 | 1.000 | 0 |
| policy in effect (‘48’, ‘16384’, ‘1’) | 1.0095 | 1.35x | 8 | 1.000 | 0 |
| fitted (‘48’, ‘16384’, ‘1’) | 1.0095 | 1.35x | 8 | 1.000 | 0 |
Large batches: the GPU also for N above gpu_max_n up to 64 in a batch of at least 1024, fitted with the product rule (per cap, the rule, the clause over it and the rule again given the clause, the best kept) (product rule alone 1.0165, worst 1.68x; chosen 1.0095, worst 1.35x).
24 of 1188 combinations are within 0.5% of the best geomean: gpu_max_n 48 .. 64, gpu_min_batch_times_n 8192 .. 16384, gpu_min_batch 1 .. 32.
xychart-beta
title "Regret by gpu_min_batch_times_n"
x-axis "gpu_min_batch_times_n" [0, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, none]
y-axis "geometric-mean regret" 1.0 --> 1.87
line [1.8511, 1.3465, 1.2350, 1.1542, 1.0982, 1.0609, 1.0323, 1.0159, 1.0092, 1.0095, 1.0487]
| gpu_min_batch_times_n | 0 | 64 | 128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 | 16384 | none |
|---|---|---|---|---|---|---|---|---|---|---|---|
| geomean | 1.8511 | 1.3465 | 1.2350 | 1.1542 | 1.0982 | 1.0609 | 1.0323 | 1.0159 | 1.0092 | 1.0095 | 1.0487 |
| worst | 119.07x | 11.75x | 10.02x | 5.88x | 3.97x | 3.00x | 2.20x | 1.54x | 1.54x | 1.35x | 2.19x |
xychart-beta
title "Regret by gpu_min_batch"
x-axis "gpu_min_batch" [1, 2, 4, 8, 16, 32]
y-axis "geometric-mean regret" 1.0 --> 1.02
line [1.0095, 1.0095, 1.0095, 1.0095, 1.0095, 1.0095]
| gpu_min_batch | 1 | 2 | 4 | 8 | 16 | 32 |
|---|---|---|---|---|---|---|
| geomean | 1.0095 | 1.0095 | 1.0095 | 1.0095 | 1.0095 | 1.0095 |
| worst | 1.35x | 1.35x | 1.35x | 1.35x | 1.35x | 1.35x |
xychart-beta
title "Regret by gpu_max_n"
x-axis "gpu_max_n" [16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, none]
y-axis "geometric-mean regret" 1.0 --> 1.37
line [1.0107, 1.0107, 1.0102, 1.0095, 1.0123, 1.0433, 1.0885, 1.1479, 1.2031, 1.2589, 1.3027, 1.3362, 1.3509, 1.3509, 1.3509, 1.3509, 1.3509, 1.3509]
| gpu_max_n | 16 | 24 | 32 | 48 | 64 | 96 | 128 | 192 | 256 | 384 | 512 | 768 | 1024 | 1536 | 2048 | 3072 | 4096 | none |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| geomean | 1.0107 | 1.0107 | 1.0102 | 1.0095 | 1.0123 | 1.0433 | 1.0885 | 1.1479 | 1.2031 | 1.2589 | 1.3027 | 1.3362 | 1.3509 | 1.3509 | 1.3509 | 1.3509 | 1.3509 | 1.3509 |
| worst | 1.42x | 1.42x | 1.42x | 1.35x | 1.52x | 3.62x | 4.59x | 6.54x | 7.61x | 10.55x | 11.13x | 14.00x | 14.00x | 14.00x | 14.00x | 14.00x | 14.00x | 14.00x |
Held-out check of a per-N boundary (a lookup table of the smallest batch at which the GPU wins, per N) against the product rule. Fitted on 112 points, scored on the other 93.
