Alright, let's put theory into practice. We've discussed how quantized operations are represented and within compiler IRs. Now, we'll focus on the critical step of lowering: transforming these high-level quantized operations into sequences of lower-level, often standard integer, arithmetic operations that explicitly handle scaling, zero-point adjustments, and type conversions. This process is fundamental for generating executable code, especially for hardware that lacks direct support for arbitrarily scaled quantized types but possesses efficient integer arithmetic units.

Imagine we have a high-level quantized operation in our IR, perhaps representing a 2D convolution or matrix multiplication. It looks something like this (using a simplified, MLIR-inspired notation):

// High-Level Representation
%input_q = "quant.cast"(%input_fp32) : tensor<1xHxWxCinxf32> -> tensor<1xHxWxCin x !quant.uniform<i8:...>>
%weight_q = "quant.cast"(%weight_fp32) : tensor<CoutxKhxKwxCinxf32> -> tensor<CoutxKhxKwxCin x !quant.uniform<i8:...>>
%bias_q = "quant.cast"(%bias_fp32) : tensor<Coutxf32> -> tensor<Cout x !quant.uniform<i32:...>> // Bias often INT32

// Represents the entire quantized convolution including output scaling
%output_q = "quant.conv2d"(%input_q, %weight_q, %bias_q) {
    strides = [...], padding = [...],
    output_scale = ..., output_zero_point = ...
} : (..., ..., ...) -> tensor<1xHoWxWoxCout x !quant.uniform<i8:...>>

This quant.conv2d operation encapsulates the core computation along with the necessary requantization logic to produce the final INT8 output. Our goal is to lower this into operations available in standard dialects (like arith for arithmetic, memref for memory access in MLIR) or into target-specific intrinsics.

The Mathematical Foundation of Lowering

Recall the affine quantization formula: $r = S \times (q - Z)$ , where $r$ is the real value, $q$ is the quantized integer value, $S$ is the scale, and $Z$ is the zero point.

For a convolution (or matrix multiplication), the core computation involves multiply-accumulate operations. Let's consider a single output element calculation, which is a sum of products: $Output_{real} = \sum (Input_{real} \times Weight_{real}) + Bias_{real}$ .

Substituting the quantization formula:

S_{out} (Output_q - Z_{out}) \approx \sum [ S_{in} (Input_q - Z_{in}) \times S_{w} (Weight_q - Z_{w}) ] + S_{bias} (Bias_q - Z_{bias})

Our target is to compute $Output_q$ using integer arithmetic. Rearranging the equation involves:

Performing the core accumulation using integer multiplication: $\sum (Input_q \times Weight_q)$ . This typically uses a wider accumulator (e.g., INT32).
Calculating corrections related to zero points.
Applying a final scaling factor (requantization scale) and adding the output zero point.

The full expansion can get complex:

S_{out} (Output_q - Z_{out}) \approx S_{in} S_{w} \sum (Input_q - Z_{in})(Weight_q - Z_{w}) + S_{bias} (Bias_q - Z_{bias})

S_{out} (Output_q - Z_{out}) \approx S_{in} S_{w} \sum [ Input_q Weight_q - Input_q Z_{w} - Z_{in} Weight_q + Z_{in} Z_{w} ] + S_{bias} (Bias_q - Z_{bias})

The compiler's task during lowering is to generate efficient integer code that computes the right-hand side and then solves for $Output_q$ .

Lowering Strategy Example: Integer Accumulation with Requantization

A common strategy involves these steps, which would be reflected in the generated lower-level IR:

Integer Accumulation: Perform the primary multiply-accumulate operations using the quantized integer inputs ( $Input_q$ $I n p u t_{q}$ , $Weight_q$ $W e i g h t_{q}$ ). This results in a wider intermediate representation, typically INT32.

// Lowered IR Snippet 1: Integer MatMul/Conv Accumulation %acc_i32 = arith.constant 0 : i32 // Loop over reduction dimensions (e.g., input channels, kernel spatial dims) affine.for %k = 0 to K { %in_val_i8 = memref.load %input_q[...] : memref<...x!quant.uniformi8:...> %wt_val_i8 = memref.load %weight_q[...] : memref<...x!quant.uniformi8:...> // Extend i8 to i32 for accumulation %in_val_i32 = arith.extsi %in_val_i8 : i8 to i32 %wt_val_i32 = arith.extsi %wt_val_i8 : i8 to i32 // Integer multiplication %mul_i32 = arith.muli %in_val_i32, %wt_val_i32 : i32 // Accumulate %acc_i32 = arith.addi %acc_i32, %mul_i32 : i32 } ```

Zero-Point Correction: Apply corrections based on input ( $Z_{in}$ $Z_{in}$ ) and weight ( $Z_w$ $Z_{w}$ ) zero points. This can be done in several ways:
- Directly in the loop: Modify the loaded values before multiplication: (Input_q - Z_in) * (Weight_q - Z_w). This adds subtractions inside the loop.
- Post-accumulation correction: Calculate the correction terms involving sums of inputs/weights multiplied by zero points outside the main accumulation loop. This often requires pre-calculating sums. For example, the term $- \sum (Input_q Z_w)$ becomes $- Z_w \sum Input_q$ . The compiler might choose based on performance heuristics.

