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Xiaojin Zhang

Publications and source records attributed to Xiaojin Zhang.

At least 73 records · Page 4Linked to original sources
arXiv2023

Variance-Dependent Best Arm Identification

We study the problem of identifying the best arm in a stochastic multi-armed bandit game. Given a set of $n$ arms indexed from $1$ to $n$, each arm $i$ is associated with an unknown reward distribution supported on $[0,1]$ with mean $θ_i$ and variance $σ_i^2$. Assume $θ_1 > θ_2 \geq \cdots \geqθ_n$. We propose an adaptive algorithm which explores the gaps and variances of the rewards of the arms and makes future decisions based on the gathered information using a novel approach called \textit{grouped median elimination}. The proposed algorithm guarantees to output the best arm with probability $(1-δ)$ and uses at most $O \left(\sum_{i = 1}^n \left(\frac{σ_i^2}{Δ_i^2} + \frac{1}{Δ_i}\right)(\ln δ^{-1} + \ln \ln Δ_i^{-1})\right)$ samples, where $Δ_i$ ($i \geq 2$) denotes the reward gap between arm $i$ and the best arm and we define $Δ_1 = Δ_2$. This achieves a significant advantage over the variance-independent algorithms in some favorable scenarios and is the first result that removes the extra $\ln n$ factor on the best arm compared with the state-of-the-art. We further show that $Ω\left( \sum_{i = 1}^n \left( \frac{σ_i^2}{Δ_i^2} + \frac{1}{Δ_i} \right) \ln δ^{-1} \right)$ samples are necessary for an algorithm to achieve the same goal, thereby illustrating that our algorithm is optimal up to doubly logarithmic terms.

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arXiv2023

Probably Approximately Correct Federated Learning

Federated learning (FL) is a new distributed learning paradigm, with privacy, utility, and efficiency as its primary pillars. Existing research indicates that it is unlikely to simultaneously attain infinitesimal privacy leakage, utility loss, and efficiency. Therefore, how to find an optimal trade-off solution is the key consideration when designing the FL algorithm. One common way is to cast the trade-off problem as a multi-objective optimization problem, i.e., the goal is to minimize the utility loss and efficiency reduction while constraining the privacy leakage not exceeding a predefined value. However, existing multi-objective optimization frameworks are very time-consuming, and do not guarantee the existence of the Pareto frontier, this motivates us to seek a solution to transform the multi-objective problem into a single-objective problem because it is more efficient and easier to be solved. To this end, we propose FedPAC, a unified framework that leverages PAC learning to quantify multiple objectives in terms of sample complexity, such quantification allows us to constrain the solution space of multiple objectives to a shared dimension, so that it can be solved with the help of a single-objective optimization algorithm. Specifically, we provide the results and detailed analyses of how to quantify the utility loss, privacy leakage, privacy-utility-efficiency trade-off, as well as the cost of the attacker from the PAC learning perspective.

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arXiv2023

A non-vanishing result on the singularity category

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomology of differential graded Leavitt algebras.

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arXiv2023

Trading Off Privacy, Utility and Efficiency in Federated Learning

Federated learning (FL) enables participating parties to collaboratively build a global model with boosted utility without disclosing private data information. Appropriate protection mechanisms have to be adopted to fulfill the opposing requirements in preserving \textit{privacy} and maintaining high model \textit{utility}. In addition, it is a mandate for a federated learning system to achieve high \textit{efficiency} in order to enable large-scale model training and deployment. We propose a unified federated learning framework that reconciles horizontal and vertical federated learning. Based on this framework, we formulate and quantify the trade-offs between privacy leakage, utility loss, and efficiency reduction, which leads us to the No-Free-Lunch (NFL) theorem for the federated learning system. NFL indicates that it is unrealistic to expect an FL algorithm to simultaneously provide excellent privacy, utility, and efficiency in certain scenarios. We then analyze the lower bounds for the privacy leakage, utility loss and efficiency reduction for several widely-adopted protection mechanisms including \textit{Randomization}, \textit{Homomorphic Encryption}, \textit{Secret Sharing} and \textit{Compression}. Our analysis could serve as a guide for selecting protection parameters to meet particular requirements.

