Search arXiv⌕ Search

arXiv subjects

Tianhao Wang

Publications and source records attributed to Tianhao Wang.

At least 19 recordsLinked to original sources
arXiv2026

Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics

Understanding how gradient descent learns features remains a central challenge in neural network theory. We study this question in a minimal multi-index setting that already exhibits rich multi-phase gradient dynamics: a well-specified two-neuron ReLU teacher-student model under isotropic Gaussian inputs, trained by population gradient flow from small initialization. We show that the dynamics are organized by two directions: the "easy" bisector direction, which carries the leading signal, and the "hard" splitting direction, which governs specialization. To characterize the loss structure and learning behavior, we analyze the population squared-loss landscape and show that every critical point with both student weights nonzero is either a global minimum or a saddle. We then track the gradient-flow trajectory, showing that student neurons first collapse toward a bisector saddle before escaping and specializing to teacher neurons. Our results provide both a landscape and dynamics account of the multi-phase symmetry-breaking behavior that arises in a simple multi-index model and under standard algorithmic and architectural choices.

cs.LG↗
arXiv2026

PrivCode: When Code Generation Meets Differential Privacy

Large language models (LLMs) have presented outstanding performance in code generation and completion. However, fine-tuning these models on private datasets can raise privacy and proprietary concerns, such as the leakage of sensitive personal information. Differentially private (DP) code generation provides theoretical guarantees for protecting sensitive code by generating synthetic datasets that preserve statistical properties while reducing privacy leakage concerns. However, DP code generation faces significant challenges due to the strict syntactic dependencies and the privacy-utility trade-off. We propose PrivCode, the first DP synthesizer specifically designed for code datasets. It incorporates a two-stage framework to improve both privacy and utility. In the first stage, termed "privacy-sanitizing", PrivCode generates DP-compliant synthetic code by training models using DP-SGD while introducing syntactic information to preserve code structure. The second stage, termed "utility-boosting", fine-tunes a larger pre-trained LLM on the synthetic privacy-free code to mitigate the utility loss caused by DP, enhancing the utility of the generated code. Extensive experiments on four LLMs show that PrivCode generates higher-utility code across various testing tasks under four benchmarks. The experiments also confirm its ability to protect sensitive data under varying privacy budgets. We provide the replication package at the code repository: https://github.com/Liuzzyg/PrivCode.

cs.CR↗
arXiv2026

CUA-Recovery: Benchmarking Error Recovery in Long-Horizon Computer-Use Agents

As computer-use agents (CUAs) move to long-horizon workflows spanning multiple applications, undetected errors can silently corrupt persistent state and cause cascading failures in subsequent actions. Reliable error detection and recovery have become critical bottlenecks for building robust CUAs. However, existing benchmarks focus primarily on end-to-end task performance, providing limited insight into whether agents can detect and recover from errors during execution. The few benchmarks that evaluate recovery rely largely on manually injected errors, which may not reflect the types or horizons of failures induced by real agent policies. We introduce CUA-Recovery, a benchmark comprising test and training suites across multiple desktop applications. We construct it using ReRail, a framework for synthesizing long-horizon tasks and trajectories in a persistent desktop environment. ReRail composes MyPCBench tasks into new long-horizon workflows with executable ground truth, collects verifier-confirmed failures from multiple policies, and turns them into verified recovery trajectories. CUA-Recovery comprises a human-verified test suite of 2,250 erroneous states over 100 tasks and a training suite of 8.6K trajectories over 377 tasks. Evaluation on the test suite shows that agents struggle both to recognize inherited errors and to complete the task. Across six agents and six takeover depths, Error Awareness Rate (EAR) reaches at most 44.0% and Pass@3 at most 25.5%. After fine-tuning Qwen3.5-35B-A3B on the training suite, its average EAR rises from 19.9% to 35.2% on held-out recovery states, and its Rubric Score goes from 5.0% to 16.1%. It also gains 15.3 points on MyPCBench and 10.8 points on Odysseys for Pass@3, showing that the training trajectories help both with recovery and with completing tasks from a clean start. Our code is available at https://github.com/xu-hu-2002/CUA-Recovery.

