Improved learning of k-parities

Arnab Bhattacharyya, Ameet Gadekar, Ninad Rajgopal

Di Yan, Yu Yu, Hanlin Liu +2 more

Elena Grigorescu, Lev Reyzin, Santosh Vempala

Sang Hyub Kim, Oliver Knitter, Jonathan Mei +4 more

We study parity features as representations that can be evaluated entirely classically once the binary or quantized input representation and parity words are fixed, particularly when labels depend on higher-order feature interactions or when discrete inference interfaces support perturbation robustness. A parity feature is a signed product over selected bits of a binary input: once the participating bits are known, evaluation requires no quantum resources. Reaching a useful parity representation requires solving two challenges. When the input is parity-ready (a meaningful binary string), the challenge is basis discovery: selecting useful parity words from a combinatorial search space. Otherwise, the challenge is encoding: constructing a binary vector on which parity computation is meaningful. We use hybrid quantum-classical training pipelines to address these: learnable Pauli word selection for basis discovery, learned projection encodings for continuous embeddings, and sPQC-Parity for discrete inputs. On three native-binary parity tasks with 5-10 qubits, the learned parity basis improves mean accuracy by 23.9% to 41.7% over logistic-regression and support-vector baselines. A model comparison shows that the improvement comes primarily from discovering the right parity basis, rather than from quantum moment computation at inference. On five continuous text benchmarks, learned projection recovers much of the loss introduced by dimensionality reduction and fixed binarization, exceeding the full continuous baseline on CR, SST-2, and SST-5. On three encoding-limited discrete datasets, when compared with PCA-bin as the baseline, sPQC-Parity reaches 94.6% improvement on mushroom, 3.0% on splice, and matches PCA-bin on promoter. We also analyze inference robustness under binary or quantized inference, where rounding gives exact invariance below half the quantization step.

Gregory J. Valiant

Given a set of n d-dimensional Boolean vectors with the promise that the vectors are chosen uniformly at random with the exception of two vectors that have Pearson-correlation ρ (Hamming distance d · 1-ρ/2), how quickly can one find the two correlated vectors? We present an algorithm which, for any constants ε, ρ >; 0 and d >;>; logn/ρ2, finds the correlated pair with high probability, and runs in time O(n 3ω/4 + ϵ)1.8), where w2-O(ρ)), given by Paturi et al. [15], Locality Sensitive Hashing (LSH) [11] and the Bucketing Codes approach [6]. Applications and extensions of this basic algorithm yield improved algorithms for several other problems: ApproximateClosest Pair: For any sufficiently small constant ϵ >; 0, given n vectors in Rd, our algorithm returns a pair of vectors whose Euclidean distance differs from that of the closest pair by a factor of at most 1+ϵ, and runs in time O(n2-Θ(√ϵ)). The best previous algorithms (including LSH) have runtime O(n2-O(ϵ)). Learning Sparse Parity with Noise: Given samples from an instance of the learning parity with noise problem where each example has length n, the true parity set has size at most kkpoly(1/1-2η)0.8kpoly(1/1-2η). This is the first algorithm with no depenJence on η in the exponent of n, aside from the trivial brute-force algorithm. Learning k-Juntas with Noise: Given uniformly random length n Boolean vectors, together with a label, which is some function of just kkpoly(1/1-2η)0.8kpoly(1/1-2η), 2 which improves on the previous best of >; nk(1-2/2k)poly( 1/1-2η ), from [8]. Learning k-Juntas without Noise:1 Our results for learning sparse parities with noise imply an algorithm for learning juntas without noise with runtime n ω+ϵ/4kpoly(n)0.6kpoly(n), which improves on the runtime n ω+1/ω poly(n) ≈ n0.7kpoly(n) of Mossel n et al. [13].

Oded Regev