# Quantization

This document outlines the design of the MLIR quantization system. While the term “quantization” is highly overloaded, in this case, it refers to a fairly narrow scope of techniques in use to enable conversion of floating-point computations to corresponding and plausible variants expressed in integer math for inference, as has historically been supported by low-bit depth inference engines such as TFLite, various accelerator hardware, and many DSPs.

Much of this is inspired by the approach taken in this paper with many extensions and adaptations folded in. It specifically documents the positions that MLIR has taken on the topic, and is not a general reference.

## Uniform quantization ¶

The primary quantization mechanism supported by MLIR is a scheme which can express fixed point and affine transformations via uniformly spaced point on the Real number line.

Further, the scheme can be applied:

*per-layer*: Applying to every value within the target type.*per-axis*(also called*per-channel*) : Applying individually to each index along a specific axis of a tensor type.

### Fixed point values ¶

Fixed point values are a
Real number divided by a *scale*.
We will call the result of the divided real the *scaled value*.

$$ real\_value = scaled\_value * scale $$

The scale can be interpreted as the distance, in real units, between neighboring scaled values. For example, if the scale is $ \pi $, then fixed point values with this scale can only represent multiples of $ \pi $, and nothing in between. The maximum rounding error to convert an arbitrary Real to a fixed point value with a given $ scale $ is $ \frac{scale}{2} $. Continuing the previous example, when $ scale = \pi $, the maximum rounding error will be $ \frac{\pi}{2} $.

Multiplication can be performed on scaled values with different scales, using the same algorithm as multiplication of real values (note that product scaled value has $$ scale_{product} = scale_{left \mbox{ } operand} * scale_{right \mbox{ } operand} $$). Addition can be performed on scaled values, so long as they have the same scale, using the same algorithm for addition of real values. This makes it convenient to represent scaled values on a computer as signed integers, and perform arithmetic on those signed integers, because the results will be correct scaled values.

### Affine values ¶

Mathematically speaking, affine values are the result of
adding a Real-valued *zero point*, to a scaled value.
Alternatively (and equivalently), subtracting a zero point from an affine value results in a
scaled value:

$$ real\_value = scaled\_value * scale = (affine\_value - zero\_point) * scale $$

Essentially, affine values are a shift of the scaled values by some constant amount. Arithmetic (i.e., addition, subtraction, multiplication, division) cannot, in general, be directly performed on affine values; they must first be converted to the equivalent scaled values.

As alluded to above, the motivation for using affine values is to more efficiently represent real values that will actually be encountered during computation. Frequently, real values that will be encountered are not symmetric around the real zero. We also make the assumption that the real zero is encountered during computation, and should thus be represented.

In this case, it is inefficient to store scaled values represented by signed integers, as some of the signed integers will never be used. In effect, the bit patterns corresponding to those signed integers are going to waste.

In order to exactly represent the real zero with an integral-valued affine value, the zero point must be an integer between the minimum and maximum affine value (inclusive). For example, given an affine value represented by an 8 bit unsigned integer, we have: $ 0 \leq zero\_point \leq 255 $. This is important, because in convolution-like operations of deep neural networks, we frequently need to zero-pad inputs and outputs, so zero must be exactly representable, or the result will be biased.

### Relation ¶

Real values, fixed point values, and affine values relate through the following equation, which demonstrates how to convert one type of number to another:

$$ real\_value = scaled\_value * scale = (affine\_value - zero\_point) * scale $$

Note that computers generally store mathematical values using a finite number of bits. Thus, while the above conversions are exact, to store the result in a finite number of bits, we must, in general, round the result of the conversion (this applies to both cases: storing using floating point and storing using fixed point). Note that a full discussion of rounding behavior is outside the scope of this document, and it is safe to assume unless otherwise stated that rounding should be according to the IEEE754 default of RNE (where hardware permits).

### Converting between real and fixed point or affine ¶

To convert a real value to a fixed point value, we must know the scale. To convert a real value to an affine value, we must know the scale and the zero point.

#### Real to affine ¶

To convert an input tensor of real-valued elements (usually represented by a floating point format, frequently Single precision) to a tensor of affine elements represented by an integral type (e.g. 8-bit unsigned integer), the following conversion can be performed (note that it is not required that all representable values of the integral type are used):

$$
\begin{align*}
af&fine\_value_{uint8 \, or \, uint16} \\

&= clampToTargetSize(roundToNearestInteger( \frac{real\_value_{Single}}{scale_{Single}})_{sint32} + zero\_point_{uint8 , or , uint16})
\end{align*}
$$

In the above, we assume that $real\_value$ is a Single, $scale$ is a Single, $roundToNearestInteger$ returns a signed 32-bit integer, and $zero\_point$ is an unsigned 8-bit or 16-bit integer. Note that bit depth and number of fixed point values are indicative of common types on typical hardware but is not constrained to particular bit depths or a requirement that the entire range of an N-bit integer is used.

