arXiv subjects

Zihao Hu

Publications and source records attributed to Zihao Hu.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Strategies for Multi-parameter Quantum Metrology

Estimating multiple unknown parameters simultaneously is essential for practical quantum sensing. However, it faces a fundamental challenge: the optimal strategy for estimating one parameter is often incompatible with that for another, making it impossible to simultaneously achieve the ultimate precision limits for all parameters. Here we develop a general and efficient computational framework that jointly optimizes probe states, control operations, and measurements across different strategy families, including parallel, sequential, and those with indefinite causal order. Our approach provides exact semidefinite-program formulations for several precision bounds, including the Holevo, Nagaoka-Hayashi, and quantum Cram\'er-Rao bounds. We demonstrate the capabilities of the framework in multiparameter magnetometry and frequency estimation, identifying optimal protocols within each class and revealing a strict hierarchy among the achievable performances of different classes in the multiparameter regime. The framework also directly incorporates resource constraints, such as energy budgets, enabling systematic investigation of experimentally realistic sensing scenarios. Furthermore, we develop a finite-memory optimization method for sequential strategies with restricted ancillary-memory dimension. By decomposing the protocol into initial probe preparation and intermediate control operations, this method provides a practical route to designing resource-constrained sequential sensing schemes. Our work establishes a versatile computational tool for determining fundamental precision limits and designing optimal quantum-sensing protocols in complex multiparameter settings.

quant-ph↗

arXiv2026

Dynamic Regret for Non-Stationary Linear Bandits via Misspecification Reductions

Many online decision-making problems involve both round-specific feasible actions and drifting reward models: eligible ad impressions, feasible prices, and available treatments can change over time, while user preferences, demand curves, and patient responses may evolve. Motivated by these applications, we study non-stationary linear bandits with round-specific feasible decision sets. Existing methods that obtain the optimal $\widetilde O(T^{2/3}P_T^{1/3})$ dependence, where $P_T$ is the path length of the reward-parameter sequence, impose an orthogonal-structure assumption on round-specific decision sets, which can be restrictive in contextual applications. We address this gap through a unified misspecification-reduction viewpoint: after partitioning the horizon into blocks, we relate each block's dynamic regret to regret against a fixed-parameter linear bandit benchmark, with the within-block parameter drift entering as bounded misspecification. Restarting algorithms with misspecification-dependent regret guarantees then yields the optimal $T^{2/3}P_T^{1/3}$ dynamic-regret dependence for both linear bandits with general compact decision sets and $K$-armed contextual linear bandits.

cs.LG↗

arXiv2026

Learning to Bid with Unknown Private Values in Budget-Constrained First-Price Auctions

We study the operational problem of automated bidding in repeated first-price auctions under budget and return-on-spend (RoS) constraints. In this setting, an auto-bidder must translate advertiser goals and constraints into real-time bids while learning two latent objects: the causal uplift value of each ad impression and the highest competing bid (HoB) needed to win it. We model uplift values and HoBs through a shared-context Linear Treatment Effect (LTE) structure and analyze both full-information and binary HoB feedback. We develop Dual-LTE, a dual-aware online learning framework that coordinates value estimation, HoB estimation, and budget/RoS control through confidence-guided exploration. We prove regret and constraint-violation guarantees that scale as $\widetilde{O}(\sqrt{T})$ under full-information HoB feedback and $\widetilde{O}(T^{2/3})$ under binary win/loss feedback, where $\widetilde{O}(\cdot)$ hides problem-dependent and logarithmic factors. Semi-synthetic experiments using real auction covariates show that Dual-LTE achieves lower regret than the baselines across budget and RoS settings, while illustrating the tradeoff between regret and constraint violation. Our results provide operational guidance for DSPs and platform auto-bidders that manage advertiser budgets or seek to meet ROAS targets. When impression values must be learned, value estimation should be coordinated with budget or ROAS control: the auto-bidder should follow the Lagrangian bidding rule only when value estimates are sufficiently accurate given the current constraint pressure and should otherwise use controlled exploration.

cs.LG↗

arXiv2026

The (Marginal) Value of a Search Ad: An Online Causal Framework for Repeated Second-price Auctions