// Lowered IR Snippet 2: Zero-Point Correction (Post-Accumulation style) // Assume %sum_inputs_i32, %sum_weights_i32, %reduction_size are pre-calculated or available %zp_in_i32 = arith.constant ... : i32 // Z_in %zp_wt_i32 = arith.constant ... : i32 // Z_w

%correction1 = arith.muli %sum_inputs_i32, %zp_wt_i32 : i32
%correction2 = arith.muli %sum_weights_i32, %zp_in_i32 : i32
%correction3 = arith.muli %reduction_size, %zp_in_i32 : i32
%correction3 = arith.muli %correction3, %zp_wt_i32 : i32 // Term Z_in * Z_w * K

%acc_i32 = arith.subi %acc_i32, %correction1 : i32
%acc_i32 = arith.subi %acc_i32, %correction2 : i32
%acc_i32 = arith.addi %acc_i32, %correction3 : i32
```

3. Bias Addition: Add the quantized bias term ( $Bias_q$ ). Remember, the bias scale $S_{bias}$ must ideally equal $S_{in} \times S_w$ for direct INT32 addition. If not, the bias must be rescaled before addition. mlir // Lowered IR Snippet 3: Bias Addition %bias_val_i32 = memref.load %bias_q[...] : memref<...x!quant.uniform<i32:...>> // Assume bias scale matches S_in * S_w, otherwise rescaling needed here %acc_i32 = arith.addi %acc_i32, %bias_val_i32 : i32

Requantization Scaling: Apply the final scale factor to convert the INT32 accumulator back towards the target output precision (e.g., INT8). The scale factor is $M = \frac{S_{in} S_w}{S_{out}}$ $M = \frac{S _{in} S _{w}}{S _{o u t}}$ . This is almost never an integer or power of two, so it's implemented using fixed-point multiplication. The compiler calculates an integer multiplier $M_0$ $M_{0}$ and a right-shift amount $N$ $N$ such that $M \approx M_0 / 2^N$ $M \approx M_{0} / 2^{N}$ .

// Lowered IR Snippet 4: Requantization Scaling %requant_mult_i32 = arith.constant ... : i32 // M0 %requant_shift_i32 = arith.constant ... : i32 // N

// Perform fixed-point multiplication: (acc * M0) >> N
// Often requires widening to i64 to avoid overflow during multiplication
%acc_i64 = arith.extsi %acc_i32 : i32 to i64
%requant_mult_i64 = arith.extsi %requant_mult_i32 : i32 to i64
%scaled_acc_i64 = arith.muli %acc_i64, %requant_mult_i64 : i64

// Apply rounding shift (add 0.5 before shifting)
%rounding_delta = arith.constant (1 << (%N - 1)) : i64
%scaled_acc_i64 = arith.addi %scaled_acc_i64, %rounding_delta : i64

// Perform the shift (arithmetic right shift)
%shifted_acc_i64 = arith.shrsi %scaled_acc_i64, %requant_shift_i32 : i64
```

5. Output Zero-Point Addition: Add the output zero point $Z_{out}$ . mlir // Lowered IR Snippet 5: Add Output Zero Point %zp_out_i64 = arith.constant ... : i64 // Z_out extended to i64 %final_acc_i64 = arith.addi %shifted_acc_i64, %zp_out_i64 : i64

Clamping and Casting: Clamp the result to the valid range of the target output type (e.g., [-128, 127] for INT8) and cast to the final type.

// Lowered IR Snippet 6: Clamp and Cast %min_val_i64 = arith.constant -128 : i64 %max_val_i64 = arith.constant 127 : i64 %clamped_acc_i64 = arith.maxsi %final_acc_i64, %min_val_i64 : i64 %clamped_acc_i64 = arith.minsi %clamped_acc_i64, %max_val_i64 : i64

// Truncate back to i8
%output_val_i8 = arith.trunci %clamped_acc_i64 : i64 to i8

// Store the final result
memref.store %output_val_i8, %output_q[...] : memref<...x!quant.uniform<i8:...>>
```

This sequence replaces the single high-level quant.conv2d operation. The compiler performs this transformation based on the specific quantization parameters (scales, zero points) associated with the operation's inputs and outputs.

Flow transforming a high-level quantized operation into a sequence of lower-level integer arithmetic and control operations during compiler lowering.

Considerations and Tooling

Compiler frameworks like MLIR use dialect conversion and rewrite patterns to automate this lowering process. The specific sequence and optimizations (e.g., how zero points are handled, choice of fixed-point multiplication method) depend on the compiler's sophistication and the target hardware capabilities. For instance, hardware with specialized instructions like Intel's VNNI or ARM's dot-product instructions might lead to a different lowered representation that directly maps to those instructions.

This hands-on perspective shows that "running a quantized model" involves significant compiler transformations. Understanding this lowering process is significant for debugging performance issues, designing custom quantized operators, or extending compilers to support new low-precision data types or hardware features. It bridges the abstract concept of quantization with the concrete integer arithmetic executed on the processor.