Affine quantizers, zero points and boundary behavior

Let a real-valued tensor element x be measured in some tensor unit. A scale s has tensor units per code, while an integer zero point z identifies the code that reconstructs to real zero. Divide x by s, round that quotient to an integer, add z and clamp to the supported code range. Reconstruction subtracts z and multiplies by s. When z lies inside the code range, zero is represented exactly, which matters for padding and sparse numerical patterns.

This lesson uses nearest-even rounding of x/s before adding the integer zero point, matching the integer rounding statement in ONNX QuantizeLinear version 21. A quotient of 2.5 rounds to 2; 3.5 rounds to 4. Specify that stage precisely: rounding the already shifted quotient can select a different tie when the integer zero point is odd. Other quantizers can choose different tie rules, but an exporter and its consuming kernel must implement the same rule.

For an input between the reconstructed endpoints, nearest rounding places it within half a scale step of its reconstruction. At a tie either adjacent reconstruction has that same absolute error, although the chosen code still affects reproducibility. Outside the endpoints, saturation prevents wraparound but can produce much larger errors. A value ten times the upper endpoint does not become accurate because its stored code is valid. Track the clipping count separately from average reconstruction error to see this failure.

Range selection trades resolution for clipping. A wider interval protects rare large values while increasing the spacing available to common small values at fixed bit width. A tighter interval does the reverse. An observed minimum and maximum do not themselves define a unique affine quantizer: endpoint inclusion, zero inclusion, rounding of the zero point and permitted signed codes still matter. Save the actual resulting parameters, rather than expecting a future runtime to infer them from a descriptive range.

A useful conformance test contains real zero, both endpoints, values just beyond each endpoint and exact half-step cases of both parities. Compare integer codes before comparing end-to-end outputs; otherwise later operations can hide a boundary mismatch. Include the storage signedness and dequantization arithmetic in the artifact manifest. Floating-point scale representation can move a supposed mathematical tie, so use exactly representable inputs when testing the rule. Backend support is separate: TensorRT 10.13 IQuantizeLayer requires zero-valued zero points. This lesson’s nonzero-zero-point examples therefore do not directly describe that backend’s supported representation.

In the modular lab, Reset and set both modulus m=8 and additive step a=3. Predict the residues after three Steps: 3, 6 and 1, starting from zero. Each Step advances half a second and one addition. A saturating unsigned code accumulator instead reaches 3, 6 and 7. The instrument deliberately wraps modulo eight; it demonstrates why a quantizer must explicitly request clipping rather than treating every integer operation as interchangeable. It does not perform affine quantization.