Existing auto-bidding algorithms in digital advertising often treat the value of an ad opportunity as the revenue obtained when an ad is shown and/or clicked, and bid accordingly. This can lead to wasteful spending because the true value is the marginal gain from paid exposure: even without winning a sponsored slot, an advertiser may still earn revenue via an organic search result (e.g., on Google or Amazon). Motivated by recent work, we model ad value as a treatment effect--the outcome difference between winning and losing the auction--and study online learning for bidding in second-price (Vickrey) auctions under this causal perspective. We develop algorithms that attain rate-optimal regret under several feedback models. A key ingredient exploits the information revealed by the second-price payment rule, which strictly improves regret relative to analogous learning problems in first-price auctions.

cs.GT↗

arXiv2025

Communication-Learning Co-Design for Differentially Private Over-the-Air Federated Distillation

The ever-growing learning model size nowadays challenges the communication efficiency and privacy preservation of the traditional federated learning (FL). In this paper, we propose a novel differentially private (DP) over-the-air federated distillation (FD) framework, where wireless devices (WDs) periodically share noise-perturbed model outputs with the parameter server by harnessing the superposition property of multi-access channels. Accordingly, over-the-air FD enables the shared responsibility of the DP preservation on the low-dimensional disclosed signals among WDs. We study the communication-learning co-design problem in differentially private over-the-air FD, aiming to maximize the learning convergence rate while meeting the transmit power and DP requirements of WDs. The main challenge is rooted in the intractable learning and privacy analysis in over-the-air FD, together with the strong coupling among the decision variables spanning two timescales. To tackle this problem, we first derive the analytical learning convergence rate and privacy losses of WDs, based on which the optimal transceiver design per FD round and long-term training rounds decision are obtained in the closed forms. Numerical results demonstrate that the proposed differentially private over-the-air FD approach achieves a better learning-privacy trade-off with largely-reduced communication overhead than the conventional FL benchmarks.

cs.IT↗

arXiv2025

Optimal Transceiver Design in Over-the-Air Federated Distillation

The rapid proliferation and growth of artificial intelligence (AI) has led to the development of federated learning (FL). FL allows wireless devices (WDs) to cooperatively learn by sharing only local model parameters, without needing to share the entire dataset. However, the emergence of large AI models has made existing FL approaches inefficient, due to the significant communication overhead required. In this paper, we propose a novel over-the-air federated distillation (FD) framework by synergizing the strength of FL and knowledge distillation to avoid the heavy local model transmission. Instead of sharing the model parameters, only the WDs' model outputs, referred to as knowledge, are shared and aggregated over-the-air by exploiting the superposition property of the multiple-access channel. We shall study the transceiver design in over-the-air FD, aiming to maximize the learning convergence rate while meeting the power constraints of the transceivers. The main challenge lies in the intractability of the learning performance analysis, as well as the non-convex nature and the optimization spanning the whole FD training period. To tackle this problem, we first derive an analytical expression of the convergence rate in over-the-air FD. Then, the closed-form optimal solutions of the WDs' transmit power and the estimator for over-the-air aggregation are obtained given the receiver combining strategy. Accordingly, we put forth an efficient approach to find the optimal receiver beamforming vector via semidefinite relaxation. We further prove that there is no optimality gap between the original and relaxed problem for the receiver beamforming design. Numerical results will show that the proposed over-the-air FD approach achieves a significant reduction in communication overhead, with only a minor compromise in testing accuracy compared to conventional FL benchmarks.

eess.SP↗

arXiv2025

Learning to Bid in Non-Stationary Repeated First-Price Auctions

First-price auctions have recently gained significant traction in digital advertising markets, exemplified by Google's transition from second-price to first-price auctions. Unlike in second-price auctions, where bidding one's private valuation is a dominant strategy, determining an optimal bidding strategy in first-price auctions is more complex. From a learning perspective, the learner (a specific bidder) can interact with the environment (other bidders, i.e., opponents) sequentially to infer their behaviors. Existing research often assumes specific environmental conditions and benchmarks performance against the best fixed policy (static benchmark). While this approach ensures strong learning guarantees, the static benchmark can deviate significantly from the optimal strategy in environments with even mild non-stationarity. To address such scenarios, a dynamic benchmark--representing the sum of the highest achievable rewards at each time step--offers a more suitable objective. However, achieving no-regret learning with respect to the dynamic benchmark requires additional constraints. By inspecting reward functions in online first-price auctions, we introduce two metrics to quantify the regularity of the sequence of opponents' highest bids, which serve as measures of non-stationarity. We provide a minimax-optimal characterization of the dynamic regret for the class of sequences of opponents' highest bids that satisfy either of these regularity constraints. Our main technical tool is the Optimistic Mirror Descent (OMD) framework with a novel optimism configuration, which is well-suited for achieving minimax-optimal dynamic regret rates in this context. We then use synthetic datasets to validate our theoretical guarantees and demonstrate that our methods outperform existing ones.

