edteams/ref-ops-v2

Reference Operations V2

Dot Product

• MXINT8 dot product

# sum(a .* b), dot product
# a.shape: [m]
# b.shape: [m]
# output.shape: scalar
output = dot_product(a, b)

• MXINT8 vector-matrix multiplication

# vector @ matrix, vector-matrix multiplication
# vector.shape: [m]
# matrix.shape: [m, k]
# output.shape: [k]
output = gevm(vector, matrix)

• MXINT8 matrix-matrix multiplication

# input @ other, matrix-matrix multiplication
# input.shape: [m, k]
# other.shape: [k, n]
# output.shape: [m, n]
output = gemm(input, other)

RoPE constants

BF16

Dumped tensors:

rope.freq_bf16:

• shape: [1, 4096, 64], 4096 denotes maximum sequence length, 64 is half of the head dimension (128/2=64)
• each element is $m \theta_i$ where $m$ is the position id, $i$ is the head dimension index.

rope.cos_bf16: cos_bf16 = cos(freq_bf16)

rope.sin_bf16: sin_bf16 = sin(freq_bf16)

FP32 for reference

HuggingFace forces the RoPE constants to be in FP32 format because both BF16 and FP16 introduce errors in the output cos and sin values. For example, BF16 cannot represent the exact value of 4088, 4089, ..., 4095 (all these values are rounded to 4096). This issue discusses this problem.

Therefore, we also provide the FP32 version of the RoPE constants for reference. This assumes all the constants are computed in FP32 format.

• rope.freq_fp32
• rope.cos_fp32
• rope.sin_fp32

Softmax

softmax_v is a function that computes the matmul(softmax(qk_T), v) operation in the attention layer. To match the HW behaviour, softmax_v is broken down into the following steps:

def softmaxed_v(qk_T, v):
    """
    qk_T.shape: [bs, num_heads, seq_q_len, seq_kv_len]
    v.shape: [bs, num_heads, seq_kv_len, head_dim]
    """

    # elementwise exp
    exp_bf16 = row_exp(qk_T, dim=-1) # shape: [bs, num_heads, seq_q_len, seq_kv_len]
    # sum exp along the head dimension
    exp_sum = reduce_sum(exp_bf16, dim=-1) # shape: [bs, num_heads, seq_q_len, 1]
    # quantize to mxint8 for vector-matrix multiplication
    exp_mxint8 = quantize(exp_bf16)
    # vector-matrix multiplication between two mxint8 tensors
    scaled_v = mxint8_gevm(exp_mxint8, v) # shape: [bs, num_heads, seq_q_len, head_dim]
    # inverse of the sum
    inv = 1.0 / exp_sum # shape: [bs, num_heads, seq_q_len, 1]
    # adjust scaled_v by the inverse
    out = scaled_v * inv # shape: [bs, num_heads, seq_q_len, head_dim]
    return out

Notation	Description
bs	batch size, 1 in this case
num_heads	number of attention heads, 32 for llama-2-7b
seq_q_len	query sequence length, 1 in this case
seq_kv_len	key-value sequence length, which increments by 1 for each decoding step
head_dim	head dimension, 128 for llama-2-7b

The folder name takes the form kv-size-<seq_kv_len>_seed-<seed>. For example, kv-size-4_seed-0 means the key-value sequence length is 4 and the random seed is 0.

The following tensors are dumped:

File Name	Description
softmaxed_v.qk_T	BF16 qk_T tensor
softmaxed_v.exp-bf16	BF16 exp_bf16 tensor
softmaxed_v.exp_sum	BF16 exp_sum tensor
softmaxed_v.v	MXINT8 v tensor
softmaxed_v.exp_mxint8	MXINT8 exp_mxint8 tensor
softmaxed_v.scaled_v	BF16 scaled_v tensor
softmaxed_v.inv	BF16 inv tensor
softmaxed_v.out	BF16 out tensor