Han Shao

Nika Haghtalab, Ariel D. Procaccia, Han Shao et al.

Recent leaderboard-based evaluations of large language models aggregate user feedback by fitting a Bradley--Terry model to pairwise comparisons, producing a single global ranking based on a latent quality score. While appealing for its simplicity, this approach is incompatible with heterogeneous preferences: when LLMs are used across diverse tasks and use cases, users who favor fundamentally different model behaviors can be systematically misrepresented when collapsed into a single quality score. To address this issue, we study \emph{pluralistic leaderboards} that aim to remain \emph{stable} with respect to heterogeneous user populations. Drawing on ideas from social choice theory, we adapt the notion of \emph{local stability}, which requires that no model outside the top-$k$ positions is collectively preferred to the top-$k$ set by more than $O(1/k)$ fraction of users. Building on techniques from the social choice literature, we design an alternative leaderboard mechanism that satisfies local stability while eliciting only $\widetilde{O}(k)$ pairwise comparisons per user, where $k$ is the size of the prefix for which stability is guaranteed. Using data from LMArena, we show that standard Bradley--Terry aggregation can violate local stability in practice, whereas our method provides substantially stronger stability guarantees.

Han Shao, Lee Cohen, Avrim Blum et al.

In classic reinforcement learning (RL) and decision making problems, policies are evaluated with respect to a scalar reward function, and all optimal policies are the same with regards to their expected return. However, many real-world problems involve balancing multiple, sometimes conflicting, objectives whose relative priority will vary according to the preferences of each user. Consequently, a policy that is optimal for one user might be sub-optimal for another. In this work, we propose a multi-objective decision making framework that accommodates different user preferences over objectives, where preferences are learned via policy comparisons. Our model consists of a Markov decision process with a vector-valued reward function, with each user having an unknown preference vector that expresses the relative importance of each objective. The goal is to efficiently compute a near-optimal policy for a given user. We consider two user feedback models. We first address the case where a user is provided with two policies and returns their preferred policy as feedback. We then move to a different user feedback model, where a user is instead provided with two small weighted sets of representative trajectories and selects the preferred one. In both cases, we suggest an algorithm that finds a nearly optimal policy for the user using a small number of comparison queries.

Minbiao Han, Kumar Kshitij Patel, Han Shao et al.

Federated learning is a machine learning protocol that enables a large population of agents to collaborate over multiple rounds to produce a single consensus model. There are several federated learning applications where agents may choose to defect permanently$-$essentially withdrawing from the collaboration$-$if they are content with their instantaneous model in that round. This work demonstrates the detrimental impact of such defections on the final model's robustness and ability to generalize. We also show that current federated optimization algorithms fail to disincentivize these harmful defections. We introduce a novel optimization algorithm with theoretical guarantees to prevent defections while ensuring asymptotic convergence to an effective solution for all participating agents. We also provide numerical experiments to corroborate our findings and demonstrate the effectiveness of our algorithm.

Shashaank Aiyer, Yishay Mansour, Shay Moran et al.

Statistical evaluation aims to estimate the generalization performance of a model using held-out i.i.d.\ test data sampled from the ground-truth distribution. In supervised learning settings such as classification, performance metrics such as error rate are well-defined, and test error reliably approximates population error given sufficiently large datasets. In contrast, evaluation is more challenging for generative models due to their open-ended nature: it is unclear which metrics are appropriate and whether such metrics can be reliably evaluated from finite samples. In this work, we introduce a theoretical framework for evaluating generative models and establish evaluability results for commonly used metrics. We study two categories of metrics: test-based metrics, including integral probability metrics (IPMs), and Rényi divergences. We show that IPMs with respect to any bounded test class can be evaluated from finite samples up to multiplicative and additive approximation errors. Moreover, when the test class has finite fat-shattering dimension, IPMs can be evaluated with arbitrary precision. In contrast, Rényi and KL divergences are not evaluable from finite samples, as their values can be critically determined by rare events. We also analyze the potential and limitations of perplexity as an evaluation method.

Shashaank Aiyer, Yishay Mansour, Shay Moran et al.