| rule | fitted on train | train geomean | test geomean | test worst | verdict |
|---|---|---|---|---|---|
| product rule | [48, 16384, 1] | 1.0099 | 1.0090 | 1.35x | baseline |
| per-N table | {“min_batch_by_n”: {“2”: 1024, “4”: 4096, “8”: null, “12”: 2048, “16”: 2048, “24”: 256, “32”: 512, “48”: 512, “64”: 2048, “96”: null, “128”: null, “192”: null, “256”: null, “384”: null, “512”: null, “768”: null, “1024”: null, “1536”: null, “2048”: null, “3072”: null, “4096”: null}} | 1.0082 | 1.0183 | 1.48x | rejected (better in 3% of resamples, median gain -0.8%) |
Best backend per point (c CPU, s simd, t threadgroup, B block, q ql, Q ql shared with the CPU, . not measured), what the whole rule picks, and the speedup of the best GPU backend over the CPU:
N \ batch 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
2 c c c c c c c c c c q c s
4 c c c c c c c c c c c c q
8 c c c c c c c c c c s s s
12 c c c c c c c c c c q q Q
16 c c c c c c c c c q q Q Q
24 c c c c c c c c q q Q Q Q
32 c c c c c c c c q q Q Q Q
48 c c c c c c c c Q Q Q Q Q
64 c c c c c c c c c Q Q Q Q
96 c c c c c c c c c c c c c
128 c c c c c c c c c c c c c
192 c c c c c c c c c c c c c
256 c c c c c c c c c c c . .
384 c c c c c c c c c c . . .
512 c c c c c c c c . . . . .
768 c c c c c c c . . . . . .
1024 c c c c c . . . . . . . .
1536 c c c . . . . . . . . . .
2048 c c c . . . . . . . . . .
3072 c . . . . . . . . . . . .
4096 c . . . . . . . . . . . .
N \ batch 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
2 c c c c c c c c c c c c c
4 c c c c c c c c c c c c t
8 c c c c c c c c c c c t t
12 c c c c c c c c c c c Q Q
16 c c c c c c c c c c Q Q Q
24 c c c c c c c c c c Q Q Q
32 c c c c c c c c c q Q Q Q
48 c c c c c c c c c q Q Q Q
64 c c c c c c c c c c Q Q Q
96 c c c c c c c c c c c c c
128 c c c c c c c c c c c c c
192 c c c c c c c c c c c c c
256 c c c c c c c c c c c . .
384 c c c c c c c c c c . . .
512 c c c c c c c c . . . . .
768 c c c c c c c . . . . . .
1024 c c c c c . . . . . . . .
1536 c c c . . . . . . . . . .
2048 c c c . . . . . . . . . .
3072 c . . . . . . . . . . . .
4096 c . . . . . . . . . . . .
N \ batch 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
2 0.01 0.02 0.04 0.03 0.06 0.12 0.20 0.38 0.61 0.75 1.21 0.85 1.01
4 0.01 0.03 0.05 0.12 0.11 0.22 0.37 0.71 0.49 0.72 0.81 0.98 1.19
8 0.03 0.05 0.10 0.14 0.23 0.45 0.44 0.66 0.67 0.81 1.03 1.22 1.42
12 0.03 0.08 0.10 0.15 0.28 0.34 0.62 0.60 0.79 0.94 1.10 1.27 1.40
16 0.05 0.09 0.10 0.17 0.30 0.45 0.54 0.62 0.86 1.06 1.21 1.34 1.57
24 0.08 0.11 0.14 0.27 0.35 0.47 0.52 0.68 1.08 1.26 1.42 1.87 1.82
32 0.11 0.13 0.14 0.24 0.38 0.42 0.45 0.71 1.06 1.11 1.56 1.71 1.73
48 0.12 0.14 0.15 0.25 0.29 0.33 0.50 0.85 0.88 1.15 1.26 1.32 1.34
64 0.12 0.13 0.13 0.20 0.23 0.33 0.51 0.51 0.66 0.86 0.86 0.86 0.87
96 0.08 0.09 0.09 0.11 0.14 0.16 0.21 0.30 0.31 0.30 0.29 0.28 0.28
128 0.08 0.09 0.10 0.10 0.12 0.18 0.22 0.27 0.24 0.23 0.23 0.23 0.22
192 0.13 0.13 0.13 0.15 0.16 0.19 0.21 0.19 0.16 0.16 0.16 0.15 0.15
256 0.19 0.18 0.19 0.18 0.19 0.19 0.18 0.15 0.14 0.14 0.13 . .
384 0.22 0.20 0.21 0.19 0.14 0.13 0.10 0.10 0.10 0.09 . . .
512 0.30 0.25 0.25 0.20 0.12 0.11 0.09 0.09 . . . . .
768 0.30 0.28 0.21 0.13 0.08 0.08 0.07 . . . . . .
