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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.17493 |
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| _version_ | 1866912913550213120 |
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| author | Elser, Veit Lal, Manish Krishan |
| author_facet | Elser, Veit Lal, Manish Krishan |
| contents | We develop a method for training neural networks on Boolean data in which the values at all nodes are strictly $\pm 1$, and the resulting models are typically equivalent to networks whose nonzero weights are also $\pm 1$. The method replaces loss minimization with a nonconvex constraint formulation. Each node implements a Boolean threshold function (BTF), and training is expressed through a divide-and-concur decomposition into two complementary constraints: one enforces local BTF consistency between inputs, weights, and output; the other imposes architectural concurrence, equating neuron outputs with downstream inputs and enforcing weight equality across training-data instantiations of the network. The reflect-reflect-relax (RRR) projection algorithm is used to reconcile these constraints.
Each BTF constraint includes a lower bound on the margin. When this bound is sufficiently large, the learned representations are provably sparse and equivalent to networks composed of simple logical gates with $\pm 1$ weights. Across a range of tasks -- including multiplier-circuit discovery, binary autoencoding, logic-network inference, and cellular automata learning -- the method achieves exact solutions or strong generalization in regimes where standard gradient-based methods struggle. These results demonstrate that projection-based constraint satisfaction provides a viable and conceptually distinct foundation for learning in discrete neural systems, with implications for interpretability and efficient inference. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning with Boolean threshold functions Elser, Veit Lal, Manish Krishan Machine Learning Artificial Intelligence We develop a method for training neural networks on Boolean data in which the values at all nodes are strictly $\pm 1$, and the resulting models are typically equivalent to networks whose nonzero weights are also $\pm 1$. The method replaces loss minimization with a nonconvex constraint formulation. Each node implements a Boolean threshold function (BTF), and training is expressed through a divide-and-concur decomposition into two complementary constraints: one enforces local BTF consistency between inputs, weights, and output; the other imposes architectural concurrence, equating neuron outputs with downstream inputs and enforcing weight equality across training-data instantiations of the network. The reflect-reflect-relax (RRR) projection algorithm is used to reconcile these constraints. Each BTF constraint includes a lower bound on the margin. When this bound is sufficiently large, the learned representations are provably sparse and equivalent to networks composed of simple logical gates with $\pm 1$ weights. Across a range of tasks -- including multiplier-circuit discovery, binary autoencoding, logic-network inference, and cellular automata learning -- the method achieves exact solutions or strong generalization in regimes where standard gradient-based methods struggle. These results demonstrate that projection-based constraint satisfaction provides a viable and conceptually distinct foundation for learning in discrete neural systems, with implications for interpretability and efficient inference. |
| title | Learning with Boolean threshold functions |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2602.17493 |