We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability. Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning: for every bounded real-valued class and every $γ>0$, uniform convergence at scale $γ$, agnostic learnability at scale $γ/2$, and finiteness of the fat-shattering dimension at every scale $γ'>γ$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press 1999) on the precise scales governing learnability, refuting a conjecture attributed there to Phil Long that a multiplicative 2-factor gap is unavoidable, and improves the upper bounds of Bartlett and Long (JCSS 1998), which incur such a loss. The key technical ingredient is a direct bound on empirical $\ell_\infty$ covering numbers, avoiding the standard detour through packing numbers. As a consequence, we obtain sharp asymptotic metric-entropy bounds in terms of the fat-shattering scale $γ$: an $O(\log^2 n)$ bound holds already at scale $γ/2$, while an $O(\log n)$ bound holds at scale $2γ$. We further show that the $O(\log^2 n)$ bound is sometimes tight. These results resolve open questions by Alon et al. (JACM 1997) and Rudelson and Vershynin (Ann. of Math. 2006). As an application, we establish a sharp dichotomy for bounded integral probability metrics: every such IPM is either estimable or cannot be weakly evaluated within any multiplicative factor $c<3$, while $3$-weak evaluability always holds, resolving an open question from Aiyer et al. (ICML 2026). We also highlight several open questions on quantitative sample complexity and evaluability.

Lee Cohen, Han Shao

In collaborative active learning, where multiple agents try to learn labels from a common hypothesis, we introduce an innovative framework for incentivized collaboration. Here, rational agents aim to obtain labels for their data sets while keeping label complexity at a minimum. We focus on designing (strict) individually rational (IR) collaboration protocols, ensuring that agents cannot reduce their expected label complexity by acting individually. We first show that given any optimal active learning algorithm, the collaboration protocol that runs the algorithm as is over the entire data is already IR. However, computing the optimal algorithm is NP-hard. We therefore provide collaboration protocols that achieve (strict) IR and are comparable with the best known tractable approximation algorithm in terms of label complexity.

Lee Cohen, Yishay Mansour, Shay Moran et al.

We consider the problem of learning an unknown subset $N_\text{target}$ of a domain in an online setting. In each round $t$, the learner predicts a set of items ${N}_t$ and receives one of two types of feedback, each with equal probability: precision feedback, in which a randomly chosen item from the predicted set $N_t$ is revealed and the learner is told whether it belongs to $N_\text{target}$ (incurring a reward if it does), or recall feedback, in which a randomly chosen item from the target set $N_\text{target}$ is revealed and the learner is told whether it belongs to $N_t$ (incurring a reward if it does). The goal is to maximize the cumulative reward over time. This simple online set learning problem abstracts a variety of learning scenarios with precision- and recall-type feedback. We show that a hypothesis class (a family of subsets of the domain) is learnable in this setting if and only if it has finite Vapnik-Chervonenkis (VC) dimension, mirroring the classical PAC characterization. However, the resulting algorithmic structure is markedly more intricate: in contrast to standard Probably Approximately Correct (PAC) learning -- where the algorithmic landscape is governed by the simple principle of Empirical Risk Minimization (ERM) -- our partial feedback model can invalidate ERM and even all proper learning rules. We develop algorithms to address the dependencies induced by the feedback, obtaining regret guarantees in both the realizable and agnostic settings. Our results provide a qualitative characterization of learnability in this model, addressing its most basic question, while pointing to a range of natural and intriguing open questions, including the determination of optimal regret rates.

Elbert Du, Cynthia Dwork, Lunjia Hu et al.

Individuals with similar qualifications and skills may vary in their demeanor, or outward manner: some tend toward self-promotion while others are modest to the point of omitting crucial information. Comparing the self-descriptions of equally qualified job-seekers with different self-presentation styles is therefore problematic. We build an interactive AI for skill elicitation that provides accurate determination of skills while simultaneously allowing individuals to speak in their own voice. Such a system can be deployed, for example, when a new user joins a professional networking platform, or when matching employees to needs during a company reorganization. To obtain sufficient training data, we train an LLM to act as synthetic humans. Elicitation mitigates endogenous bias arising from individuals' own self-reports. To address systematic model bias we enforce a mathematically rigorous notion of equitability ensuring that the covariance between self-presentation manner and skill evaluation error is small.