cs.LG↗

arXiv2024

Fully-Optimized Quantum Metrology: Framework, Tools, and Applications

This tutorial introduces a systematic approach for addressing the key question of quantum metrology: For a generic task of sensing an unknown parameter, what is the ultimate precision given a constrained set of admissible strategies. The approach outputs the maximal attainable precision (in terms of the maximum of quantum Fisher information) as a semidefinite program and optimal strategies as feasible solutions thereof. Remarkably, the approach can identify the optimal precision for different sets of strategies, including parallel, sequential, quantum SWITCH-enhanced, causally superposed, and generic indefinite-causal-order strategies. The tutorial consists of a pedagogic introduction to the background and mathematical tools of optimal quantum metrology, a detailed derivation of the main approach, and various concrete examples. As shown in the tutorial, applications of the approach include, but are not limited to, strict hierarchy of strategies in noisy quantum metrology, memory effect in non-Markovian metrology, and designing optimal strategies. Compared with traditional approaches, the approach here yields the exact value of the optimal precision, offering more accurate criteria for experiments and practical applications. It also allows for the comparison between conventional strategies and the recently discovered causally-indefinite strategies, serving as a powerful tool for exploring this new area of quantum metrology.

quant-ph↗

arXiv2023

Extragradient Type Methods for Riemannian Variational Inequality Problems

Riemannian convex optimization and minimax optimization have recently drawn considerable attention. Their appeal lies in their capacity to adeptly manage the non-convexity of the objective function as well as constraints inherent in the feasible set in the Euclidean sense. In this work, we delve into monotone Riemannian Variational Inequality Problems (RVIPs), which encompass both Riemannian convex optimization and minimax optimization as particular cases. In the context of Euclidean space, it is established that the last-iterates of both the extragradient (EG) and past extragradient (PEG) methods converge to the solution of monotone variational inequality problems at a rate of $O\left(\frac{1}{\sqrt{T}}\right)$ (Cai et al., 2022). However, analogous behavior on Riemannian manifolds remains an open question. To bridge this gap, we introduce the Riemannian extragradient (REG) and Riemannian past extragradient (RPEG) methods. We demonstrate that both exhibit $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence. Additionally, we show that the average-iterate convergence of both REG and RPEG is $O\left(\frac{1}{{T}}\right)$, aligning with observations in the Euclidean case (Mokhtari et al., 2020). These results are enabled by judiciously addressing the holonomy effect so that additional complications in Riemannian cases can be reduced and the Euclidean proof inspired by the performance estimation problem (PEP) technique or the sum-of-squares (SOS) technique can be applied again.

math.OC↗

arXiv2023

Riemannian Optimistic Algorithms

In this paper, we consider Riemannian online convex optimization with dynamic regret. First, we propose two novel algorithms, namely the Riemannian Online Optimistic Gradient Descent (R-OOGD) and the Riemannian Adaptive Online Optimistic Gradient Descent (R-AOOGD), which combine the advantages of classical optimistic algorithms with the rich geometric properties of Riemannian manifolds. We analyze the dynamic regrets of the R-OOGD and R-AOOGD in terms of regularity of the sequence of cost functions and comparators. Next, we apply the R-OOGD to Riemannian zero-sum games, leading to the Riemannian Optimistic Gradient Descent Ascent algorithm (R-OGDA). We analyze the average iterate and best-iterate of the R-OGDA in seeking Nash equilibrium for a two-player, zero-sum, g-convex-concave games. We also prove the last-iterate convergence of the R-OGDA for g-strongly convex-strongly concave problems. Our theoretical analysis shows that all proposed algorithms achieve results in regret and convergence that match their counterparts in Euclidean spaces. Finally, we conduct several experiments to verify our theoretical findings.

