v811_spc009/external/gemmlowp/doc/low-precision.md

The low-precision paradigm in gemmlowp, and how it's implemented

Introduction

"Low-precision" means that the input and output matrix entries are integers on at most 8 bits. The scalar type is uint8_t.

This isn't the same as just doing plain matrix arithmetic over uint8_t, because that would overflow. To avoid overflow, we internally accumulate results on more than 8 bits, and at the end we keep only some significant 8 bits. This relies on the caller providing suitable offset/multiplier/shift parameters, which effectively govern how we extract some significant 8 bit from our more-than-8bit temporary accumulators.

Low-precision paradigms

gemmlowp is flexible enough to support multiple low-precision paradigms, i.e. multiple ways that a meaning is attached to 8bit values so that a computation can rely on a 8bit GEMM provided by gemmlowp.

The current flexible design with arbitrary "output pipelines".

See output.md for more details about output pipelines. This is a mechanism by which gemmlowp becomes generic enough to support multiple 8bit computation paradigms, by allowing the user to set up a chain of transformations to be performed on internal 32bit accumulators to obtain the final outputs.

The public entry point in public/gemmlowp.h allowing to set up an arbitrary output pipeline is GemmWithOutputPipeline.

Refer to quantization.md for details of how one gets from first principles to the actual output pipelines to assemble for successful real-world quantized calculations.

For the scope of the present document, it suffices to say that quantized matrix multiplication takes the following parameters:

• The lhs matrix of uint8 quantized values.
• The rhs matrix of uint8 quantized values.
• A int32 lhs_offset, that will be added to each entry of the lhs matrix.
• A int32 rhs_offset, that will be added to each entry of the rhs matrix.
• An output pipeline, that will process int32 accumulators into final outputs.

The overall computation goes through the following steps:

1. Cast lhs entries from uint8 to int32 and add lhs_offset to each of them.
2. Cast rhs entries from uint8 to int32 and add rhs_offset to each of them.
3. Compute the int32 matrix product of the resulting lhs times rhs.
4. Apply the output pipeline on these int32 accumulators, to obtain the final outputs.

The legacy low-precision paradigm

This older paradigm is the one exposed by the following entry points:

• In public/gemmlowp.h, the Gemm entry point.
• The deprecateed eight_bit_int_gemm directory.

Originally, gemmlowp started an implementation of the (now deprecated) EightBitIntGemm paradigm, where quantized matrix multiplication takes the following input parameters: - the lhs matrix of uint8 quantized values - the rhs matrix of uint8 quantized values - the following int32 "quantization parameters", which control how the uint8 quantized values in the matrices are to be interpreted during the matrix computation: - lhs_offset - rhs_offset - result_offset - result_mult_int - result_shift

In that legacy paradigm, the mathematical expression to be computed is the result of the following steps:

1. Cast lhs entries from uint8 to int32 and add lhs_offset to each of them.
2. Cast rhs entries from uint8 to int32 and add rhs_offset to each of them.
3. Compute the int32 matrix product of the resulting lhs times rhs.
4. Add result_offset to each entry of the result.
5. Multiply each entry of the result by the following fraction, and round to the nearest integer:

result_mult_int
---------------                             (1)
2^result_shift

1. Clamp the resulting int32 values to the [0..255] range and cast to uint8.

Again, this paradigm is not recommended for new usage. See quantization.md for how reasoning from first principles, one arrives to a substantially different quantization paradigm.

In addition, note that the integer multiplication by the numerator in the above step 5. risks overflowing. That concern is avoided in the currently recommended output stages by performing a fixed-point multiplication instead of an ordinary integer multiplication.

Efficient handling of offsets

At first glance it may seem like the above-described quantized computation scheme requires adding the lhs_offset and rhs_offset to each of the lhs and rhs matrix entries.

Doing that in the GEMM kernel would incur substantial overhead: - It would mean extra arithmetic work in the GEMM kernel; - It would require storing the lhs_offset and rhs_offset in registers, which would eat into the register space available for the rest of the GEMM kernel.

One may then consider adding the lhs_offset and rhs_offset once and for all to lhs and rhs blocks, in a GEMM implementation operating on one lhs block and one rhs block at a time. However, doing so would require storing lhs and rhs blocks in 32 bit (or at least in 16 bit in real-world cases), which would partially negate the memory bandwidth benefits of low-precision computation.

Fortunately, there is another way to handle these offsets that has none of the costs of the approaches described above. The idea is as follows.

Let P denote the matrix shaped like lhs, but filled with 1's.

Let Q denote the matrix shaped like rhs, but filled with 1's.

Adding lhs_offset to each entry of lhs, means adding lhs_offset * P to lhs.

Adding rhs_offset to each entry of rhs, means adding rhs_offset * Q to rhs.

Thus, as far as handling lhs_offset and rhs_offset goes, the matrix product to be computed is:

(lhs + lhs_offset * P) * (rhs + rhs_offset * Q)

Expanding this (using distributivity of matrix multiplication over addition), we see that the above product is equal to the following sum of 4 terms:

  lhs * rhs                                 (2)
+ lhs_offset * P * rhs
+ lhs * rhs_offset * Q
+ lhs_offset * rhs_offset * P * Q

The first term, lhs * rhs, is just the matrix multiplication ignoring the offsets, i.e. as if lhs_offset==rhs_offset==0. Our claim here is that this is all what we have to compute in the GEMM kernel.

In the second term, lhs_offset * P * rhs, notice that since P is filled with 1's, P * rhs has all its rows equal to each other, and equal to the row-vector of sums of all the entries in each column of rhs.

Thus, we can compute the second term, lhs_offset * P * rhs, by summing each column of rhs. This produces a single row-vector, and in order to add the second term, we simply need to add this row-vector (multiplied by lhs_offset) to each row of the result. This is just a rank one update of the result (equivalently, the second term is a rank one matrix), and we can efficiently store it as a single vector.

The third term, lhs * rhs_offset * Q, is entirely similar to the second one, and can be similarly computed by summing each row of lhs, storing this in a single column-vector, and later multiplying these sums by rhs_offset.

The fourth term is a single constant, repeated into all the entries of the matrix. The matrix P * Q is filled with the single constant value 'depth' (the depth of the matrix product i.e. the number of columns of the lhs). Thus the fourth term is simply the rank zero update adding this constant to each matrix entry:

lhs_offset * rhs_offset * depth

Implementation of this technique in gemmlowp

In gemmlowp, at the packing stage (where we traverse blocks of the lhs and rhs to prepare them for efficient repeated traversal by the kernel), we compute the sum of each row of the lhs block and the sum of each column of the rhs block.

See in internal/pack.h, in the PackedSideBlock class, the following member:

// Handle on the additional buffer backing the vector of sums of slices
// associated with this block. Owned.
Allocator::Handle sums_of_each_slice_handle_;

sums_of_each_slice_handle_ is the handle to the buffer allocated to store the vector containing sums of rows of lhs, or of sums of columns of rhs.

After these rank one updates have been computed at the packing stage, they are ignored at the compute kernel stage, since that stage is only concerned with the first of the four terms in (2); they are only used at the unpacking stage. See the default/reference implementation, UnpackResultImpl, in internal/unpack.h.