Multi-armed Bandit Model Research Articles

A Strategy for Advertisement Placement based on the Multi-Armed Tiger Problem

The purpose of this study is to investigate the effectiveness of using Multi armed bandit model, which contains -greedy, Upper Confidence Bound (UCB), and Thompson sampling algorithms, to optimize online advertisement placement. Through simulating different types of ad placements using different algorithms and comparing them, this paper intends to demonstrate the feasibility of the Multi-Armed Robber model for the ad placement problem. The results show that the multi-armed bandit model can improve the ad click rate compared with the traditional ad placement strategy. Thompson sampling algorithm outperforms the -greedy algorithm and UCB algorithm in this paper's experiments, which can better balance exploration and application and reduce regret. The algorithm provides a more efficient method of allocating ad resources. These findings provide new insights into the field of digital marketing and may have an impact on the development of actual ad placement strategies.

Combinatorial-restless-bandit-based transmitter–receiver online selection of distributed MIMO radar with non-stationary channels

In a distributed multiple-input multiple-output (MIMO) radar for tracking moving targets, optimizing sensible selections of the transmitter–receiver pairs is crucial for maximizing the sum of signal-to-interference-plus-noise ratios (SINRs), as it directly affects the tracking accuracy. In solving the trade-off between exploitation and exploration in non-stationary channels, the optimization problem is modeled by a restless multi-armed bandits model. This paper regards the estimated SINR mean reward as the state of an arm (transceiver channel). The SINR reward of each arm is estimated based on whether it is probed. A closed loop is established between SINR rewards and the dynamic states of targets, which are estimated via the interacting multiple model-unscented Kalman filter. The combinatorial optimized selection of transmitter–receiver pairs at each time is accomplished by using the binary particle swarm optimization with the SINR index fitness function, where the index represents the upper bound on the confidence of the SINR reward. Above all, a multi-group combinatorial-restless-bandit closed-loop (MG-CRB-CL) algorithm is proposed. Simulation results for different scenarios are provided to verify the effectiveness and superior performance of MG-CRB-CL.

A survey of the application and technical improvement of the multi-armed bandit

In recent years, the multi-armed bandit (MAB) model has been widely used and has shown excellent performance. This article provides an overview of the applications and technical improvements of the multi-armed bandit machine problem. First, an overview of the multi-armed bandit problem is presented, including the explanation of a general modeling approach and several existing common algorithms, such as -greedy, ETC, UCB, and Thompson sampling. Then, the real-life applications of the multi-armed bandit model are explored, covering the fields of recommender systems, healthcare, and finance. Then, some improved algorithms and models are summarized by addressing the problems encountered in different application domains, including the multi-armed bandit considering multiple objectives, the mortal multi-armed bandits, the multi-armed bandit considering contextual side information, combinatorial multi-armed bandits. Finally, the characteristics, trends of changes among different algorithms, and applicable scenarios are summarized and discussed.

Multi-Armed Bandit-Based User Network Node Selection.

In the scenario of an integrated space-air-ground emergency communication network, users encounter the challenge of rapidly identifying the optimal network node amidst the uncertainty and stochastic fluctuations of network states. This study introduces a Multi-Armed Bandit (MAB) model and proposes an optimization algorithm leveraging dynamic variance sampling (DVS). The algorithm posits that the prior distribution of each node's network state conforms to a normal distribution, and by constructing the distribution's expected value and variance, it maximizes the utilization of sample data, thereby maintaining an equilibrium between data exploitation and the exploration of the unknown. Theoretical substantiation is provided to illustrate that the Bayesian regret associated with the algorithm exhibits sublinear growth. Empirical simulations corroborate that the algorithm in question outperforms traditional ε-greedy, Upper Confidence Bound (UCB), and Thompson sampling algorithms in terms of higher cumulative rewards, diminished total regret, accelerated convergence rates, and enhanced system throughput.