1024 0.39 0.34 0.19 0.12 0.11 . . . . . . . .
1536 0.40 0.24 0.12 . . . . . . . . . .
2048 0.41 0.26 0.16 . . . . . . . . . .
3072 0.32 . . . . . . . . . . . .
4096 0.49 . . . . . . . . . . . .
The same rule for eigvalsh, with its own thresholds (values_gpu_max_n, values_gpu_min_batch_times_n, values_gpu_min_batch), fitted on the _vals timings of 205 points given the split above. The CPU computes eigenvalues alone by LAPACK’s two-stage reduction from N = 128, so the boundary need not be eigh’s.
| rule | geomean regret | worst | >10% | total time / oracle | est. picks |
|---|---|---|---|---|---|
| policy in effect | 1.0137 | 1.46x | 12 | 1.004 | 0 |
| eigh’s fitted boundary | 1.0092 | 1.46x | 9 | 1.001 | 0 |
| fitted (‘48’, ‘16384’, ‘1’) | 1.0137 | 1.46x | 12 | 1.004 | 0 |
18 combinations are within 0.5% of the best geomean: values_gpu_max_n 32 .. 64, values_gpu_min_batch_times_n 16384 .. 16384, values_gpu_min_batch 1 .. 32.
Held out: fitted on 112 points (‘48’, ‘16384’, ‘1’), scored on the other 93: geomean 1.0083x, worst 1.32x, against 1.0083x, worst 1.32x for eigh’s boundary on the same points.
Where the rule above chooses the CPU, the tridiag backend from a threshold N on (0: never), for batches up to a cap (0: any; it solves a batch one matrix after another, the CPU path spreads one over every core), fitted over the measured N and batches against the best of all backends, tridiag included, on the points where tridiag was timed (N >= 128, within the cost cap): the region the threshold decides.
| threshold | batch cap | geomean regret | worst | without tridiag: geomean | worst | held out (fitted on half) | |
|---|---|---|---|---|---|---|---|
| with eigenvectors | 1024 | 4 | 1.0069 | 1.28x | 1.1156 | 5.62x | from 768, batch <= 2: 1.0225 vs 1.0350 |
| eigenvalues alone | 1536 | 2 | 1.0081 | 1.46x | 1.0650 | 2.23x | from 1536, batch <= 2: 1.0137 vs 1.0137 |
tridiag over the CPU (with eigenvectors), N x batch: 128x1 0.32x, 128x2 0.21x, 128x4 0.11x, 128x8 0.06x, 128x16 0.04x, 128x32 0.03x, 128x64 0.03x, 128x128 0.03x, 128x256 0.03x, 128x512 0.03x, 192x1 0.47x, 192x2 0.28x, 192x4 0.16x, 192x8 0.09x, 192x16 0.06x, 192x32 0.05x, 192x64 0.05x, 192x128 0.05x, 192x256 0.05x, 256x1 0.65x, 256x2 0.38x, 256x4 0.22x, 256x8 0.13x, 256x16 0.09x, 256x32 0.08x, 256x64 0.07x, 256x128 0.07x, 256x256 0.07x, 384x1 0.72x, 384x2 0.49x, 384x4 0.31x, 384x8 0.18x, 384x16 0.12x, 384x32 0.12x, 384x64 0.11x, 384x128 0.10x, 512x1 1.00x, 512x2 0.68x, 512x4 0.44x, 512x8 0.29x, 512x16 0.18x, 512x32 0.17x, 512x64 0.16x, 512x128 0.16x, 768x1 1.15x, 768x2 0.89x, 768x4 0.54x, 768x8 0.36x, 768x16 0.27x, 768x32 0.26x, 768x64 0.26x, 1024x1 1.58x, 1024x2 1.29x, 1024x4 0.78x, 1024x8 0.57x, 1024x16 0.55x, 1536x1 1.91x, 1536x2 1.44x, 1536x4 0.95x, 2048x1 2.48x, 2048x2 2.35x, 2048x4 1.66x, 3072x1 3.13x, 4096x1 5.62x
