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Optimization of the quantization of dense neural networks from an exact QUBO formulation

Sergio Muñiz Subiñas, Manuel L. González, Jorge Ruiz Gómez, Alejandro Mata Ali, Jorge Martínez Martín, Miguel Franco Hernando, Ángel Miguel García-Vico

TL;DR

The paper tackles post-training quantization of dense neural networks by reformulating the ADAROUND rounding strategy as a quadratic unconstrained binary optimization (QUBO) problem that minimizes the Frobenius distance between pre-activation outputs. A novel decomposition exploits the $n$ independent subproblems of size $f+1$ arising from the QUBO coefficient structure, enabling efficient heuristic solutions via simulated annealing. Empirical results on MNIST, Fashion-MNIST, EMNIST, and CIFAR-10 show that QUBO-ADAROUND can outperform round-to-nearest at aggressive quantization levels (notably int2), though performance degrades at int1 and depends on heuristic solvers. The work provides a rigorous, QUBO-based PTQ framework with potential for extension to broader architectures and future quantum optimization approaches, offering a principled path to preserving model behavior under low-bit quantization.

Abstract

This work introduces a post-training quantization (PTQ) method for dense neural networks via a novel ADAROUND-based QUBO formulation. Using the Frobenius distance between the theoretical output and the dequantized output (before the activation function) as the objective, an explicit QUBO whose binary variables represent the rounding choice for each weight and bias is obtained. Additionally, by exploiting the structure of the coefficient QUBO matrix, the global problem can be exactly decomposed into $n$ independent subproblems of size $f+1$, which can be efficiently solved using some heuristics such as simulated annealing. The approach is evaluated on MNIST, Fashion-MNIST, EMNIST, and CIFAR-10 across integer precisions from int8 to int1 and compared with a round-to-nearest traditional quantization methodology.

Optimization of the quantization of dense neural networks from an exact QUBO formulation

TL;DR

The paper tackles post-training quantization of dense neural networks by reformulating the ADAROUND rounding strategy as a quadratic unconstrained binary optimization (QUBO) problem that minimizes the Frobenius distance between pre-activation outputs. A novel decomposition exploits the independent subproblems of size arising from the QUBO coefficient structure, enabling efficient heuristic solutions via simulated annealing. Empirical results on MNIST, Fashion-MNIST, EMNIST, and CIFAR-10 show that QUBO-ADAROUND can outperform round-to-nearest at aggressive quantization levels (notably int2), though performance degrades at int1 and depends on heuristic solvers. The work provides a rigorous, QUBO-based PTQ framework with potential for extension to broader architectures and future quantum optimization approaches, offering a principled path to preserving model behavior under low-bit quantization.

Abstract

This work introduces a post-training quantization (PTQ) method for dense neural networks via a novel ADAROUND-based QUBO formulation. Using the Frobenius distance between the theoretical output and the dequantized output (before the activation function) as the objective, an explicit QUBO whose binary variables represent the rounding choice for each weight and bias is obtained. Additionally, by exploiting the structure of the coefficient QUBO matrix, the global problem can be exactly decomposed into independent subproblems of size , which can be efficiently solved using some heuristics such as simulated annealing. The approach is evaluated on MNIST, Fashion-MNIST, EMNIST, and CIFAR-10 across integer precisions from int8 to int1 and compared with a round-to-nearest traditional quantization methodology.
Paper Structure (11 sections, 41 equations, 2 figures, 2 tables)

This paper contains 11 sections, 41 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related works and preliminaries
  3. Quadratic Unconstrained Binary Optimization (QUBO) formulation
  4. Related work of the ADAROUND quantization method
  5. Mathematical formulation of quantization of variables
  6. QUBO-ADAROUND problem of a neural network layer
  7. Division of the problem into subproblems
  8. Algorithm implementation
  9. Experimentation
  10. Conclusions
  11. Mathematical derivation

Figures (2)

  • Figure 1: Accuracy vs $C_{QUBO} = \mathbf{v}^{T}\cdot{\mathbb{E}[M]}\cdot\mathbf{v}$ for a neural network of one layer with 10 neurons for the MNIST dataset quantized to int2. Random quantizations are shown in blue, ADAROUND in red, and rounding-to-the-nearest in green.
  • Figure 2: Accuracy vs $C_{QUBO} = \mathbf{v}^{T}\cdot{\mathbb{E}[M]}\cdot\mathbf{v}$ for a neural network of one layer with 10 neurons for the MNIST dataset quantized to int8. Random quantizations are shown in blue, ADAROUND in red, and rounding-to-the-nearest in green.