Chase Hutton, Adam Melrod, Han Shao

Online strategic classification studies settings in which agents strategically modify their features to obtain favorable predictions. For example, given a classifier that determines loan approval based on credit scores, applicants may open or close credit cards and bank accounts to obtain a positive prediction. The learning goal is to achieve low mistake or regret bounds despite such strategic behavior. While randomized algorithms have the potential to offer advantages to the learner in strategic settings, they have been largely underexplored. In the realizable setting, no lower bound is known for randomized algorithms, and existing lower bound constructions for deterministic learners can be circumvented by randomization. In the agnostic setting, the best known regret upper bound is $O(T^{3/4}\log^{1/4}T|\mathcal H|)$, which is far from the standard online learning rate of $O(\sqrt{T\log|\mathcal H|})$. In this work, we provide refined bounds for online strategic classification in both settings. In the realizable setting, we extend, for $T > \mathrm{Ldim}(\mathcal{H}) Δ^2$, the existing lower bound $Ω(\mathrm{Ldim}(\mathcal{H}) Δ)$ for deterministic learners to all learners. This yields the first lower bound that applies to randomized learners. We also provide the first randomized learner that improves the known (deterministic) upper bound of $O(\mathrm{Ldim}(\mathcal H) \cdot Δ\log Δ)$. In the agnostic setting, we give a proper learner using convex optimization techniques to improve the regret upper bound to $O(\sqrt{T \log |\mathcal{H}|} + |\mathcal{H}| \log(T|\mathcal{H}|))$. We show a matching lower bound up to logarithmic factors for all proper learning rules, demonstrating the optimality of our learner among proper learners. As such, improper learning is necessary to further improve regret guarantees.

Han Shao

Machine learning has witnessed remarkable breakthroughs in recent years. As machine learning permeates various aspects of daily life, individuals and organizations increasingly interact with these systems, exhibiting a wide range of social and adversarial behaviors. These behaviors may have a notable impact on the behavior and performance of machine learning systems. Specifically, during these interactions, data may be generated by strategic individuals, collected by self-interested data collectors, possibly poisoned by adversarial attackers, and used to create predictors, models, and policies satisfying multiple objectives. As a result, the machine learning systems' outputs might degrade, such as the susceptibility of deep neural networks to adversarial examples (Shafahi et al., 2018; Szegedy et al., 2013) and the diminished performance of classic algorithms in the presence of strategic individuals (Ahmadi et al., 2021). Addressing these challenges is imperative for the success of machine learning in societal settings.

Lee Cohen, Yishay Mansour, Shay Moran et al.

In contrast with standard classification tasks, strategic classification involves agents strategically modifying their features in an effort to receive favorable predictions. For instance, given a classifier determining loan approval based on credit scores, applicants may open or close their credit cards to fool the classifier. The learning goal is to find a classifier robust against strategic manipulations. Various settings, based on what and when information is known, have been explored in strategic classification. In this work, we focus on addressing a fundamental question: the learnability gaps between strategic classification and standard learning. We essentially show that any learnable class is also strategically learnable: we first consider a fully informative setting, where the manipulation structure (which is modeled by a manipulation graph $G^\star$) is known and during training time the learner has access to both the pre-manipulation data and post-manipulation data. We provide nearly tight sample complexity and regret bounds, offering significant improvements over prior results. Then, we relax the fully informative setting by introducing two natural types of uncertainty. First, following Ahmadi et al. (2023), we consider the setting in which the learner only has access to the post-manipulation data. We improve the results of Ahmadi et al. (2023) and close the gap between mistake upper bound and lower bound raised by them. Our second relaxation of the fully informative setting introduces uncertainty to the manipulation structure. That is, we assume that the manipulation graph is unknown but belongs to a known class of graphs. We provide nearly tight bounds on the learning complexity in various unknown manipulation graph settings. Notably, our algorithm in this setting is of independent interest and can be applied to other problems such as multi-label learning.

Omar Montasser, Han Shao, Emmanuel Abbe

Learning with identical train and test distributions has been extensively investigated both practically and theoretically. Much remains to be understood, however, in statistical learning under distribution shifts. This paper focuses on a distribution shift setting where train and test distributions can be related by classes of (data) transformation maps. We initiate a theoretical study for this framework, investigating learning scenarios where the target class of transformations is either known or unknown. We establish learning rules and algorithmic reductions to Empirical Risk Minimization (ERM), accompanied with learning guarantees. We obtain upper bounds on the sample complexity in terms of the VC dimension of the class composing predictors with transformations, which we show in many cases is not much larger than the VC dimension of the class of predictors. We highlight that the learning rules we derive offer a game-theoretic viewpoint on distribution shift: a learner searching for predictors and an adversary searching for transformation maps to respectively minimize and maximize the worst-case loss.