cs.AI↗
arXiv2026

Borel Combinatorics of Schreier Graphs of $\mathbb{Z}$-actions

In this paper we consider the Borel combinatorics of Schreier graphs of $\mathbb{Z}$-actions with arbitrary finite generating sets. We focus mostly on the problem of computing the Borel chromatic numbers of the Schreier graphs on the free part of the shift space $2^\mathbb{Z}$ with a finite symmetric generating set $S$ of $\mathbb{Z}$. Our main results include some general lower and upper bounds, as well as the exact values in some special cases. These values are sometimes the same as the chromatic numbers of the distance graphs on $\mathbb{Z}$ given by $S$, and more often different from them.

math.CO↗
arXiv2026

Honesty over Accuracy: Trustworthy Language Models through Reinforced Hesitation

Modern language models fail a fundamental requirement of trustworthy intelligence: knowing when not to answer. Despite achieving impressive accuracy on benchmarks, these models produce confident hallucinations, even when wrong answers carry catastrophic consequences. Our evaluations on GSM8K, MedQA and GPQA show frontier models almost never abstain despite explicit warnings of severe penalties, suggesting that prompts cannot override training that rewards any answer over no answer. As a remedy, we propose Reinforced Hesitation (RH): a modification to Reinforcement Learning from Verifiable Rewards (RLVR) to use ternary rewards (+1 correct, 0 abstention, -$λ$ error) instead of binary. Controlled experiments on logic puzzles reveal that varying $λ$ produces distinct models along a Pareto frontier, where each training penalty yields the optimal model for its corresponding risk regime: low penalties produce aggressive answerers, high penalties conservative abstainers. The same frontier holds on MATH Levels 4--5 and on medical QA, where it transfers to an unseen dataset. We then introduce two inference strategies that exploit trained abstention as a coordination signal: cascading routes queries through models with decreasing risk tolerance, while self-cascading re-queries the same model on abstention. Both outperform majority voting with lower computational cost. These results establish abstention as a first-class training objective that transforms ``I don't know'' from failure into a coordination signal, enabling models to earn trust through calibrated honesty about their limits.

cs.LG↗
arXiv2026

Provable Benefit of SignGD: A Minimal Model Under Heavy-Tailed Class Imbalance

Adaptive and non-Euclidean optimizers often outperform Euclidean methods such as stochastic gradient descent (SGD) in language modeling by a large margin. Existing theory usually explains this gap by assuming favorable smoothness geometry or noise structure tailored to the specific optimizer. We instead ask whether such geometry can be induced from a concrete learning setting. Starting from an optimizer gap that persists across realistic language-modeling experiments, we progressively remove sequence dependence, architectural complexity, and stochasticity. We find that the gap exists in a minimal setting: the softmax unigram model with heavy-tailed data. This model exposes a simple deterministic mechanism under heavy-tailed class imbalance. We prove that GD learns rare tokens slowly because the corresponding logits receive only tiny updates, while SignGD removes this magnitude dependence and moves rare and common coordinates on a more comparable scale. We make this precise with upper and lower bounds for the convergence rate of GD and upper bounds for the convergence of SignGD. Our stochastic bounds contain additional noise-dependent terms that can obscure this advantage in the convergence guarantees and can be reduced by increasing the batch size

cs.LG↗
arXiv2026

Realistic Characterization and Modeling of Vehicle-Mounted Antenna Radiation Patterns Based on Large-Scale Full-Vehicle Measurements

Vehicle-mounted antenna radiation patterns can be significantly altered after installation due to interactions with the vehicle body. However, existing studies mainly focus on simulations or isolated antenna characterization, while large-scale experimental investigations remain limited. This paper presents a comprehensive experimental analysis of vehicle-mounted antenna radiation patterns based on full-vehicle spherical near-field measurements. A large-scale dataset covering ten commercial vehicle platforms, multiple antenna mounting locations, and various wireless functions, including 4G, 5G, V2X, and GNSS, is established under a unified measurement framework. Based on the measured results, representative case studies and statistical characterization are performed to investigate the impacts of vehicle installation on antenna radiation behavior. The results reveal substantial variations across mounting locations and wireless functions, highlighting the strong influence of vehicle integration on realistic antenna performance. Furthermore, a novel statistical model is proposed to generate realistic vehicle-mounted antenna patterns using a small set of physically interpretable parameters. The obtained dataset, characterization results, and modeling approach provide a foundation for realistic antenna evaluation and wireless system simulations in vehicular communication applications.

