Boolean Function Learning

Interpretable machine learning is essential in high-stakes domains where decision-making requires accountability, transparency, and trust. While rule-based models offer global and exact interpretability, learning rule sets that simultaneously achieve high predictive performance and low, human-understandable complexity remains challenging. To address this, we introduce TT-Sparse, a flexible neural building block that leverages differentiable truth tables as nodes to learn sparse, effective connections. A key contribution of our approach is a new soft TopK operator with straight-through estimation for learning discrete, cardinality-constrained feature selection in an end-to-end differentiable manner. Crucially, the forward pass remains sparse, enabling efficient computation and exact symbolic rule extraction. As a result, each node (and the entire model) can be transformed exactly into compact, globally interpretable DNF/CNF Boolean formulas via Quine-McCluskey minimization. Extensive empirical results across 28 datasets spanning binary, multiclass, and regression tasks show that the learned sparse rules exhibit superior predictive performance with lower complexity compared to existing state-of-the-art methods.

The complexity of learning a concept class under Gaussian marginals in the difficult agnostic model is closely related to its $L_1$-approximability by low-degree polynomials. For any concept class with Gaussian surface area at most $Γ$, Klivans et al. (2008) show that degree $d = O(Γ^2 / \varepsilon^4)$ suffices to achieve an $\varepsilon$-approximation. This leads to the best-known bounds on the complexity of learning a variety of concept classes. In this note, we improve their analysis by showing that degree $d = \tilde O (Γ^2 / \varepsilon^2)$ is enough. In light of lower bounds due to Diakonikolas et al. (2021), this yields (near) optimal bounds on the complexity of agnostically learning polynomial threshold functions in the statistical query model. Our proof relies on a direct analogue of a construction of Feldman et al. (2020), who considered $L_1$-approximation on the Boolean hypercube.

Recent work has shown the surprising power of low-degree sandwiching polynomial approximators in the context of challenging learning settings such as learning with distribution shift, testable learning, and learning with contamination. A pair of sandwiching polynomials approximate a target function in expectation while also providing pointwise upper and lower bounds on the function's values. In this paper, we give a new method for constructing low-degree sandwiching polynomials that yield greatly improved degree bounds for several fundamental function classes and marginal distributions. In particular, we obtain degree $\mathrm{poly}(k)$ sandwiching polynomials for functions of $k$ halfspaces under the Gaussian distribution, improving exponentially over the prior $2^{O(k)}$ bound. More broadly, our approach applies to function classes that are low-dimensional and have smooth boundary. In contrast to prior work, our proof is relatively simple and directly uses the smoothness of the target function's boundary to construct sandwiching Lipschitz functions, which are amenable to results from high-dimensional approximation theory. For low-dimensional polynomial threshold functions (PTFs) with respect to Gaussians, we obtain doubly exponential improvements without applying the FT-mollification method of Kane used in the best previous result.

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.

In the classic point location problem, one is given an arbitrary dataset $X \subset \mathbb{R}^d$ of $n$ points with query access to an unknown halfspace $f : \mathbb{R}^d \to \{0,1\}$, and the goal is to learn the label of every point in $X$. This problem is extremely well-studied and a nearly-optimal $\widetilde{O}(d \log n)$ query algorithm is known due to Hopkins-Kane-Lovett-Mahajan (FOCS 2020). However, their algorithm is granted the power to query arbitrary points outside of $X$ (point synthesis), and in fact without this power there is an $Ω(n)$ query lower bound due to Dasgupta (NeurIPS 2004). In this work our goal is to design efficient algorithms for learning halfspaces without point synthesis. To circumvent the $Ω(n)$ lower bound, we consider learning halfspaces whose normal vectors come from a set of size $D$, and show tight bounds of $Θ(D + \log n)$. As a corollary, we obtain an optimal $O(d + \log n)$ query deterministic learner for axis-aligned halfspaces, closing a previous gap of $O(d \log n)$ vs. $Ω(d + \log n)$. In fact, our algorithm solves the more general problem of learning a Boolean function $f$ over $n$ elements which is monotone under at least one of $D$ provided orderings. Our technical insight is to exploit the structure in these orderings to perform a binary search in parallel rather than considering each ordering sequentially, and we believe our approach may be of broader interest. Furthermore, we use our exact learning algorithm to obtain nearly optimal algorithms for PAC-learning. We show that $O(\min(D + \log(1/\varepsilon), 1/\varepsilon) \cdot \log D)$ queries suffice to learn $f$ within error $\varepsilon$, even in a setting when $f$ can be adversarially corrupted on a $c\varepsilon$-fraction of points, for a sufficiently small constant $c$. This bound is optimal up to a $\log D$ factor, including in the realizable setting.

Traditional neural networks have an impressive classification performance, but what they learn cannot be inspected, verified or extracted. Neural Logic Networks on the other hand have an interpretable structure that enables them to learn a logical mechanism relating the inputs and outputs with AND and OR operations. We generalize these networks with NOT operations and biases that take into account unobserved data and develop a rigorous logical and probabilistic modeling in terms of concept combinations to motivate their use. We also propose a novel factorized IF-THEN rule structure for the model as well as a modified learning algorithm. Our method improves the state-of-the-art in Boolean networks discovery and is able to learn relevant, interpretable rules in tabular classification, notably on examples from the medical and industrial fields where interpretability has tangible value.