Lee Cohen, Yishay Mansour, Shay Moran et al.

Precision and Recall are fundamental metrics in machine learning tasks where both accurate predictions and comprehensive coverage are essential, such as in multi-label learning, language generation, medical studies, and recommender systems. A key challenge in these settings is the prevalence of one-sided feedback, where only positive examples are observed during training--e.g., in multi-label tasks like tagging people in Facebook photos, we may observe only a few tagged individuals, without knowing who else appears in the image. To address learning under such partial feedback, we introduce a Probably Approximately Correct (PAC) framework in which hypotheses are set functions that map each input to a set of labels, extending beyond single-label predictions and generalizing classical binary, multi-class, and multi-label models. Our results reveal sharp statistical and algorithmic separations from standard settings: classical methods such as Empirical Risk Minimization provably fail, even for simple hypothesis classes. We develop new algorithms that learn from positive data alone, achieving optimal sample complexity in the realizable case, and establishing multiplicative--rather than additive-approximation guarantees in the agnostic case, where achieving additive regret is impossible.

Han Shao, Shuo Xie, Kunhe Yang

Strategic classification addresses a learning problem where a decision-maker implements a classifier over agents who may manipulate their features in order to receive favorable predictions. In the standard model of online strategic classification, in each round, the decision-maker implements and publicly reveals a classifier, after which agents perfectly best respond based on this knowledge. However, in practice, whether to disclose the classifier is often debated -- some decision-makers believe that hiding the classifier can prevent misclassification errors caused by manipulation. In this paper, we formally examine how limiting the agents' access to the current classifier affects the decision-maker's performance. Specifically, we consider an extended online strategic classification setting where agents lack direct knowledge about the current classifier and instead manipulate based on a weighted average of historically implemented classifiers. Our main result shows that in this setting, the decision-maker incurs $(1-γ)^{-1}$ or $k_{\text{in}}$ times more mistakes compared to the full-knowledge setting, where $k_{\text{in}}$ is the maximum in-degree of the manipulation graph (representing how many distinct feature vectors can be manipulated to appear as a single one), and $γ$ is the discount factor indicating agents' memory of past classifiers. Our results demonstrate how withholding access to the classifier can backfire and degrade the decision-maker's performance in online strategic classification.

Cynthia Dwork, Lunjia Hu, Han Shao

We study a fundamental question of domain generalization: given a family of domains (i.e., data distributions), how many randomly sampled domains do we need to collect data from in order to learn a model that performs reasonably well on every seen and unseen domain in the family? We model this problem in the PAC framework and introduce a new combinatorial measure, which we call the domain shattering dimension. We show that this dimension characterizes the domain sample complexity. Furthermore, we establish a tight quantitative relationship between the domain shattering dimension and the classic VC dimension, demonstrating that every hypothesis class that is learnable in the standard PAC setting is also learnable in our setting.

Melissa Dutz, Han Shao, Avrim Blum et al.

Strategic litigation involves bringing a legal case to court with the goal of having a broader impact beyond resolving the case itself: for example, creating precedent which will influence future rulings. In this paper, we explore strategic litigation from the perspective of machine learning theory. We consider an abstract model of a common-law legal system where a lower court decides new cases by applying a decision rule learned from a higher court's past rulings. In this model, we explore the power of a strategic litigator, who strategically brings cases to the higher court to influence the learned decision rule, thereby affecting future cases. We explore questions including: What impact can a strategic litigator have? Which cases should a strategic litigator bring to court? Does it ever make sense for a strategic litigator to bring a case when they are sure the court will rule against them?

Han Shao, Avrim Blum, Omar Montasser

We study the fundamental mistake bound and sample complexity in the strategic classification, where agents can strategically manipulate their feature vector up to an extent in order to be predicted as positive. For example, given a classifier determining college admission, student candidates may try to take easier classes to improve their GPA, retake SAT and change schools in an effort to fool the classifier. Ball manipulations are a widely studied class of manipulations in the literature, where agents can modify their feature vector within a bounded radius ball. Unlike most prior work, our work considers manipulations to be personalized, meaning that agents can have different levels of manipulation abilities (e.g., varying radii for ball manipulations), and unknown to the learner. We formalize the learning problem in an interaction model where the learner first deploys a classifier and the agent manipulates the feature vector within their manipulation set to game the deployed classifier. We investigate various scenarios in terms of the information available to the learner during the interaction, such as observing the original feature vector before or after deployment, observing the manipulated feature vector, or not seeing either the original or the manipulated feature vector. We begin by providing online mistake bounds and PAC sample complexity in these scenarios for ball manipulations. We also explore non-ball manipulations and show that, even in the simplest scenario where both the original and the manipulated feature vectors are revealed, the mistake bounds and sample complexity are lower bounded by $Ω(|H|)$ when the target function belongs to a known class $H$.