eess.SP↗
arXiv2026

INFUSER: Influence-Guided Self-Evolution Improves Reasoning

Self-evolution offers a scalable path to stronger reasoning: a pretrained language model improves itself with only minimal external supervision. Yet existing methods either depend on extensively curated or teacher-generated training data, or, when the generator runs unsupervised, reward it by a difficulty heuristic that need not improve the solver. We introduce INFUSER, an iterative co-training framework with two co-evolving roles: a Generator that drafts questions and reference golden answers from a pool of unstructured, automatically collected documents, and a Solver that improves by training on them. The solver is trained with standard correctness rewards against the generator-provided answers, while the generator is rewarded by an optimizer-aware influence score that measures whether each proposed question would actually improve the solver on the target distribution. Because this continuous, noisy influence score is poorly served by standard GRPO, we propose DuGRPO, a dual-normalized variant of GRPO, for generator training. Together, these turn the document pool into an adaptive curriculum that favors questions useful to the current solver, not just hard ones. On Qwen3-8B-Base, INFUSER outperforms strong self-evolution baselines with over 20% relative improvement on Olympiad and SuperGPQA benchmarks, and an 8B INFUSER co-evolving generator outperforms a frozen 32B thinking generator on math and coding. Ablations confirm each design choice is necessary, and two extensions, applying INFUSER to an instruction-finetuned anchor and augmenting it with rule-verifiable RLVR data, further demonstrate the flexibility and generalizability of the framework. Code is available at https://github.com/FFishy-git/INFUSER.

cs.LG↗
arXiv2026

Towards Understanding Momentum Acceleration in River-Valley Loss Landscape

The empirical success of pretraining large language models has inspired a deeper investigation into the underlying loss landscapes and the optimization dynamics. Recent empirical and theoretical study suggest that the training loss landscape often exhibits a "river-valley" structure, which features a low-loss manifold (river) flanked by sharp orthogonal directions with higher loss (mountains). In the long term, the optimization progress is determined primarily by the progress along the river. Within such a landscape, gradient descent with large learning rates can move faster along the river despite high apparent loss due to vertical oscillations, while a subsequent sharp decay in the learning rate suppresses these oscillations, revealing genuine optimization progress. This explains the recent success of warmup-stable-decay (WSD) learning rate scheduler which, unlike cosine scheduling, keeps stable high learning rate and decays before producing intermediate checkpoints. Building on this foundation, in this work we take a step further and study the role of momentum within such a loss landscape. We establish theoretical analysis that characterizes how momentum accelerates optimization by stabilizing large learning rates that can not be tolerated by vanilla GD without deviating significantly from the river. The enabled large learning rate in-turn gives greater speed along the river and makes faster essential progress in the long run. Another intriguing observation from theory is that for a river-valley landscape with very flat and slow-spinning river, the momentum itself does not contribute directly to acceleration in terms of the speed of tracking the river, while the main acceleration comes from the admissible larger learning rate.

cs.LG↗
arXiv2026

Tight Privacy Audit in One Run

In this paper, we study the problem of privacy audit in one run and show that our method achieves tight audit results for various differentially private protocols. This includes obtaining tight results for auditing $(\varepsilon,δ)$-DP algorithms where all previous work fails to achieve in any parameter setups. We first formulate a framework for privacy audit \textit{in one run} with refinement compared with previous work. Then, based on modeling privacy by the $f$-DP formulation, we study the implications of our framework to obtain a theoretically justified lower bound for privacy audit. In the experiment, we compare with previous work and show that our audit method outperforms the rest in auditing various differentially private algorithms. We also provide experiments that give contrasting conclusions to previous work on the parameter settings for privacy audits in one run. Our implementation is at https://github.com/zihangxiang/tight-privacy-audit-in-one-run.git

