File size: 7,573 Bytes
6ec9472 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | """Projection of a 1T-parameter sparse MoE onto the measured machine.
Methodology: every hardware quantity is measured on the host; the expert-cache
behaviour is modelled with the Che approximation for LRU under an
independent-reference model, *validated against the real OLMoE trace at E=64*
before being extrapolated to the 1T configuration's E=320.
"""
import json, os, sys
import numpy as np
RES = os.path.join(os.path.dirname(__file__), "..", "results")
# ---------------------------------------------------------------- 1T config
T1 = dict(name="T1-1046B", layers=64, d_model=8192, n_experts=320, topk=8,
d_ff_expert=2048, n_shared=1, kv_dim=1024, vocab=129280)
def config_params(c):
attn = 2 * c["d_model"] ** 2 + 2 * c["d_model"] * c["kv_dim"]
expert = 3 * c["d_model"] * c["d_ff_expert"]
per_layer = attn + expert * (c["n_experts"] + c["n_shared"])
total = per_layer * c["layers"] + 2 * c["vocab"] * c["d_model"]
active = (attn + expert * (c["topk"] + c["n_shared"])) * c["layers"] \
+ c["vocab"] * c["d_model"]
return dict(total=total, active=active, expert=expert,
attn_total=attn * c["layers"],
shared_total=expert * c["n_shared"] * c["layers"],
routed_total=expert * c["n_experts"] * c["layers"],
n_slots=c["n_experts"] * c["layers"])
# ---------------------------------------------------------------- cache model
def che_hit_rate(p, capacity):
"""LRU hit rate under IRM via Che's approximation."""
p = np.asarray(p, dtype=np.float64)
p = p / p.sum()
if capacity >= len(p):
return 1.0
if capacity <= 0:
return 0.0
lo, hi = 1e-6, 1e12
for _ in range(200):
t = (lo * hi) ** 0.5
occ = (1.0 - np.exp(-p * t)).sum()
if occ < capacity:
lo = t
else:
hi = t
t = (lo * hi) ** 0.5
return float((p * (1.0 - np.exp(-p * t))).sum())
def zipf_fit(freq):
"""Least-squares Zipf exponent of a measured popularity vector."""
f = np.sort(np.asarray(freq, dtype=np.float64))[::-1]
f = f[f > 0]
r = np.arange(1, len(f) + 1)
a, _ = np.polyfit(np.log(r), np.log(f), 1)
return float(-a)
def zipf_pmf(n, s):
r = np.arange(1, n + 1, dtype=np.float64)
p = r ** (-s)
return p / p.sum()
def distinct_per_layer(p, n_draws):
"""Expected distinct experts touched by n_draws independent selections."""
p = np.asarray(p, dtype=np.float64)
p = p / p.sum()
return float((1.0 - (1.0 - p) ** n_draws).sum())
# ---------------------------------------------------------------- throughput
def io_bandwidth(io, block_bytes, threads=4):
"""Interpolate measured unbuffered random-read bandwidth at a block size."""
pts = [(r["block_kb"] * 1024, r["mb_s"]) for r in io["random"]
if r["threads"] == threads]
pts.sort()
xs = np.log2([p[0] for p in pts]); ys = [p[1] for p in pts]
return float(np.interp(np.log2(block_bytes), xs, ys)) * 1e6
def analytic_static(p, cap):
"""Hit rate of a popularity-pinned cache: mass of the top-`cap` slots.
Exact given the popularity vector (validated to 0.00 pp on the real trace)."""
q = np.sort(np.asarray(p, dtype=np.float64))[::-1]
q = q / q.sum()
return float(q[:min(cap, len(q))].sum())
def hit_rate_for(cfg, cap_slots, zipf_s, bias_pp=0.0):
"""Popularity-pinned hit rate for a configuration with E experts per layer."""