Han Shao, Omar Montasser, Avrim Blum

Transformation invariances are present in many real-world problems. For example, image classification is usually invariant to rotation and color transformation: a rotated car in a different color is still identified as a car. Data augmentation, which adds the transformed data into the training set and trains a model on the augmented data, is one commonly used technique to build these invariances into the learning process. However, it is unclear how data augmentation performs theoretically and what the optimal algorithm is in presence of transformation invariances. In this paper, we study PAC learnability under transformation invariances in three settings according to different levels of realizability: (i) A hypothesis fits the augmented data; (ii) A hypothesis fits only the original data and the transformed data lying in the support of the data distribution; (iii) Agnostic case. One interesting observation is that distinguishing between the original data and the transformed data is necessary to achieve optimal accuracy in setting (ii) and (iii), which implies that any algorithm not differentiating between the original and transformed data (including data augmentation) is not optimal. Furthermore, this type of algorithms can even "harm" the accuracy. In setting (i), although it is unnecessary to distinguish between the two data sets, data augmentation still does not perform optimally. Due to such a difference, we propose two combinatorial measures characterizing the optimal sample complexity in setting (i) and (ii)(iii) and provide the optimal algorithms.

Avrim Blum, Nika Haghtalab, Richard Lanas Phillips et al.

In recent years, federated learning has been embraced as an approach for bringing about collaboration across large populations of learning agents. However, little is known about how collaboration protocols should take agents' incentives into account when allocating individual resources for communal learning in order to maintain such collaborations. Inspired by game theoretic notions, this paper introduces a framework for incentive-aware learning and data sharing in federated learning. Our stable and envy-free equilibria capture notions of collaboration in the presence of agents interested in meeting their learning objectives while keeping their own sample collection burden low. For example, in an envy-free equilibrium, no agent would wish to swap their sampling burden with any other agent and in a stable equilibrium, no agent would wish to unilaterally reduce their sampling burden. In addition to formalizing this framework, our contributions include characterizing the structural properties of such equilibria, proving when they exist, and showing how they can be computed. Furthermore, we compare the sample complexity of incentive-aware collaboration with that of optimal collaboration when one ignores agents' incentives.

Avrim Blum, Steve Hanneke, Jian Qian et al.

We study the problem of robust learning under clean-label data-poisoning attacks, where the attacker injects (an arbitrary set of) correctly-labeled examples to the training set to fool the algorithm into making mistakes on specific test instances at test time. The learning goal is to minimize the attackable rate (the probability mass of attackable test instances), which is more difficult than optimal PAC learning. As we show, any robust algorithm with diminishing attackable rate can achieve the optimal dependence on $ε$ in its PAC sample complexity, i.e., $O(1/ε)$. On the other hand, the attackable rate might be large even for some optimal PAC learners, e.g., SVM for linear classifiers. Furthermore, we show that the class of linear hypotheses is not robustly learnable when the data distribution has zero margin and is robustly learnable in the case of positive margin but requires sample complexity exponential in the dimension. For a general hypothesis class with bounded VC dimension, if the attacker is limited to add at most $t>0$ poison examples, the optimal robust learning sample complexity grows almost linearly with $t$.

Avrim Blum, Han Shao

We study the problem of online learning with primary and secondary losses. For example, a recruiter making decisions of which job applicants to hire might weigh false positives and false negatives equally (the primary loss) but the applicants might weigh false negatives much higher (the secondary loss). We consider the following question: Can we combine "expert advice" to achieve low regret with respect to the primary loss, while at the same time performing {\em not much worse than the worst expert} with respect to the secondary loss? Unfortunately, we show that this goal is unachievable without any bounded variance assumption on the secondary loss. More generally, we consider the goal of minimizing the regret with respect to the primary loss and bounding the secondary loss by a linear threshold. On the positive side, we show that running any switching-limited algorithm can achieve this goal if all experts satisfy the assumption that the secondary loss does not exceed the linear threshold by $o(T)$ for any time interval. If not all experts satisfy this assumption, our algorithms can achieve this goal given access to some external oracles which determine when to deactivate and reactivate experts.