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arXiv2022

No Free Lunch Theorem for Security and Utility in Federated Learning

In a federated learning scenario where multiple parties jointly learn a model from their respective data, there exist two conflicting goals for the choice of appropriate algorithms. On one hand, private and sensitive training data must be kept secure as much as possible in the presence of \textit{semi-honest} partners, while on the other hand, a certain amount of information has to be exchanged among different parties for the sake of learning utility. Such a challenge calls for the privacy-preserving federated learning solution, which maximizes the utility of the learned model and maintains a provable privacy guarantee of participating parties' private data. This article illustrates a general framework that a) formulates the trade-off between privacy loss and utility loss from a unified information-theoretic point of view, and b) delineates quantitative bounds of privacy-utility trade-off when different protection mechanisms including Randomization, Sparsity, and Homomorphic Encryption are used. It was shown that in general \textit{there is no free lunch for the privacy-utility trade-off} and one has to trade the preserving of privacy with a certain degree of degraded utility. The quantitative analysis illustrated in this article may serve as the guidance for the design of practical federated learning algorithms.

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arXiv2022

A Bijection theorem for Gorenstein projective τ-tilting modules

We introduce the notions of Gorenstein projective $τ$-rigid modules, Gorenstein projective support $τ$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support $τ$-tilting modules. More precisely, for an algebra $Λ$, We prove that there is a bijection between the set of Gorenstein projective support $τ$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$τ$-tilting finite algebras and show that $Λ$ is CM-$τ$-tilting finite if and only if $Λ^{\rm {op}}$ is CM-$τ$-tilting finite. Moreover, we show that the Bongartz completion of a Gorenstein projective $τ$-rigid module need not be a Gorenstein projective $τ$-tilting module.

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arXiv2022

On the number of tilting modules over a class of Auslander algebras

Let $Λ$ be a radical square zero algebra of a Dynkin quiver and let $Γ$ be the Auslander algebra of $Λ$. Then the number of tilting right $Γ$-modules is $2^{m-1}$ if $Λ$ is of $A_{m}$ type for $m\geq 1$. Otherwise, the number of tilting right $Γ$-modules is $2^{m-3}\times14$ if $Λ$ is either of $D_{m}$ type for $m\geq 4$ or of $E_{m}$ type for $m=6,7,8$.

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arXiv2022

Adaptive Double-Exploration Tradeoff for Outlier Detection

We study a variant of the thresholding bandit problem (TBP) in the context of outlier detection, where the objective is to identify the outliers whose rewards are above a threshold. Distinct from the traditional TBP, the threshold is defined as a function of the rewards of all the arms, which is motivated by the criterion for identifying outliers. The learner needs to explore the rewards of the arms as well as the threshold. We refer to this problem as "double exploration for outlier detection". We construct an adaptively updated confidence interval for the threshold, based on the estimated value of the threshold in the previous rounds. Furthermore, by automatically trading off exploring the individual arms and exploring the outlier threshold, we provide an efficient algorithm in terms of the sample complexity. Experimental results on both synthetic datasets and real-world datasets demonstrate the efficiency of our algorithm.

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arXiv2022

Contextual Combinatorial Conservative Bandits

The problem of multi-armed bandits (MAB) asks to make sequential decisions while balancing between exploitation and exploration, and have been successfully applied to a wide range of practical scenarios. Various algorithms have been designed to achieve a high reward in a long term. However, its short-term performance might be rather low, which is injurious in risk sensitive applications. Building on previous work of conservative bandits, we bring up a framework of contextual combinatorial conservative bandits. An algorithm is presented and a regret bound of $\tilde O(d^2+d\sqrt{T})$ is proven, where $d$ is the dimension of the feature vectors, and $T$ is the total number of time steps. We further provide an algorithm as well as regret analysis for the case when the conservative reward is unknown. Experiments are conducted, and the results validate the effectiveness of our algorithm.

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arXiv2022

Tau-tilting modules over trivial extenstions

We study (support) $τ$-tilting modules over the trivial extensions of finite dimensional algebras. More precisely, we construct two classes of (support)$τ$-tilting modules in terms of the adjoint functors which extend and generalize the results on (support) $τ$-tilting modules over triangular matrix rings given by Gao-Huang.

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arXiv2022

A construction of Gorenstein projective tau-tilting modules

We give a construction of Gorenstein projective $τ$-tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective $τ$-tilting modules. Moreover, we show that a finite dimensional algebra $Λ$ over an algebraically closed field is $CM$-$τ$-tilting finite if $T_n(Λ)$ is $CM$-$τ$-tilting finite which gives a partial answer to a question on $CM$-$τ$-tilting finite algebras posed by Xie and Zhang.