Optimizing video click-through rates with bandit algorithms

In recent years, videos have increasingly influenced public perception, making video platforms a focal point of digital consumption. One critical challenge for platform operators is identifying videos that resonate most with users, as user ratings directly reflect viewer preferences and experiences. This study explores the use of bandit algorithms to predict and strategize the overall ratings of various anime videos on the Bilibili platform. Bandit algorithms, a subset of the multi-armed bandit model, dynamically adjust selection strategies based on prior feedback to maximize cumulative rewards. Our empirical research assessed multiple gambling algorithms, including the -greedy, Upper Confidence Bound (UCB), Explore-then-Commit (ETC), and Thompson Sampling (TS) algorithms. The findings indicate that the Thompson Sampling algorithm, in particular, achieved the lowest cumulative regret in selecting optimal videos on the Bilibili platform, showcasing its superior performance. This study highlights the potential of bandit algorithms to enhance video selection processes, ensuring that platforms can effectively cater to user preferences and enhance viewer satisfaction.

Selecting workers like expert for crowdsourcing by integration evaluation of individual and collaborative abilities

Team-based worker selection has been extensively studied for Mobile Crowdsourcing (MCS), in which a set of workers are recruited to form a team to complete complex tasks collaboratively. However, existing studies face two typical challenges: 1) how to dynamically evaluate workers’ individual abilities and collaborative contributions to the team; 2) how to select unknown workers to form a team with high quality at low cost. To tackle the above challenges, this paper proposes an Integration of Individual and Collaborative Abilities based Dynamic Worker Selection (IICA-DWS) algorithm to recruit excellent workers as a team in a high-quality and low-cost style. In the IICA-DWS algorithm, each worker’s individual ability and collaborative contribution to the team are evaluated more accurately using the Approximate Shapley Value (ASV). In addition, a high-quality team formation method is established to complete complex tasks at low cost. This involves the selection of both team leaders and team members. In this process, the Multi-Armed Bandit (MAB) model is adopted to dynamically select excellent workers using exploration and exploitation phases. Lastly, the IICA-DWS algorithm is evaluated through theoretical analysis and experimental results. The results show that the IICA-DWS algorithm can improve the data quality of tasks by 47.3% and reduce the cost by 61.7% on average. Moreover, the IICA-DWS algorithm has a high probability of approximating optimal results, which shows the best performance among the comparative algorithms.

In-depth Exploration and Implementation of Multi-Armed Bandit Models Across Diverse Fields

This paper presents an in-depth analysis of the Multi-Armed Bandit (MAB) problem, tracing its evolution from its origins in the gambling domain of the 1940s to its current prominence in machine learning and artificial intelligence. The analysis begins with a historical overview, noting key developments like Herbert Robbins' probabilistic framework and the expansion of the problem into strategic decision-making in the 1970s. The emergence of algorithms like the Upper Confidence Bound (UCB) and Thompson Sampling in the late 20th century is highlighted, demonstrating the MAB problem's transition to practical applications. The integration of MAB algorithms with machine learning, particularly in the era of reinforcement learning, is explored, emphasizing their application in various domains such as online advertising, financial market trading, and clinical trials. The paper discusses the critical role of decision theory and probabilistic models in MAB problems, focusing on the balance between exploration and exploitation strategies. Recent advancements in Contextual Bandits, non-stationary reward distributions, and Multi-agent Bandits are examined, showcasing the ongoing evolution and adaptability of MAB problems.

Multiobjective Lipschitz Bandits under Lexicographic Ordering

This paper studies the multiobjective bandit problem under lexicographic ordering, wherein the learner aims to simultaneously maximize ? objectives hierarchically. The only existing algorithm for this problem considers the multi-armed bandit model, and its regret bound is O((KT)^(2/3)) under a metric called priority-based regret. However, this bound is suboptimal, as the lower bound for single objective multi-armed bandits is Omega(KlogT). Moreover, this bound becomes vacuous when the arm number K is infinite. To address these limitations, we investigate the multiobjective Lipschitz bandit model, which allows for an infinite arm set. Utilizing a newly designed multi-stage decision-making strategy, we develop an improved algorithm that achieves a general regret bound of O(T^((d_z^i+1)/(d_z^i+2))) for the i-th objective, where d_z^i is the zooming dimension for the i-th objective, with i in {1,2,...,m}. This bound matches the lower bound of the single objective Lipschitz bandit problem in terms of T, indicating that our algorithm is almost optimal. Numerical experiments confirm the effectiveness of our algorithm.