cs.CR↗
arXiv2026

On Universality of Non-Separable Approximate Message Passing Algorithms

Mean-field characterizations of first-order iterative algorithms -- including Approximate Message Passing (AMP), stochastic and proximal gradient descent, and Langevin diffusions -- have enabled a precise understanding of learning dynamics in many statistical applications. For algorithms whose non-linearities have a coordinate-separable form, it is known that such characterizations enjoy a degree of universality with respect to the underlying data distribution. However, mean-field characterizations of non-separable algorithm dynamics have largely remained restricted to i.i.d. Gaussian or rotationally-invariant data. In this work, we initiate a study of universality for non-separable AMP algorithms. We identify a general condition for AMP with polynomial non-linearities, in terms of a Bounded Composition Property (BCP) for their representing tensors, to admit a state evolution that holds universally for matrices with non-Gaussian entries. We then formalize a condition of BCP-approximability for Lipschitz AMP algorithms to enjoy a similar universal guarantee. We demonstrate that many common classes of non-separable non-linearities are BCP-approximable, including local denoisers, spectral denoisers for generic signals, and compositions of separable functions with generic linear maps, implying the universality of state evolution for AMP algorithms employing these non-linearities.

math.ST↗
arXiv2026

Beyond One-Size-Fits-All: Neural Networks for Differentially Private Tabular Data Synthesis

In differentially private (DP) tabular data synthesis, the consensus is that statistical models are better than neural network (NN)-based methods. However, we argue that this conclusion is incomplete and overlooks the challenge of densely correlated datasets, where intricate dependencies can overwhelm statistical models. In such complex scenarios, neural networks are more suitable due to their capacity to fit complex distributions by learning directly from samples. Despite this potential, existing NN-based algorithms still suffer from significant limitations. We therefore propose MargNet, incorporating successful algorithmic designs of statistical models into neural networks. MargNet applies an adaptive marginal selection strategy and trains the neural networks to generate data that conforms to the selected marginals. On sparsely correlated datasets, our approach achieves utility close to the best statistical method while offering an average 7$\times$ speedup over it. More importantly, on densely correlated datasets, MargNet establishes a new state-of-the-art, reducing fidelity error by up to 26\% compared to the previous best. We release our code on GitHub.\footnote{https://github.com/KaiChen9909/margnet}

cs.LG↗
arXiv2026

Differentially Private Paired Table-Image Multimodal Synthesis

Differentially private (DP) synthesis has been extensively studied for tabular and image data separately, yet many real-world datasets contain images paired with multivariate tabular records. Synthesizing such data is particularly challenging under DP, as the two modalities favor different private learning mechanisms while their dependence must also be preserved. To address this challenge, we propose DP-TabImage, a modality-specialized framework for private paired synthesis. DP-TabImage instantiates the factorization $p(x,y)=p_T(y)p_I(x\;|\;y)$ using a private Probabilistic Graphical Model for the multivariate table distribution and a table-conditioned diffusion model trained with DP-SGD for the conditional image distribution. To facilitate conditional learning under clipped and noisy gradients, we further pretrain the model on private table-image prototypes, pairing privately constructed attribute-conditioned images with tabular vectors derived from the already private tabular model at no additional privacy cost. Experiments on three real-world datasets show that DP-TabImage achieves a strong balance among tabular fidelity, image fidelity, and cross-modal alignment. Our analysis further reveals that visual warm-up primarily improves marginal image fidelity, whereas aligned table-image warm-up is critical for improving cross-modal correspondence. Our source code is available in the GitHub repository, https://github.com/KaiChen9909/TabImage_Syn.

