arXiv subjects

Guangjian Zhang

Publications and source records attributed to Guangjian Zhang.

8 recordsLinked to original sources

arXiv2026

Strictly Replicable Realizable Learning: Unknown Marginals, Finite-Class Interpolation, and the Cost of Shared Randomness

Strict replicability requires two independent training samples, processed with the same random seed, to produce exactly the same hypothesis. We study realizable Boolean learning under unknown and known instance marginals. For unknown marginals, we obtain the worst-case sample complexity $Θ(((\log M)^2/ρ^2+\log(1/δ))/\varepsilon)$ for classes of cardinality at most $M$, throughout a fixed small-parameter range. This resolves the distribution-free finite-class sample-complexity question up to universal constants and rules out a general $ρ^{-1}$ dependence. The upper bound uses a truncated-power distribution with a hard empirical-error budget. For known marginals, a control-variate learner uses $O(((1+\log M)^{3/2}/ρ+\log(1/δ))/\varepsilon)$ samples, and explicit uniform-marginal instances establish the matching class-size exponent. On a product of $d$ finite threshold chains of length $k$, we determine the exact rate $Θ((dk\log(1+\sqrt d/(kρ))+\log(1/δ))/\varepsilon)$. We also characterize deterministic learning, the very-small-$ρ$ regime for arbitrary finite classes, and every cardinality-replicability regime for finite classes of VC dimension one. A VC-covering-number bound yields the fixed-accuracy exponent $3/2$ independently of cardinality. Finally, we separate statistical complexity from implementation costs: we prove a sharp fixed-block trade-off, short-seed existence bounds, and a shared rejection-filter reduction from optimal learning of the full Boolean class to efficient public sampling of ordinary Bernoulli product distributions. Detailed proofs include the geometric lower bounds, the discrete posterior-kernel comparison, and the finite-bit constructions.

stat.ML↗

arXiv2026

The Optimal Sample Complexity of Pointwise Replicable Agnostic Learning

Pointwise replicability asks that two independently trained predictors, using the same random seed, agree at each fixed query with high probability. We determine its optimal agnostic sample complexity on finite and countable domains. For every nonempty binary class of finite VC dimension $d\ge1$, the required number of independent examples is $Θ\!\left(\frac{d}{ρ^2α^2}+\frac{\log(1/β)}{α^2}\right)$, uniformly over independent accuracy, failure, and disagreement parameters $α,β,ρ\in(0,1/10)$. The learner returns one binary function, may be improper, and uses unrestricted shared randomness. Its sample budget is fixed before observing data, and the replicability guarantee includes queries of zero marginal mass. The main construction combines an ensemble of empirical-risk minimizers with a smooth, signal-dependent risk decomposition. A single shared phase selects the final binary predictor. Record times certify its weak-signal risk, while a uniform bound protects the strong-signal risk at that same selected time. A phase-comparison theorem controls the actual shared-seed execution under simultaneous changes in the marginals and risk coefficients. A rare repair step then makes the confidence cost additive. Together with matching lower bounds, this removes the remaining logarithmic overhead and parameter restrictions in the general countable-domain sample-complexity problem. This extended version also retains the earlier explicit constructions and their proofs, including proper learning for VC-one classes, the full finite-atom class, a faster realizable bound, and counterexamples that separate restrictions of particular algorithms from restrictions of the learning problem.

cs.LG↗

arXiv2026

Bandit Multiclass PAC Learning: Structural Baselines and a Confidence Direct Sum

How much does high confidence cost when a multiclass learner sees only whether its prediction is correct? We separate two behaviors in realizable bandit PAC learning. An anchored dimension gives a general lower bound that is attained, up to universal constants, by a family of product-block classes. In contrast, for $n\ge16$ divisible by four, an affine multiplexer with bandit DS dimension $2n-1$ has sample complexity $Θ((n\min\{n,\ln(1/δ)\}+\ln(1/δ))/\varepsilon)$ in the standard accuracy and confidence regime. Its lower bound applies to adaptive randomized learners with improper outputs. The obstruction is that evidence collected against one candidate group cannot be reused against the others. The better of two algorithms matches this cost. One finds the correct group through a positive response. The other exhausts the rank of successive groups. The resulting confidence profile rules out a universal additive-confidence bound linear in the bandit DS dimension. Moreover, classes with identical anchored dimension, bandit DS dimension, and pointwise label width can have polynomially different sample complexities. We also sharpen the confidence accounting in ListCascade and show how one positive response replaces the bandit DS dimension by the anchored dimension in a general upper bound.

cs.LG↗

arXiv2026

An Agnostic Sample Compression Scheme for Squared Loss of Near-Linear Size in the Fat-Shattering Dimension

