The paper proves that training a binary quantized neural network (2-QNNT) is W[1]-hard when parameterized solely by the sum of input and output dimensions, α+ω. This hardness result holds even for zero training error on a specially constructed dataset where each input equals its target and the examples form a coordinate‑wise prefix chain. The proof reduces from DAG edge‑disjoint paths, employing a one‑flip routing equivalence that links activation transitions to vertex‑disjoint paths in the network.
arXiv:2505. 18113v2 Announce Type: replace Abstract: Training quantized neural networks requires addressing the non-differentiable and discrete nature of the underlying optimization problem.
The paper proves that several decision and approximation problems for ReLU neural networks are computationally hard. For any number of layers λ≥2, deciding whether a network’s output is positive (and thus whether it is surjective) is W[ℓ−1]-hard when parameterized by the input dimension d. In particular, for two-layer networks, the related geometric problem of zonotope non‑containment is W[1]-hard in the ambient dimension, and computing or approximating the Lp‑Lipschitz constant is NP‑hard and W[ℓ−1]-hard with respect to d. The results also show that these problems remain hard when parameterized by the number of layers for constant d, implying that naive enumeration algorithms running in n^{(ℓ−1)d}·poly(N) time are essentially optimal under the Exponential Time Hypothesis.
arXiv:2607. 20811v1 Announce Type: new Abstract: In spite of the fundamental role of neural networks in contemporary machine learning research, our understanding of the computational complexity of optimally training neural networks remains incomplete even when dealing with the simplest kinds of activation functions.
arXiv:2606. 26705v1 Announce Type: cross Abstract: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes.
arXiv:2607. 16568v1 Announce Type: new Abstract: Function-preserving network growth techniques such as Net2Net and progressive stacking expand a model's capacity without destroying its learned function, but existing formulations either tolerate numerical perturbations or require a full rebuild of the training program.
The paper presents a Quadratic Constrained Binary Optimization (QCBO) framework that provides provable guarantees for training quantized neural networks. It characterizes the topology of zero‑loss level sets, compiles finite‑depth architectures into bounded QCBOs, and introduces a sample‑wise Decomposed Lower‑Bound Optimization (DLBO) to scale Ising‑based optimization. Experiments on a coherent Ising machine show high accuracy on binary Fashion‑MNIST at 1.1‑bit precision and validate the approach on multi‑class datasets.
arXiv:2608.24865v1 Announce Type: cross Abstract: Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult e...
arXiv:2608. 04252v1 Announce Type: cross Abstract: The dynamical Lie algebraic (DLA) theory of variational quantum algorithms (VQAs) predicts commonplace exponentially vanishing loss and gradient variances for sufficiently deep parametrized circuits.
arXiv:2606. 12211v1 Announce Type: cross Abstract: A central principle in quantum machine learning is that an ansatz should be expressive enough to represent the quantum data of interest.
In spite of the fundamental role of neural networks in contemporary machine learning research, our understanding of the computational complexity of optimally training neural networks remains incomplete even when dealing with the simplest kinds of activation functions. Indeed, while there has been a number of very recent results that establish ever-tighter lower bounds for the problem under linear and ReLU activation functions, less progress has been made towards the identification of novel polynomial-time tractable network architectures.
arXiv:2602.13106v2 Announce Type: replace-cross Abstract: In recent years, there has been growing interest in understanding neural architectures' ability to learn to execute discrete algorithms, a li...
arXiv:2609.01441v1 Announce Type: new Abstract: Graph neural networks (GNN) based on message passing are provably no more powerful than the one-dimensional Weisfeiler--Leman colour-refinement test (1...