cs.CR↗
arXiv2026

Revisiting Continuous Noise Sampling for Multi-Party Differential Privacy

Combining secure multi-party computation (MPC) with differential privacy (DP) enables multiple parties to release aggregate statistics without a trusted curator, and the core primitive is the protocol to sample noise from a continuous distribution under finite-precision arithmetic. In this paper, we revisit the continuous noise sampling protocols and present several improvements in both security and efficiency. We start by identifying a vulnerability in widely used sample-and-scale constructions. We demonstrate that the scaling operation in arithmetic circuits confines the noise to a sparse, publicly known set of values, so that an adversary can observe the released noisy queries and decide which dataset produced them. As concrete demonstrations, we instantiate attacks on two systems employing such ``flawed'' sampling protocols: Orchard (OSDI'20) for DP secure aggregation and DP-BREM$^+$ (USENIX Sec'25) for DP federated learning. We report a near-$100\%$ attack success rate on both systems, under any noise scaler $s\geq 2$ used in practice. The leakage we reveal is intrinsic to the scaling operation, and direct repairs either substantially sacrifice utility or add significant precision bits to make the sampling more expensive. To address the security and efficiency issues together, we turn to discrete sampling at the granularity of individual biased bits. We make several optimizations to the sampler and prove its security. Our implementation achieves $4\times \sim 612\times$ speedup over existing secure discrete samplers and orders-of-magnitude speedup over the insecure sample-and-scale paradigm, with negligible utility loss compared to the ideal continuous mechanism.

cs.CR↗
arXiv2026

An Exact Counting Formula for the Mutual Position of Two Plane Conics

Let $q$ be an odd prime power, and $\mathcal{C}, \mathcal{D}$ be two smooth plane conics defined over $\mathbb{F}_q$ with transversal intersection. We present an exact formula for the number of points in $\mathbb{P}^2(\mathbb{F}_q)$ that are internal/external to $\mathcal{C}$ and internal/external to $\mathcal{D}$. This refines the $\frac{q^2}{4}+O(q^{3/2})$ asymptotic estimate for this quantity due to Asgarli and Yip \cite[Theorem 1.2]{Asgarli}. In particular, we show that the error term is of size at most $q+\sqrt{q} + 1$. By studying the geometry of the incidence variety related to this problem, we link the exact point counts directly to the Frobenius traces of two associated elliptic curves, and the number of $\mathbb{F}_q$-rational intersection points of $\mathcal{C}$ and $\mathcal{D}$ and of the corresponding dual conics $\mathcal{C}^*$ and $\mathcal{D}^*$. Lastly, We provide a remark explaining the challenges in generalizing this method to the study of higher-dimensional quadrics.

math.CO↗
arXiv2026

Geometry-Aware MCTS for Extremal Problems in Combinatorial Geometry

We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from $O(n^3)$ to $O(n^2)$. To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly $1.8 n$ for grids of size $82 \le n \le 119$. For the Smallest Complete Set problem, we find configurations of size roughly $0.95 n$, providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

cs.AI↗
arXiv2026

Strong marker sets for arbitrary generating sets

We prove the existence of clopen strong marker sets in $F(2^{\mathbb{Z}^n})$ for arbitrary finite generating sets. Specifically, for any positive integers $n, d_0\geq 1$ and any finite generating set $S\subseteq \mathbb{Z}^n$, we construct a clopen set $M\subseteq F(2^{\mathbb{Z}^n})$ and a positive integer $Δ$ such that (1) for any distinct $x,y\in M$ in the same orbit, $ρ(x,y)\geq d_0$; (2) for any $v\in S$ and any $x\in F(2^{\mathbb{Z}^n})$, there are non-negative integers $a,b\leq Δ$ such that $av\cdot x\in M$ and $-bv\cdot x\in M$. Here $ρ$ denotes the Euclidean metric. The same result then holds for the standard supremum-norm metric $ρ_\infty$ (with an adjusted constant), by the equivalence of norms on $\mathbb{Z}^n$. The proof introduces polyhedral packages in $\mathbb{R}^n$ as a generalization of the rectangular packages used in earlier work, enabling the construction to handle generating vectors with arbitrary coordinate patterns. As an application, we obtain a continuous proper edge $(2|S|+1)$-coloring of the Schreier graph on $F(2^{\mathbb{Z}^n})$ with generating set $S$, recovering a result of Gao--Wang--Wang.

math.LO↗
arXiv2026

Can Neural Networks Achieve Optimal Computational-statistical Tradeoff? An Analysis on Single-Index Model

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal computational-statistical tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $Ω(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model. However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time. Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches. We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $θ^\star$, with sample complexity $\widetilde{O} (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$. Furthermore, we extend our approach to the setting where $θ^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\widetildeΩ(k^{s^\star})$, matched by our method up to a polylogarithmic factor. Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

cs.LG↗