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Neural networks (NNs) have been extremely successful across many tasks in machine learning. Quantization of NN weights has become an important topic due to its impact on their energy efficiency, inference time and deployment on hardware. Although post-training quantization is well-studied, training optimal quantized NNs involves combinatorial non-convex optimization problems which appear intractable. In this work, we introduce a convex optimization strategy to train quantized NNs with polynomial activations. Our method leverages hidden convexity in twolayer neural networks from the recent literature, semidefinite lifting, and Grothendieck’s identity. Surprisingly, we show that certain quantized NN problems can be solved to global optimality provably in polynomial time in all relevant parameters via tight semidefinite relaxations. We present numerical examples to illustrate the effectiveness of our method.
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This content will become publicly available on December 1, 2026
Quantum sequel of neural network training
Abstract Training of neural networks (NNs) has emerged as a major consumer of both computational and energy resources. Quantum computers were coined as a root to facilitate training, but no experimental evidence has been presented so far. Here we demonstrate that quantum annealing platforms, such as D-Wave, can enable fast and efficient training of classical NNs, which are then deployable on conventional hardware. From a physics perspective, NN training can be viewed as a dynamical phase transition: the system evolves from an initial spin glass state to a highly ordered, trained state. This process involves eliminating numerous undesired minima in its energy landscape. The advantage of annealing devices is their ability to rapidly find multiple deep states. We found that this quantum training achieves superior performance scaling compared to classical backpropagation methods, with a clearly higher scaling exponent (1.01 vs. 0.78). It may be further increased up to a factor of 2 with a fully coherent quantum platform using a variant of the Grover algorithm. Furthermore, we argue that even a modestly sized annealer can be beneficial to train a deep NN by being applied sequentially to a few layers at a time.
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- Award ID(s):
- 2338819
- PAR ID:
- 10651320
- Publisher / Repository:
- Nature
- Date Published:
- Journal Name:
- Communications Physics
- Volume:
- 8
- Issue:
- 1
- ISSN:
- 2399-3650
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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