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Algorithm 1 SSM (S4)Algorithm 2 SSM + Selection (S6)
Input: x : (B,L,D)Input: x : (B,L,D)
Output: y:(B,L,D)Output: y: (B,L,D)
1:A:(D,N)←Parameter1:A:(D,N)←Parameter
Represents structured N × N matrix>Represents structured N × N matrix
2:B:(D,N)←Parameter2: B:(B,L,N)← SB(𝑥)
3: C:(D,N)←Parameter 3: C:(B,L,N)←sc(x)
4:△:(D)←T△(Parameter)4: △:(B,L,D)←T△(Parameter+s△(x))
5: A,B:(D,N)←discretize(���,A,B) 5: A,B:(B,L,D,N)←discretize(△,A,B)
6: y←SSM(A,B,C)(x)6: y←SSM(A,B,C)(x)
> Time-invariant: recurrence or convolution 7: return yTime-varying: recurrence (scan) only 7: return y
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MODELPPL↓PPL↓TOKEN. PILE LAMBADA LAMBADA HellSwag PIQA Arc-E Arc-C WinoGrande Average ACC个ACC个ACC个ACC↑ACC↑ACC个ACC↑
Pythia-160MNeoX29.64 38.1033.030.261.443.224.151.940.6
Mamba-130M NeoX10.5616.0744.335.364.548.024.351.944.7
Pythia-410MNeoX9.9510.8451.440.666.952.124.653.848.2
Mamba-370M NeoX8.288.1455.646.569.555.128.055.350.0
Pythia-1BNeoX7.827.9256.147.270.757.027.153.551.9
Mamba-790M NeoX7.336.0262.755.172.161.229.556.157.1
GPT-Neo 1.3BGPT2一7.5057.248.971.156.225.954.952.4
OPT-1.3BOPT6.6458.053.772.456.729.659.555.0
Pythia-1.4BNeoX7.516.0861.752.171.060.528.557.255.2
RWKV-1.5BNeoX7.707.0456.452.572.460.529.454.654.3
Mamba-1.4BNeoX6.805.0465.059.174.265.532.861.559.7
GPT-Neo 2.7BGPT25.6362.255.872.161.130.257.656.5
OPT-2.7BOPT5.1263.660.674.860.831.361.058.7
Pythia-2.8BNeoX6.735.0464.759.374.064.132.959.759.1
RWKV-3BNeoX7.005.2463.959.673.767.833.159.659.6
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WaveNet4.2M1.9255.082.275.801.47
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MODELBLOCKSSMLAYERPERPLEXITY
HyenaH3Hyena10.24
-H3S4 (real)10.34
H3H3S4 (complex)10.30
H3S68.95
=MambaHyena10.75
MambaS4 (real)10.56
MambaS4 (complex) 10.54
MambaMambaS68.69
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Selective△ Selective B Selective C Perplexity
X10.93
Xxx<x> 10.15
x√9.98
x<xx> √9.81 8.71
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An InitializationFieldPerplexity
A=+miComplex 9.16
An=-Real8.85
An=-(n+1)Real8.71
An ~exp(N (0,1))Real8.71
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State dim. N Params Perplexity
1367.1M 9.88
2367.4M 9.86
4368.0M 9.82
8369.1M 9.82
16371.5M 9.81
1367.1M 9.73
2367.4M 9.40
4368.0M 9.09
8369.1M 8.84
16371.5M 8.71
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Rank of△ proj. ParamsPerplexity
= 358.9M 9.12
1 359.1M 8.97
2 359.3M 8.97
359.7M 8.91
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16 362.1M 8.84
32 365.2M 8.80
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Params n_layers dmodel n_heads /d_head Training steps Learning Rate Batch SizeTokens
125M1276812/6448006e-40.5M tokens2.5B
350M24102416/64135003e-40.5Mtokens7B
760M24153616/96290002.5e-40.5Mtokens15B
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ModelParamsTest Accuracy (%) at Sequence Length
26272829210211212213214 215 216 217 218 219220
MHA-Abs137K99.6100.0 58.6 26.6 18.8 9.810.97.8XX×XX×
MHA-RoPE 137K√100.0 83.6 31.318.4 8.69.05.5X×XXX
MHA-xPos137K√100.0 99.6 67.6 25.4 7.09.07.8 x×X×X
H3153K100.080.9 39.5 23.814.8 8.25.9 6.68.24.78.2 6.3 7.4
Hyena69M*97.7100.044.1 12.5 6.65.17.0 5.9 6.6 6.6 5.96.3 9.8
Mamba74K100.0√√√√√√
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15×2048=30720 32983040
8×2048=16384 641048576
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