math.OC↗

arXiv2023

Minimizing Dynamic Regret on Geodesic Metric Spaces

In this paper, we consider the sequential decision problem where the goal is to minimize the general dynamic regret on a complete Riemannian manifold. The task of offline optimization on such a domain, also known as a geodesic metric space, has recently received significant attention. The online setting has received significantly less attention, and it has remained an open question whether the body of results that hold in the Euclidean setting can be transplanted into the land of Riemannian manifolds where new challenges (e.g., curvature) come into play. In this paper, we show how to get optimistic regret bound on manifolds with non-positive curvature whenever improper learning is allowed and propose an array of adaptive no-regret algorithms. To the best of our knowledge, this is the first work that considers general dynamic regret and develops "optimistic" online learning algorithms which can be employed on geodesic metric spaces.

cs.LG↗

arXiv2023

Randomized Quantization is All You Need for Differential Privacy in Federated Learning

Federated learning (FL) is a common and practical framework for learning a machine model in a decentralized fashion. A primary motivation behind this decentralized approach is data privacy, ensuring that the learner never sees the data of each local source itself. Federated learning then comes with two majors challenges: one is handling potentially complex model updates between a server and a large number of data sources; the other is that de-centralization may, in fact, be insufficient for privacy, as the local updates themselves can reveal information about the sources' data. To address these issues, we consider an approach to federated learning that combines quantization and differential privacy. Absent privacy, Federated Learning often relies on quantization to reduce communication complexity. We build upon this approach and develop a new algorithm called the \textbf{R}andomized \textbf{Q}uantization \textbf{M}echanism (RQM), which obtains privacy through a two-levels of randomization. More precisely, we randomly sub-sample feasible quantization levels, then employ a randomized rounding procedure using these sub-sampled discrete levels. We are able to establish that our results preserve ``Renyi differential privacy'' (Renyi DP). We empirically study the performance of our algorithm and demonstrate that compared to previous work it yields improved privacy-accuracy trade-offs for DP federated learning. To the best of our knowledge, this is the first study that solely relies on randomized quantization without incorporating explicit discrete noise to achieve Renyi DP guarantees in Federated Learning systems.

cs.LG↗

arXiv2023

Riemannian Projection-free Online Learning

The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), for enforcing constraints and achieving optimal regret bounds. However, it suffers from computational complexity limitations in high-dimensional settings or when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue by replacing the projection oracle with more efficient optimization subroutines. But to date, these methods have been developed primarily in the Euclidean setting, and while there has been growing interest in optimization on Riemannian manifolds, there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions are generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization on curved spaces for two scenarios: when we have access to (a) a separation oracle or (b) a linear optimization oracle. For geodesically convex losses, and when a separation oracle is available, our algorithms achieve $O(T^{1/2}\:)$ and $O(T^{3/4}\;)$ adaptive regret guarantees in the full information setting and the bandit setting, respectively. When a linear optimization oracle is available, we obtain regret rates of $O(T^{3/4}\;)$ for geodesically convex losses and $O(T^{2/3}\; log T )$ for strongly geodesically convex losses.

cs.LG↗

arXiv2023

Faster Margin Maximization Rates for Generic and Adversarially Robust Optimization Methods

First-order optimization methods tend to inherently favor certain solutions over others when minimizing an underdetermined training objective that has multiple global optima. This phenomenon, known as implicit bias, plays a critical role in understanding the generalization capabilities of optimization algorithms. Recent research has revealed that in separable binary classification tasks gradient-descent-based methods exhibit an implicit bias for the $\ell_2$-maximal margin classifier. Similarly, generic optimization methods, such as mirror descent and steepest descent, have been shown to converge to maximal margin classifiers defined by alternative geometries. While gradient-descent-based algorithms provably achieve fast implicit bias rates, corresponding rates in the literature for generic optimization methods are relatively slow. To address this limitation, we present a series of state-of-the-art implicit bias rates for mirror descent and steepest descent algorithms. Our primary technique involves transforming a generic optimization algorithm into an online optimization dynamic that solves a regularized bilinear game, providing a unified framework for analyzing the implicit bias of various optimization methods. Our accelerated rates are derived by leveraging the regret bounds of online learning algorithms within this game framework. We then show the flexibility of this framework by analyzing the implicit bias in adversarial training, and again obtain significantly improved convergence rates.