New Approach to Equitable Intervention Planning to Improve Engagement and Outcomes in a Digital Health Program: Simulation Study.

Digital health programs provide individualized support to patients with chronic diseases and their effectiveness is measured by the extent to which patients achieve target individual clinical outcomes and the program's ability to sustain patient engagement. However, patient dropout and inequitable intervention delivery strategies, which may unintentionally penalize certain patient subgroups, represent challenges to maximizing effectiveness. Therefore, methodologies that optimize the balance between success factors (achievement of target clinical outcomes and sustained engagement) equitably would be desirable, particularly when there are resource constraints. Our objectives were to propose a model for digital health program resource management that accounts jointly for the interaction between individual clinical outcomes and patient engagement, ensures equitable allocation as well as allows for capacity planning, and conducts extensive simulations using publicly available data on type 2 diabetes, a chronic disease. We propose a restless multiarmed bandit (RMAB) model to plan interventions that jointly optimize long-term engagement and individual clinical outcomes (in this case measured as the achievement of target healthy glucose levels). To mitigate the tendency of RMAB to achieve good aggregate performance by exacerbating disparities between groups, we propose new equitable objectives for RMAB and apply bilevel optimization algorithms to solve them. We formulated a model for the joint evolution of patient engagement and individual clinical outcome trajectory to capture the key dynamics of interest in digital chronic disease management programs. In simulation exercises, our optimized intervention policies lead to up to 10% more patients reaching healthy glucose levels after 12 months, with a 10% reduction in dropout compared to standard-of-care baselines. Further, our new equitable policies reduce the mean absolute difference of engagement and health outcomes across 6 demographic groups by up to 85% compared to the state-of-the-art. Planning digital health interventions with individual clinical outcome objectives and long-term engagement dynamics as considerations can be both feasible and effective. We propose using an RMAB sequential decision-making framework, which may offer additional capabilities in capacity planning as well. The integration of an equitable RMAB algorithm further enhances the potential for reaching equitable solutions. This approach provides program designers with the flexibility to switch between different priorities and balance trade-offs across various objectives according to their preferences.

Investigation of frontier Multi-Armed Bandit algorithms and applications

Since the amount of content online is growing exponentially and people's time is limited, there is an urgent need for a high-performance algorithm that can make effective recommendations. This paper will introduce a recommendation system model, a sequential decision model, which is called Multi-Armed Bandit. The main idea of the Multi-Armed Bandit model is that at the beginning of the algorithm, all the recommended items are set to the same weight. In the subsequent recommendation process, the model explores the distribution of each item while changing the weight of each item according to the average revenue of each item, and selects more items with larger weight. This paper will introduce three cutting-edge Multi-Armed Bandit algorithms, their algorithmic ideas and their respective characteristics. The idea of Explore-Then-Commit (ETC) algorithm is to explore each item a certain number of times, and then select the best item for subsequent recommendation. The idea of the Upper Confidence Bound (UCB) algorithm is to represent the "exploration" and "exploitation" of each item by numerical values and add them to the UCB value, and select the item with the largest UCB value each time. The idea of TS is to first assume the distribution of each item, and then change the parameters of the distribution of each item based on the reward. At the end, this paper will introduce several scenarios where Multi-Armed Bandit algorithms can be used to give the reader an idea of how to use Multi-Armed Bandit algorithms.

