arXiv subjects

Huibo Xu

Publications and source records attributed to Huibo Xu.

3 recordsLinked to original sources

arXiv2026

Feature Priming in Online Linear Regression: Sparse-Regret Lower Bounds and Tight Coordinatewise Rates

In high-dimensional online prediction, sparse comparators motivate regret bounds that depend on sparsity rather than ambient dimension. Feature priming seeks such adaptation by reweighting features using past data and refitting a minimum-norm predictor. At COLT 2023, Warmuth and Amid posed the open problem of whether the univariate, Pearson, or multivariate priming rules admit competitive online regret guarantees. Under the natural past-only Moore--Penrose protocol, we establish sparse-regret lower bounds that refute the corresponding sparse-logarithmic guarantee. The key obstruction is cheap nuisance interpolation, which permits exact interpolation of the history while assigning insufficient weight to the truly predictive coordinate. An exact target-mass identity and a two-sign argument convert this obstruction into clipped prediction loss. Hadamard constructions yield $Ω(\min\{T,\sqrt d\})$ clipped regret for each of the three unit-power rules against a zero-loss one-sparse comparator. For every fixed power $α\ge1$, one shared paired construction further yields linear regret simultaneously for all three powered rules and selectors among them in sufficiently high dimension. A rank upper bound is tight for powered univariate priming, even with Euclidean-unit inputs, and for unit-power Pearson priming with coordinatewise bounded inputs and target-preserving totalization. A separate algebraic construction gives $Ω(\min\{T,d^{1/4}\})$ regret for unit-power multivariate priming under Euclidean-unit inputs. The univariate lower bound persists under any nonnegative second-stage ridge schedule, while a paired ridge construction yields linear lower bounds for all three powered rules. Exploratory diagnostics on frozen language-model activations are consistent with the same qualitative mechanism. The exact multivariate frontier remains open.

stat.ML↗

arXiv2026

Synchronization Strings over the Optimal Alphabet

Synchronization strings provide deterministic position labels for recovering coordinates after insertions and deletions. Haeupler and Shahrasbi introduced these objects, and subsequent work proved that four symbols suffice for some fixed parameter epsilon < 1, whereas two symbols cannot support arbitrarily long synchronization strings. We resolve the remaining ternary case: every length admits a ternary 2001/2002-synchronization string. Thus three is the exact minimum constant alphabet size. A computer-assisted refinement based on a larger 54-uniform family yields ternary epsilon-synchronization strings for every epsilon > 215/216. The previous four-symbol construction uses a ternary square-free backbone to exclude short repetitions and a fourth symbol to carry long-range synchronization marks. Our main technical contribution is a local-entropy transfer theorem: every square-free block-local source with a positive interval conditional min-entropy rate supports synchronization strings with a fixed gap. We instantiate this theorem using occurrence-wise branching in a Brinkhuis family. Every outcome remains ternary and square-free, while every long interval retains linear conditional min-entropy after all choices outside it are exposed. A deletion-ball estimate converts this entropy into an exponentially small probability of a near-complete common subsequence between adjacent intervals, and an asymmetric Lovasz Local Lemma enforces all interval constraints simultaneously. The same framework also yields exponentially many valid words, synchronization circles, and synchronization within a class of extremal square-free words. Adding constraints on distant intervals gives a Las Vegas construction in expected O(n^2 log^3(n+2)) time.

cs.IT↗

arXiv2026

Restricted Eigenvalues Beyond Gaussian Width: Threshold Occupancy under Heavy Tails

Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law follows for heavy-tailed designs from a uniform small-ball condition alone. We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form, and the missing obstruction is simultaneous threshold occupancy. A constant-width polyhedral descent cone with fixed small-ball constants has zero empirical RE on every sample path up to half the ambient dimension. More generally, every finite range space admits exact threshold encoding in an arbitrarily narrow spherical cap and a lift to a full polyhedral descent-cone section. For every fixed threshold VC dimension $d$, as $β\downarrow0$, the sharp worst-case sample complexity is $Θ(β^{-1}[d\log(1/β)+\log(1/δ)])$. The separation persists under exact isotropy and all finite moments: on the same constant-width cone, Gaussian measurements succeed with $O(1+\log(1/δ))$ samples, whereas an isotropic heavy-tailed design fails pathwise for $n\lesssim\sqrt{p/\log p}$. Gaussian smoothing yields an everywhere-positive $C^\infty$ density while retaining arbitrarily poor RE. Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.

cs.LG↗