#### Affine to real ¶

To convert an output tensor of affine elements represented by uint8 or uint16 to a tensor of real-valued elements (usually represented with a floating point format, frequently Single precision), the following conversion can be performed:

$$
\begin{align*}
re&al\_value_{Single} \\

&= roundToNearestFloat((affine\_value_{uint8 \, or \, uint16} - zero\_point_{uint8 \, or \, uint16})_{sint32})_{Single} * scale_{Single}
\end{align*}
$$

In the above, we assume that the result of subtraction is in 32-bit signed integer format, and that $roundToNearestFloat$ returns a Single.

#### Affine to fixed point ¶

When the affine and fixed point scales are the same, subtract the zero point from the affine value to get the equivalent fixed point value.

$$ \begin{align*} scaled\_value = affine\_value_{non\mbox{-}negative} - zero\_point_{non\mbox{-}negative} \end{align*} $$

#### Fixed point to affine ¶

When the affine and fixed point scales are the same, add the zero point to the fixed point value to get the equivalent affine value.

$$ \begin{align*} affine\_value_{non\mbox{-}negative} = scaled\_value + zero\_point_{non\mbox{-}negative} \end{align*} $$

## Usage within MLIR ¶

There are several components to the quantization system being developed within MLIR:

*Quantization*dialect containing:- A family of
QuantizedTypes which represent the
mapping between
*expressed*values (typically of a floating point computer type) and*storage*values (typically of an integral computer type). - Type conversion ops for converting
between types based on a QuantizedType and its
*expressed*and*storage*sub-types. - Instrumentation ops for assigning instrumentation points within the computation where runtime statistics may help guide the quantization process.

- A family of
QuantizedTypes which represent the
mapping between
- The TFLite op-set natively supports uniform-quantized variants.
- Passes and tools exist to convert directly from the
*TensorFlow*dialect to the TFLite quantized operation set.

Not every application of quantization will use all of these facilities. Specifically, the TensorFlow to TensorFlow Lite conversion uses the QuantizedTypes but has its own operations for type conversion and expression of the supporting math.

## Quantization Dialect ¶

### Quantized type ¶

TODO: Flesh this section out.

- QuantizedType base class
- UniformQuantizedType

### Quantized type conversion operations ¶

- qcast : Convert from an expressed type to QuantizedType
- dcast : Convert from a QuantizedType to its expressed type
- scast : Convert between a QuantizedType and its storage type

### Instrumentation and constraint operations ¶

- const_fake_quant : Emulates the logic of the historic TensorFlow fake_quant_with_min_max_args operation.
- stats_ref : Declares that statistics should be gathered at this point with a unique key and made available to future passes of the solver.
- stats : Declares inline statistics (per layer and per axis) for the point in the computation. stats_ref ops are generally converted to statistical operations once trial runs have been performed.
- coupled_ref : Declares points in the computation to be coupled from a type inference perspective based on a unique key.

## Integration with simulated quantization at training time ¶

TensorFlow has historically used the tf.quantization.fake_quant_* family of operations to simulate the effect of quantization at training time.

As originally implemented, TensorFlow Lite was the primary user of such operations at inference time. When quantized inference was enabled, if every eligible tensor passed through an appropriate fake_quant node (the rules of which tensors can have fake_quant applied are somewhat involved), then TensorFlow Lite would use the attributes of the fake_quant operations to make a judgment about how to convert to use kernels from its quantized operations subset.

In MLIR-based quantization, fake_quant_* operations are handled by converting them to a sequence of *qcast* (quantize) followed by *dcast* (dequantize) with an appropriate *UniformQuantizedType* as the target of the qcast operation.

This allows subsequent compiler passes to preserve the knowledge that quantization was simulated in a certain way, while giving the compiler flexibility to move the casts as it simplifies the computation and converts it to a form based on integral arithmetic.

This scheme also naturally allows computations that are *partially quantized*
where the parts which could not be reduced to integral operations are still carried out
in floating point with appropriate conversions at the boundaries.

## TFLite native quantization ¶

TODO: Flesh this out

### General algorithm ¶

- Take input min/max information and set the ArrayInfo (which really is InputOrOutputArrayInfo.
- In LegalizeTF, convert ArrayInfo min/max to tf.Quantize and tf.Dequantize nodes. (or tf.FakeQuant) Convert all constant FakeQuants to (tf.FQ -> tfl.Q -> tfl.DQ).
- Hardcode logic/propagation needs to happen here.
- Run TF constant folding.
- In PrepareTFL, convert all tf.FQ to (tfl.Q -> tfl.DQ).
- Run quantization pass that take (tfl.DQ (for both input and weights) -> op -> tfl.Q) and replaces with (op). Also replace (constant_float -> tfl.Q) with (constant_quant).