EMS: Erasure-Coded Multi-Source Streaming for UHD Videos Within Cloud Native 5G Networks

Ultra-High-Definition (UHD) videos have been getting increasing attention. However, existing video streaming solutions fail to deliver them due to the extremely high bandwidth requirement. The emerging cloud native 5G networks have opened up the possibility of enhancing UHD video quality by leveraging in-network video streaming. Unfortunately, the restricted storage and bandwidth of in-network servers could become the main bottleneck. To this end, we present <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>${\sf EMS}$</tex-math></inline-formula> , a novel UHD video streaming framework, by integrating <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</u> rasure-coded storage with <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</u> ulti-source <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</u> treaming. We respectively introduce a deadline-aware and a latency-sensitive metric to indicate the service quality of video servers and advocate a federated learning paradigm for the adaptive service quality update, including a reinforcement learning based multi-server selection (i.e., user local training) and a global service quality aggregation. To facilitate user local training without sacrificing streaming Quality-of-Experience (QoE), we cast the multi-server selection associated with the restriction on the average number of selected servers per video chunk into two kinds of Multi-Armed Bandit (MAB) models in terms of the proposed service quality metrics. We design lightweight Upper Confidence Bound (UCB) based algorithms with a theoretical performance guarantee. We implement a prototype of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>${\sf EMS}$</tex-math></inline-formula> , and extensive experiments confirm the superiority of the proposed algorithms.

Tracking people across ultra populated indoor spaces by matching unreliable Wi-Fi signals with disconnected video feeds

Tracking in dense indoor environments where several thousands of people move around is an extremely challenging problem. In this paper, we present a system — DenseTrack for tracking people in such environments. DenseTrack leverages data from the sensing modalities that are already present in these environments — Wi-Fi (from enterprise network deployments) and Video (from surveillance cameras). We combine Wi-Fi information with video data to overcome the individual errors induced by these modalities. More precisely, the locations derived from video are used to overcome the localization errors inherent in using Wi-Fi signals where precise Wi-Fi MAC IDs are used to locate the same devices across different levels and locations inside a building. Typically, localization in dense environments is a computationally expensive process when done with just video data; hence hard to scale. DenseTrack combines Wi-Fi and video data to improve the accuracy of tracking people that are represented by video objects from non-overlapping video feeds. DenseTrack is a scalable and device-agnostic solution as it does not require any app installation on user smartphones or modifications to the Wi-Fi system. At the core of DenseTrack, is our algorithm — inCremental Association of Independent Variables under Uncertainty (CAIVU). CAIVU is inspired by the multi-armed bandits model and is designed to handle various complex features of practical real-world environments. CAIVU matches the devices reported by an off-the-shelf Wi-Fi system using connectivity information to specific video blobs obtained through a computationally efficient analysis of video data. By exploiting data from heterogeneous sources, DenseTrack offers an effective real-time solution for individual tracking in heavily populated indoor environments. We emphasize that no other previous system targeted nor was validated in such dense indoor environments. We tested DenseTrack extensively using both simulated data, as well as two real-world validations using data from an extremely dense convention center and a moderately dense university environment. Our simulation results show that DenseTrack achieves an average video-to-Wi-Fi matching accuracy of up to 90% in dense environments with a matching latency of 60 s on the simulator. When tested in a real-world extremely dense environment with over 500,000 people moving between different non-overlapping camera feeds, DenseTrack achieved an average match accuracy of 83% to within a 2-people distance with an average latency of 48 s.

Truthful User Recruitment for Cooperative Crowdsensing Task: A Combinatorial Multi-Armed Bandit Approach

Mobile Crowdsensing (MCS) is a promising paradigm that recruits users to cooperatively perform a sensing task. When recruiting users, existing works mainly focus on users objective abilities, e.g., the probability or frequency of covering the task locations. However, we argue that the task completion effect depends not only on the user's objective ability, but also on their subjective collaboration likelihood with each other. Even though we can find a well-behaved group of users in the single-round scenario, while in the multi-round scenario without enough prior knowledge, we still face the problem of exploiting users or exploring users. Additionally, we consider that users have different costs and may report false costs to gain more benefits. To address these problems, we first convert the single-round user recruitment problem into the min-cut problem and propose a graph theory based algorithm to find the approximate solution. Then, in the multi-round scenario, we propose a multi-round User Recruitment strategy under the budget constraint based on the combinatorial Multi-armed Bandit model (URMB), which achieves a tight regret bound. Next, we propose a graph-based payment strategy to ensure the truthfulness and individual rationality. Finally, experiments based on real-world datasets show that URMB always outperforms the state-of-the-art strategies.