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arXiv2021

Improved Algorithm for Tolerant Junta Testing

In this paper, we consider the problem of tolerant junta testing for boolean function. Compared with the prior work by Blais et al., we provide improved results in terms of both the parameter gap and query complexity. Compared with the prior work by De et al., our work did not close the parameter gap, but we removed the $2^k$ term of the query complexity.

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arXiv2021

Self-orthogonal tau-tilting modules and tilting modules

Let $Λ$ be an artin algebra and $T$ a $τ$-tilting $Λ$-module. We prove that $T$ is a tilting module if and only if ${\rm Ext}_Λ^{i}(T,\Fac T)=0$ for all $i\geq 1$, where $\Fac T$ is the full subcategory consisting of modules generated by $T$. Consequently, a $τ$-tilting module $T$ of finite projective dimension is a tilting module if and only if ${\rm Ext}_Λ^{i}(T, T)=0$ for all $i\geq 1$. Moreover, we also give an example to show that a support $τ$-tilting but not $τ$-tilting module $M$ of finite projective dimension satisfying ${\rm Ext}_Λ^{i}(M, M)=0$ for all $i\geq1$ need not be a partial tilting module.

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arXiv2021

Achieving Near Instance-Optimality and Minimax-Optimality in Stochastic and Adversarial Linear Bandits Simultaneously

In this work, we develop linear bandit algorithms that automatically adapt to different environments. By plugging a novel loss estimator into the optimization problem that characterizes the instance-optimal strategy, our first algorithm not only achieves nearly instance-optimal regret in stochastic environments, but also works in corrupted environments with additional regret being the amount of corruption, while the state-of-the-art (Li et al., 2019) achieves neither instance-optimality nor the optimal dependence on the corruption amount. Moreover, by equipping this algorithm with an adversarial component and carefully-designed testings, our second algorithm additionally enjoys minimax-optimal regret in completely adversarial environments, which is the first of this kind to our knowledge. Finally, all our guarantees hold with high probability, while existing instance-optimal guarantees only hold in expectation.

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arXiv2020

Tilting modules over Auslander-Gorenstein Algebras

For a finite dimensional algebra $Λ$ and a non-negative integer $n$, we characterize when the set $\tilt_nΛ$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalize results of Happel-Unger. Moreover, for an $n$-Gorenstein algebra $Λ$ with $n\geq 1$, we construct a minimal element in $\tilt_{n}Λ$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $Λ$ and its factor algebra $Γ=Λ/(e)$, we show that there is a bijection between $\tilt_1Λ$ and the set $\sttiltΓ$ of isomorphism classes of basic support $τ$-tilting $Γ$-modules, where $e$ is an idempotent such that $eΛ$ is the additive generator of projective-injective $Λ$-modules.

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arXiv2020

Classifying $τ$-tilting modules over the Auslander algebra of $K[x]/(x^n)$

We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $Γ$ of Dynkin type $A_n$ as a factor algebra of $Λ$, we show that the tensor functor $-\otimes_ΛΓ$ induces a bijection between $\sttiltΛ\to\sttiltΓ$. This recover Mizuno's bijection $\mathfrak{S}_{n+1}\to\sttiltΓ$ for type $A_n$.

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arXiv2019

Automatic Ensemble Learning for Online Influence Maximization

We consider the problem of selecting a seed set to maximize the expected number of influenced nodes in the social network, referred to as the \textit{influence maximization} (IM) problem. We assume that the topology of the social network is prescribed while the influence probabilities among edges are unknown. In order to learn the influence probabilities and simultaneously maximize the influence spread, we consider the tradeoff between exploiting the current estimation of the influence probabilities to ensure certain influence spread and exploring more nodes to learn better about the influence probabilities. The exploitation-exploration trade-off is the core issue in the multi-armed bandit (MAB) problem. If we regard the influence spread as the reward, then the IM problem could be reduced to the combinatorial multi-armed bandits. At each round, the learner selects a limited number of seed nodes in the social network, then the influence spreads over the network according to the real influence probabilities. The learner could observe the activation status of the edge if and only if its start node is influenced, which is referred to as the edge-level semi-bandit feedback. Two classical bandit algorithms including Thompson Sampling and Epsilon Greedy are used to solve this combinatorial problem. To ensure the robustness of these two algorithms, we use an automatic ensemble learning strategy, which combines the exploration strategy with exploitation strategy. The ensemble algorithm is self-adaptive regarding that the probability of each algorithm could be adjusted based on the historical performance of the algorithm. Experimental evaluation illustrates the effectiveness of the automatically adjusted hybridization of exploration algorithm with exploitation algorithm.

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