p = np.tile(zipf_pmf(cfg["n_experts"], zipf_s) / cfg["layers"], cfg["layers"])
return max(0.0, analytic_static(p, cap_slots) - bias_pp / 100.0)
def project(rate_bits, hit_rate, io, cfg=T1, dram_gb=24.0, vram_gb=3.4,
batch=1, zipf_s=None):
P = config_params(cfg)
Bpp = rate_bits / 8.0
expert_bytes = P["expert"] * Bpp
total_bytes = P["total"] * Bpp
resident = (P["attn_total"] + P["shared_total"]) * Bpp
vram_free = max(0.0, vram_gb * 1e9 - resident)
dram_slots = int(dram_gb * 1e9 // expert_bytes)
vram_slots = int(vram_free // expert_bytes)
bw = io_bandwidth(io, expert_bytes)
if zipf_s is not None:
p = zipf_pmf(cfg["n_experts"], zipf_s)
u = distinct_per_layer(p, cfg["topk"] * batch)
else:
u = cfg["topk"] * batch
fetch_per_token = cfg["layers"] * u * (1.0 - hit_rate) / batch
bytes_per_token = fetch_per_token * expert_bytes
t_io = bytes_per_token / bw
return dict(rate_bits=rate_bits, hit_rate=hit_rate, batch=batch,
total_gb=total_bytes / 1e9, expert_mb=expert_bytes / 1e6,
dram_slots=dram_slots, vram_slots=vram_slots,
cache_frac=dram_slots / P["n_slots"],
resident_gb=resident / 1e9,
io_bw_gbs=bw / 1e9,
bytes_per_token_mb=bytes_per_token / 1e6,
tok_s=1.0 / t_io if t_io > 0 else float("inf"))
def main():
io = json.load(open(os.path.join(RES, "io_bench.json")))
P = config_params(T1)
out = {"config": T1, "params": {k: float(v) for k, v in P.items()}}
print(f"{T1['name']}: {P['total']/1e9:.1f}B total, {P['active']/1e9:.1f}B active/token, "
f"{P['n_slots']} expert slots")
# ---- cache-capacity amplification from quantisation
amp = []
for r in [16, 4, 3, 2, 1.5, 1.0]:
eb = P["expert"] * r / 8
amp.append(dict(bits=r, model_gb=P["total"] * r / 8 / 1e9,
expert_mb=eb / 1e6,
dram_experts=int(24e9 // eb),
frac=int(24e9 // eb) / P["n_slots"]))
out["amplification"] = amp
print("\nrate model_GB expert_MB experts_in_24GB cache_frac")
for a in amp:
print(f"{a['bits']:>4.1f} {a['model_gb']:8.0f} {a['expert_mb']:9.2f} "
f"{a['dram_experts']:15d} {a['frac']*100:9.2f}%")
# ---- cache policy calibrated on the measured trace, then extrapolated
cp = json.load(open(os.path.join(RES, "cache_policy.json")))
s_hat = cp["zipf_s"]
bias = cp["mae_analytic_zipf"] * 100 # Zipf-fit optimism, in points
out["zipf_s"] = s_hat
out["zipf_bias_pp"] = bias
ws = T1["topk"] * T1["layers"] # per-token working set, in slots
out["token_working_set"] = ws
print(f"\nper-token working set: {ws} expert slots; Zipf s={s_hat:.3f}; "
f"Zipf-fit optimism {bias:.2f} pp")
rows = []
for r in [1.0, 1.5, 2.0, 3.0, 4.0, 16.0]:
eb = P["expert"] * r / 8
cap = int(24e9 // eb)
h = hit_rate_for(T1, cap, s_hat, bias_pp=bias)
for B in [1, 8, 32]:
x = project(r, h, io, batch=B, zipf_s=s_hat)
x["cap_slots"] = cap
x["lru_viable"] = cap >= ws
rows.append(x)
out["projection"] = rows
print("\nbits batch slots LRUok hit% model_GB expert_MB IO_GB/s MB/token tok/s")
for x in rows:
print(f"{x['rate_bits']:>4.1f} {x['batch']:>5d} {x['cap_slots']:>5d} "
f"{str(x['lru_viable']):>5} {x['hit_rate']*100:4.1f} "
f"{x['total_gb']:8.0f} {x['expert_mb']:9.2f} {x['io_bw_gbs']:7.2f} "
f"{x['bytes_per_token_mb']:8.1f} {x['tok_s']:6.2f}")
# ---- sensitivity: tok/s across the whole hit-rate range at 1.5 bit
sens = [project(1.5, float(h), io, batch=1, zipf_s=s_hat)
for h in np.arange(0.0, 0.99, 0.05)]
out["sensitivity_1p5bit"] = sens
json.dump(out, open(os.path.join(RES, "projection.json"), "w"), indent=2)
print("\nsaved results/projection.json")
if __name__ == "__main__":
main()
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