Decentralized Stochastic Multi-Player Multi-Armed Walking Bandits

Multi-player multi-armed bandit is an increasingly relevant decision-making problem, motivated by applications to cognitive radio systems. Most research for this problem focuses exclusively on the settings that players have full access to all arms and receive no reward when pulling the same arm. Hence all players solve the same bandit problem with the goal of maximizing their cumulative reward. However, these settings neglect several important factors in many real-world applications, where players have limited access to a dynamic local subset of arms (i.e., an arm could sometimes be ``walking'' and not accessible to the player). To this end, this paper proposes a multi-player multi-armed walking bandits model, aiming to address aforementioned modeling issues. The goal now is to maximize the reward, however, players can only pull arms from the local subset and only collect a full reward if no other players pull the same arm. We adopt Upper Confidence Bound (UCB) to deal with the exploration-exploitation tradeoff and employ distributed optimization techniques to properly handle collisions. By carefully integrating these two techniques, we propose a decentralized algorithm with near-optimal guarantee on the regret, and can be easily implemented to obtain competitive empirical performance.

Understanding the stochastic dynamics of sequential decision-making processes: A path-integral analysis of multi-armed bandits.

The multi-armed bandit (MAB) model is one of the most classical models to study decision-making in an uncertain environment. In this model, a player chooses one of K possible arms of a bandit machine to play at each time step, where the corresponding arm returns a random reward to the player, potentially from a specific unknown distribution. The target of the player is to collect as many rewards as possible during the process. Despite its simplicity, the MAB model offers an excellent playground for studying the trade-off between exploration vs exploitation and designing effective algorithms for sequential decision-making under uncertainty. Although many asymptotically optimal algorithms have been established, the finite-time behaviors of the stochastic dynamics of the MAB model appear much more challenging to analyze due to the intertwine between the decision-making and the rewards being collected. In this paper, we employ techniques in statistical physics to analyze the MAB model, which facilitates the characterization of the distribution of cumulative regrets at a finite short time, the central quantity of interest in an MAB algorithm, as well as the intricate dynamical behaviors of the model. Our analytical results, in good agreement with simulations, point to the emergence of an interesting multimodal regret distribution, with large regrets resulting from excess exploitation of sub-optimal arms due to an initial unlucky output from the optimal one.

A novel combinatorial multi-armed bandit game to identify online the changing top-[formula omitted] flows in software-defined networks

Identifying the top-K flows that require much more bandwidth resources in a large-scale Software-Defined Network (SDN) is essential for many network management tasks, such as load balancing, anomaly detection, and traffic engineering. However, identifying such top-K flows is not trivial, not only because of the fluctuations in flow bandwidth requirements but also because of the combinatorial explosion of problem instance sizes. In this paper, we weaken the tradeoff between exploration and exploitation and innovatively define the online top-K flows identification problem as identifying the top-K arms in a Combinatorial Multi-Armed Bandit (CMAB) model. Then, we propose a general greedy selection mechanism with some identification strategies that focus on temporal variations in the rewards. Extensive simulation experiments based on real traffic data are conducted to evaluate the performance of different strategies. In addition, the results of numerical simulations demonstrate that our proposed greedy selection mechanism significantly outperforms existing counterparts on top-K arms identification.