Xuedong Shang, Han Shao, Jian Qian

Multi-armed bandits are widely applied in scenarios like recommender systems, for which the goal is to maximize the click rate. However, more factors should be considered, e.g., user stickiness, user growth rate, user experience assessment, etc. In this paper, we model this situation as a problem of $K$-armed bandit with multiple losses. We define relative loss vector of an arm where the $i$-th entry compares the arm and the optimal arm with respect to the $i$-th loss. We study two goals: (a) finding the arm with the minimum $\ell^\infty$-norm of relative losses with a given confidence level (which refers to fixed-confidence best-arm identification); (b) minimizing the $\ell^\infty$-norm of cumulative relative losses (which refers to regret minimization). For goal (a), we derive a problem-dependent sample complexity lower bound and discuss how to achieve matching algorithms. For goal (b), we provide a regret lower bound of $Ω(T^{2/3})$ and provide a matching algorithm.

Rémy Degenne, Han Shao, Wouter M. Koolen

We study reward maximisation in a wide class of structured stochastic multi-armed bandit problems, where the mean rewards of arms satisfy some given structural constraints, e.g. linear, unimodal, sparse, etc. Our aim is to develop methods that are flexible (in that they easily adapt to different structures), powerful (in that they perform well empirically and/or provably match instance-dependent lower bounds) and efficient in that the per-round computational burden is small. We develop asymptotically optimal algorithms from instance-dependent lower-bounds using iterative saddle-point solvers. Our approach generalises recent iterative methods for pure exploration to reward maximisation, where a major challenge arises from the estimation of the sub-optimality gaps and their reciprocals. Still we manage to achieve all the above desiderata. Notably, our technique avoids the computational cost of the full-blown saddle point oracle employed by previous work, while at the same time enabling finite-time regret bounds. Our experiments reveal that our method successfully leverages the structural assumptions, while its regret is at worst comparable to that of vanilla UCB.

Han Shao, Tassilo Kugelstadt, Torsten Hädrich et al.

Iterative solvers are widely used to accurately simulate physical systems. These solvers require initial guesses to generate a sequence of improving approximate solutions. In this contribution, we introduce a novel method to accelerate iterative solvers for physical systems with graph networks (GNs) by predicting the initial guesses to reduce the number of iterations. Unlike existing methods that aim to learn physical systems in an end-to-end manner, our approach guarantees long-term stability and therefore leads to more accurate solutions. Furthermore, our method improves the run time performance of traditional iterative solvers. To explore our method we make use of position-based dynamics (PBD) as a common solver for physical systems and evaluate it by simulating the dynamics of elastic rods. Our approach is able to generalize across different initial conditions, discretizations, and realistic material properties. Finally, we demonstrate that our method also performs well when taking discontinuous effects into account such as collisions between individual rods. Finally, to illustrate the scalability of our approach, we simulate complex 3D tree models composed of over a thousand individual branch segments swaying in wind fields. A video showing dynamic results of our graph learning assisted simulations of elastic rods can be found on the project website available at http://computationalsciences.org/publications/shao-2021-physical-systems-graph-learning.html .

Han Shao, Xiaotian Yu, Irwin King et al.

In linear stochastic bandits, it is commonly assumed that payoffs are with sub-Gaussian noises. In this paper, under a weaker assumption on noises, we study the problem of \underline{lin}ear stochastic {\underline b}andits with h{\underline e}avy-{\underline t}ailed payoffs (LinBET), where the distributions have finite moments of order $1+ε$, for some $ε\in (0,1]$. We rigorously analyze the regret lower bound of LinBET as $Ω(T^{\frac{1}{1+ε}})$, implying that finite moments of order 2 (i.e., finite variances) yield the bound of $Ω(\sqrt{T})$, with $T$ being the total number of rounds to play bandits. The provided lower bound also indicates that the state-of-the-art algorithms for LinBET are far from optimal. By adopting median of means with a well-designed allocation of decisions and truncation based on historical information, we develop two novel bandit algorithms, where the regret upper bounds match the lower bound up to polylogarithmic factors. To the best of our knowledge, we are the first to solve LinBET optimally in the sense of the polynomial order on $T$. Our proposed algorithms are evaluated based on synthetic datasets, and outperform the state-of-the-art results.