06. Custom Quant Schemes

Implementing a custom quantization scheme requires defining three components: the quantization function mapping float values to integers, the dequantization function reversing this mapping, and optimized computational kernels operating on the quantized representation.

The Q4_K scheme exemplifies a sophisticated custom quantization design. It combines block-wise quantization with different bit allocations for different parameter groups, achieving competitive accuracy at approximately 4.5 bits per parameter through asymmetric handling of high and low magnitude values.

import torch
import numpy as np

def quantize_q4_k(weight_block, scale, offset):
    """
    Quantize a weight block using Q4_K scheme.
    
    Q4_K format:
    - 2 bits per block: scale quantization (4 levels)
    - 6 bits for shared offset (global for small models, per-block for large)
    - Remaining bits: 4-bit weight values in interleaved pattern
    
    Args:
        weight_block: [block_size] float32 weight values
        scale: float32 scale factor
        offset: float32 shared offset
    
    Returns:
        bytes: quantized representation
    """
    block_size = 32
    assert len(weight_block) == block_size
    
    # Align weights to nearest value
    aligned = weight_block - offset
    quantized = aligned / scale
    
    # Find scale delta quantization index
    scale_min = quantized.abs().max().item()
    scale_delta = scale_min / 8.0  # 3 bits for scale delta
    
    # Quantize to 4-bit with sign
    q_vals = torch.clamp(torch.round(quantized), -8, 7).to(torch.int8)
    
    # Pack: lower 4 bits of first value, upper 4 bits of second value, etc.
    packed = []
    for i in range(0, block_size, 2):
        low_nibble = (q_vals[i] & 0x0F).item()
        high_nibble = ((q_vals[i + 1] & 0x0F) << 4).item()
        packed.append(low_nibble | high_nibble)
    
    return bytes([scale_delta_index] + packed)

Custom schemes must co-design the quantization representation and the inference kernel. The quantization determines storage efficiency; the kernel determines runtime speed. An ideal custom scheme minimizes the computational work required for dequantization during matrix multiplication.

// Q4_K dequantization kernel for tensor core matmul
__global__ void dequantize_q4_k_kernel(
    const uint8_t* __restrict__ qdata,      // quantized data
    const float* __restrict__ scales,        // per-block scales
    const float* __restrict__ offsets,      // shared offsets
    float* __restrict__ output,             // dequantized output
    int block_count
) {
    int block_idx = blockIdx.x;
    int thread_idx = threadIdx.x;
    int warp_idx = thread_idx / 32;
    int lane_idx = thread_idx % 32;
    
    // Each warp processes one 32-element block
    if (warp_idx * 32 < block_count) {
        // Load scale and compute scale times delta
        float block_scale = scales[block_idx];
        float block_offset = offsets[block_idx];
        float scale_delta = decode_scale_delta(qdata[0], block_scale);
        
        // Load and dequantize 16 bytes (32 values)
        int half_offset = warp_idx * 16;
        
        uint8_t packed = qdata[1 + half_offset + lane_idx / 2];
        int low_val = (packed & 0x0F) - 8;  // Sign-extend
        int high_val = ((packed >> 4) & 0x0F) - 8;
        
        float val0 = (float)low_val * scale_delta + block_offset;
        float val1 = (float)high_val * scale_delta + block_offset;
        
        output[block_idx * 32 + lane_idx * 2] = val0;
        output[block_idx * 32 + lane_idx * 2 + 1] = val1;
    }
}

The quality of a quantization scheme depends on two factors: reconstruction error and kernel efficiency. Reconstruction error measures how well quantized weights represent original weights—the lower, the better. Kernel efficiency measures how quickly these weights can be converted back to floating point and used in computation. A scheme with perfect reconstruction but inefficient kernels provides no practical benefit.

Mixed-precision quantization zones extend custom schemes by applying different quantization granularities to different network components. Critical layers receiving full float16 precision while embedding layers use aggressive 2-bit quantization maintains model quality while reducing memory footprint where it matters least for accuracy.