Multi-armed bandit problem with online clustering as side information

We consider the sequential resource allocation problem under the multi-armed bandit model in the non-stationary stochastic environment. Motivated by many real applications, where information can naturally be grouped, we consider a variation of the contextual multi-armed bandit with online clustering representing side information. We assume a stochastic environment in which the reward of each action, conditioned on a cluster, follows a Bernoulli distribution with unknown parameters. Additionally, we assume that the nature of the problem changes over time and the clusters drift incrementally, making the reward process non-stationary. In this setting, we propose a new algorithm based on a two-stage approach. The first stage is a sequential modification of the traditional k-means clustering algorithm, in which the algorithm deals with the continuous data stream and acts on a subset of data rather than a single batch. In the second stage, we incorporate the current information about clusters into the Thompson Sampling policy with discounting mechanism to track changes in the underlying reward and account for a potential cluster misclassification.

Automated Quantum Circuit Design With Nested Monte Carlo Tree Search

Quantum algorithms based on variational approaches are one of the most promising methods to construct quantum solutions and have found a myriad of applications in the last few years. Despite the adaptability and simplicity, their scalability and the selection of suitable ansatzes remain key challenges. In this work, we report an algorithmic framework based on nested Monte-Carlo Tree Search (MCTS) coupled with the combinatorial multi-armed bandit (CMAB) model for the automated design of quantum circuits. Through numerical experiments, we demonstrate our algorithm applied to various kinds of problems, including the ground energy problem in quantum chemistry, quantum optimisation on a graph, solving systems of linear equations, and finding encoding circuits for quantum error detection codes. Compared to the existing approaches, the results indicate that our circuit design algorithm can explore larger search spaces and optimise quantum circuits for larger systems, showing both versatility and scalability.

Earning While Learning: An Adversarial Multi-Armed Bandit Based Real-Time Bidding Scheme in Deregulated Electricity Market

As the specific incarnation of cyber-physical-social systems, in deregulated electricity market, the market gaming behaviors may have significantly affected the costs of electricity delivered to the market. Especially, from the supply side, the primary goal of power generating companies (PGCs) is to develop strategic biddings to maximize their profits in long-term trading, when facing intrinsic uncertainty. Typically, in such repeated and dynamic settings, one fundamental challenge is that, any PGC neither has prior knowledge about all unknown opponents’ incentives, nor observes their strategies and obtained profits. Especially, the common setting is that, once the bidding auction has occurred, the PGC only observes the market clearing price (MCP) at each round, and winning or losing status. While it is typical to assume some perfect or bounded rationality model of the PGCs, their real behaviors do not follow such assumptions due to lack of complete information, computational intractability, or lack of perfect execution, etc. We formulate the problem of sequentially optimizing any PGC's bids with an adversarial multi-armed bandit (MAB) model. Specifically, at each round, a PGC chooses to play against all other opponents from an infinite set of possible strategies that are split into continuous intervals by sequentially occurred MCPs. Then at the end of each round, the PGC observes the outcome of the auction and updates its estimation on the expected bid's fitness for each interval (i.e., how much the expected profit of the interval could be achieved), and selects the bid for the next round using the proposed algorithm Exp3C (i.e., exponential-weight for exploration and exploitation with continuous value). The experimental results based on real dataset demonstrate that Exp3C performs better than other heuristic schemes including pure greedy, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\boldsymbol{\varepsilon}}$</tex-math></inline-formula> -greedy and MCP predication based bidding schemes. Moreover, we theoretically prove the upper bound of average Exp3C regret per round follows <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\boldsymbol{O}} ({2/\sqrt {\boldsymbol{T}} })$</tex-math></inline-formula> , where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> is the number of total rounds. In summary, the proposed Exp3C has two distinguished advantages. First it is distributed, since its decisions uniquely depend on its past decisions and profits. Second, it is rational, since a PGC is given guarantees on its own accumulated profit regardless of other PGCs’ behaviors.

Minimax Off-Policy Evaluation for Multi-Armed Bandits

We study the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards, and develop minimax rate-optimal procedures under three settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger—relative to the oracle estimator equipped with the knowledge of the behavior policy— by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. Numerical experiments on both simulated and real data corroborate our theoretical findings.