We study the squared-loss sample compression question posed by Attias, Hanneke, Kontorovich, and Sadigurschi in their ICML 2024 paper. For every nonempty class $\mathcal{F}\subseteq[0,1]^{\mathcal{X}}$ and accuracy $0<α\le1$ with $d=\operatorname{fat}(\mathcal{F},c'α)<\infty$, where $c'>0$ is universal, we prove the existence of a deterministic agnostic scheme of total size $O(d\log^3(2/α))$. Its code contains original labeled observations and auxiliary bits, and its reconstruction has empirical loss at most $α$ above the infimum over $\mathcal{F}$ on every finite sample. This establishes the polylogarithmic formulation in Section 5 of arXiv:1810.01864v2. The construction approximates a near-optimal internal target by a fixed-step boosting ensemble and takes its equal-weight median. Average loss permits an accuracy-dependent number of rounds, so the entire ensemble can be retained without dual-dimension sparsification. Quantized target constraints and round-membership bits determine the reconstructed function exactly. We also prove that, when retained observations must come from the sample, a fixed class of fat-shattering dimension one requires total size $Ω(\log(1/α))$. With observations and auxiliary bits charged together, this rules out the proceedings version's log-free formulation under its original-example restriction. The framework extends to average power loss at scale $α/p$.

cs.LG↗

arXiv2026

Exact Risk Ratios for Weighted Data Selection in Linear Regression

How much data must a fixed learner retain? Hanneke, Moran, Shlimovich and Yehudayoff (COLT 2025) posed this question for linear regression with the minimum-norm empirical risk minimizer. A selector sees a finite dataset $D\subseteq R^d\times R$, keeps at most $n$ examples with nonnegative weights, and $F_w(d,n)$ is the worst-case ratio between the full-data loss of the trained predictor and the optimal loss. The value is $\infty$ for $n<d$, $d+1$ at $n=d$ and $1$ for $n\ge2d$, and the regime $d<n<2d$ was left open. We settle several cases. For every $d$ we prove $F_w(d,2d-1)=1+1/d$, which confirms a claim stated without proof in the original note. We also prove $F_w(3,4)=5/3$, $F_w(4,5)=2$ and $F_w(4,6)=3/2$, the three smallest cells not covered by that formula. For every intermediate budget $n=d+k$ we prove the lower bound $F_w(d,d+k)\ge1+Γ_{d,k}$, where $Γ_{d,k}$ is an explicit harmonic quantity over balanced partitions of $d$. This bound is the exact minimax value on the class of datasets whose whitened systems split into orthogonal circuit blocks. All proved values equal $1+Γ_{d,k}$, and we conjecture that this holds throughout the open regime. Our upper bounds combine a rigidity theorem for positive spanning configurations of loss gradients with normal forms of the small positive bases in $R^3$ and $R^4$. These forms are special cases of the classification of Cornaz, Kerleau and Royer; we also control all gradients outside the basis. A dimension-free extremal-basis argument converts sign-cone geometry into selections of $d+1$ points. Explicit counterexamples rule out several shorter routes. Every upper bound is constructive, with selection procedures polynomial in the number of points for fixed dimension. Every numerical claim about a specific instance is an exact rational or algebraic identity, recomputed in exact arithmetic in the supplementary material.

cs.LG↗

arXiv2026

Exact Recovery Thresholds for Weighted Data Selection in Vector-Valued Linear Regression

We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. We study vector-valued linear regression with square loss $\ell_{(x,y)}(W)=\lVert Wx-y\rVert_2^2$, where $x\in\mathbb{R}^d$ and $y\in\mathbb{R}^m$. The learner returns the minimum-Frobenius-norm empirical risk minimizer. We prove that the minimal budget of weighted examples for recovering the full-data loss on every finite dataset is exactly $n^{\star}(d,m)=(m+1)d$. We determine the weighted selection profile $F_{\mathrm{weighted}}(d,m,n)$ at the near-threshold budget: $F_{\mathrm{weighted}}(d,m,(m+1)d-1)=1+1/(dm^2)$. We recover the known spanning-budget value $F_{\mathrm{weighted}}(d,m,d)=d+1$ for every $m$, and $F_{\mathrm{weighted}}(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_{\mathrm{weighted}}(2,2,3)\in[13/8,15/8]$ and $F_{\mathrm{weighted}}(2,2,4)\in[5/4,3/2]$. We reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms and assemble structural evidence for it. The upper bounds use a fixed-basis conic compression lemma, a determinant--facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems. These tools may be of independent interest. We also exhibit an explicit six-point integer dataset with $d=m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. Thus the scalar sufficient budget $2d$ does not extend to vector-valued outputs. Our new regression-profile results for $m\ge2$ extend the scalar theory for $m=1$.

cs.LG↗

arXiv2023

Exploiting Point-Wise Attention in 6D Object Pose Estimation Based on Bidirectional Prediction

Traditional geometric registration based estimation methods only exploit the CAD model implicitly, which leads to their dependence on observation quality and deficiency to occlusion. To address the problem,the paper proposes a bidirectional correspondence prediction network with a point-wise attention-aware mechanism. This network not only requires the model points to predict the correspondence but also explicitly models the geometric similarities between observations and the model prior. Our key insight is that the correlations between each model point and scene point provide essential information for learning point-pair matches. To further tackle the correlation noises brought by feature distribution divergence, we design a simple but effective pseudo-siamese network to improve feature homogeneity. Experimental results on the public datasets of LineMOD, YCB-Video, and Occ-LineMOD show that the proposed method achieves better performance than other state-of-the-art methods under the same evaluation criteria. Its robustness in estimating poses is greatly improved, especially in an environment with severe occlusions.

cs.CV↗

arXiv2017

On subcritically Stein fillable 5-manifolds

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the 5-sphere the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

math.GT↗