cs.LG↗

arXiv2023

Optimal Strategies of Quantum Metrology with a Strict Hierarchy

One of the main quests in quantum metrology is to attain the ultimate precision limit with given resources, where the resources are not only of the number of queries, but more importantly of the allowed strategies. With the same number of queries, the restrictions on the strategies constrain the achievable precision. In this work, we establish a systematic framework to identify the ultimate precision limit of different families of strategies, including the parallel, the sequential, and the indefinite-causal-order strategies, and provide an efficient algorithm that determines an optimal strategy within the family of strategies under consideration. With our framework, we show there exists a strict hierarchy of the precision limits for different families of strategies.

quant-ph↗

arXiv2022

Adaptive Oracle-Efficient Online Learning

The classical algorithms for online learning and decision-making have the benefit of achieving the optimal performance guarantees, but suffer from computational complexity limitations when implemented at scale. More recent sophisticated techniques, which we refer to as oracle-efficient methods, address this problem by dispatching to an offline optimization oracle that can search through an exponentially-large (or even infinite) space of decisions and select that which performed the best on any dataset. But despite the benefits of computational feasibility, oracle-efficient algorithms exhibit one major limitation: while performing well in worst-case settings, they do not adapt well to friendly environments. In this paper we consider two such friendly scenarios, (a) "small-loss" problems and (b) IID data. We provide a new framework for designing follow-the-perturbed-leader algorithms that are oracle-efficient and adapt well to the small-loss environment, under a particular condition which we call approximability (which is spiritually related to sufficient conditions provided by Dudík et al., [2020]). We identify a series of real-world settings, including online auctions and transductive online classification, for which approximability holds. We also extend the algorithm to an IID data setting and establish a "best-of-both-worlds" bound in the oracle-efficient setting.

cs.LG↗

arXiv2020

Sequential Batch Learning in Finite-Action Linear Contextual Bandits

We study the sequential batch learning problem in linear contextual bandits with finite action sets, where the decision maker is constrained to split incoming individuals into (at most) a fixed number of batches and can only observe outcomes for the individuals within a batch at the batch's end. Compared with both standard online contextual-bandit learning and offline policy learning in contextual bandits, this sequential batch learning problem provides a finer-grained formulation of many personalized sequential decision making problems in practical applications, including medical treatment in clinical trials, product recommendation in e-commerce and adaptive experiment design in crowdsourcing. We study two settings of the problem: one where the contexts are arbitrarily generated and the other where the context vectors are mutually independent across actions and time and follow a common Gaussian distribution. In each setting, we establish a regret lower bound and provide an algorithm, whose regret upper bound nearly matches the lower bound. As an important insight revealed therefrom, in the former setting, we show that the number of batches required to achieve the fully online performance is polynomial in the time horizon, while for the latter setting, a pure-exploitation algorithm with a judicious batch partition scheme achieves the fully online performance even when the number of batches is less than logarithmic in the time horizon. In the stochastic context setting, we additionally provide tight margin-based (i.e. instance-dependent) upper and lower regret bounds that delineate performance in terms of how difficult the problem instance is. Together, our results provide a near-complete characterization of sequential decision making in linear contextual bandits when batch constraints are present.

cs.LG↗

arXiv2017

Supervised Hashing based on Energy Minimization

Recently, supervised hashing methods have attracted much attention since they can optimize retrieval speed and storage cost while preserving semantic information. Because hashing codes learning is NP-hard, many methods resort to some form of relaxation technique. But the performance of these methods can easily deteriorate due to the relaxation. Luckily, many supervised hashing formulations can be viewed as energy functions, hence solving hashing codes is equivalent to learning marginals in the corresponding conditional random field (CRF). By minimizing the KL divergence between a fully factorized distribution and the Gibbs distribution of this CRF, a set of consistency equations can be obtained, but updating them in parallel may not yield a local optimum since the variational lower bound is not guaranteed to increase. In this paper, we use a linear approximation of the sigmoid function to convert these consistency equations to linear systems, which have a closed-form solution. By applying this novel technique to two classical hashing formulations KSH and SPLH, we obtain two new methods called EM (energy minimizing based)-KSH and EM-SPLH. Experimental results on three datasets show the superiority of our methods.

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