tridiag over the CPU (eigenvalues alone), N x batch: 128x1 0.23x, 128x2 0.13x, 128x4 0.07x, 128x8 0.05x, 128x16 0.03x, 128x32 0.02x, 128x64 0.02x, 128x128 0.02x, 128x256 0.02x, 128x512 0.02x, 192x1 0.33x, 192x2 0.18x, 192x4 0.10x, 192x8 0.06x, 192x16 0.04x, 192x32 0.04x, 192x64 0.03x, 192x128 0.03x, 192x256 0.03x, 256x1 0.41x, 256x2 0.23x, 256x4 0.13x, 256x8 0.08x, 256x16 0.05x, 256x32 0.05x, 256x64 0.04x, 256x128 0.04x, 256x256 0.04x, 384x1 0.58x, 384x2 0.33x, 384x4 0.19x, 384x8 0.12x, 384x16 0.08x, 384x32 0.07x, 384x64 0.06x, 384x128 0.06x, 512x1 0.77x, 512x2 0.45x, 512x4 0.26x, 512x8 0.16x, 512x16 0.10x, 512x32 0.10x, 512x64 0.09x, 512x128 0.08x, 768x1 1.15x, 768x2 0.66x, 768x4 0.39x, 768x8 0.23x, 768x16 0.16x, 768x32 0.14x, 768x64 0.12x, 1024x1 1.46x, 1024x2 0.83x, 1024x4 0.52x, 1024x8 0.31x, 1024x16 0.20x, 1536x1 1.84x, 1536x2 1.26x, 1536x4 0.65x, 2048x1 2.12x, 2048x2 1.42x, 2048x4 0.76x, 3072x1 2.15x, 4096x1 2.23x
For eigenvalues alone, where the rules above choose the CPU or tridiag, the band backend (the two-stage reduction: A to a band on the GPU in blocks whose work is matrix products, the band to tridiagonal on the CPU’s cores, then bisection on the GPU) from a threshold N on (0: never), within tridiag’s batch cap, fitted on the 28 points where it was timed (N >= 512) against the CPU, tridiag and band.
Chosen: 4096 (in effect: never): 1.0212 geometric-mean regret, worst 1.46x; without band 1.0308, worst 1.46x.
band over tridiag, N x batch: 512x1 0.74x, 512x2 0.78x, 512x4 0.82x, 512x8 0.83x, 512x16 0.84x, 512x32 0.85x, 512x64 0.84x, 512x128 0.85x, 768x1 0.70x, 768x2 0.77x, 768x4 0.81x, 768x8 0.83x, 768x16 0.84x, 768x32 0.85x, 768x64 0.86x, 1024x1 0.75x, 1024x2 0.79x, 1024x4 0.86x, 1024x8 0.88x, 1024x16 0.90x, 1536x1 0.84x, 1536x2 0.95x, 1536x4 1.00x, 2048x1 0.82x, 2048x2 0.99x, 2048x4 1.10x, 3072x1 1.07x, 4096x1 1.30x
band over the CPU, N x batch: 512x1 0.57x, 512x2 0.35x, 512x4 0.21x, 512x8 0.13x, 512x16 0.09x, 512x32 0.08x, 512x64 0.07x, 512x128 0.07x, 768x1 0.81x, 768x2 0.51x, 768x4 0.31x, 768x8 0.19x, 768x16 0.13x, 768x32 0.12x, 768x64 0.11x, 1024x1 1.10x, 1024x2 0.66x, 1024x4 0.45x, 1024x8 0.27x, 1024x16 0.18x, 1536x1 1.55x, 1536x2 1.19x, 1536x4 0.65x, 2048x1 1.74x, 2048x2 1.41x, 2048x4 0.83x, 3072x1 2.30x, 4096x1 2.91x
Pass-to-pass ratio (max/min of the same measurement across passes), 1692 measurements: median 1.017, p90 1.120, max 3.77. The held-out verdicts use a bootstrap rather than this figure, since a mean over many points is far less noisy than one measurement.
| runtime | n | median | p90 | max |
|---|---|---|---|---|
| <1 ms | 809 | 1.034 | 1.172 | 3.77 |
| 1-3 ms | 185 | 1.008 | 1.069 | 1.62 |
| 3-10 ms | 204 | 1.010 | 1.042 | 1.36 |
| 10-30 ms | 152 | 1.012 | 1.048 | 1.46 |
| 30-100 ms | 151 | 1.010 | 1.038 | 1.23 |
| >100 ms | 191 | 1.011 | 1.046 | 1.15 |