For the model of constrained multi-armed bandit, we show that by construction there exists an index-based deterministic asymptotically optimal algorithm. The optimality is achieved by the convergence of the probability of choosing an optimal feasible arm to one over infinite horizon. The algorithm is built upon Locatelli et al.'s "anytime parameter-free thresholding" algorithm under the assumption that the optimal value is known. We provide a finite-time bound to the probability of the asymptotic optimality given as 1-O(|A|Te^{-T}) where T is the horizon size and A is the set of the arms in the bandit. We then study a relaxed-version of the algorithm in a general form that estimates the optimal value and discuss the asymptotic optimality of the algorithm after a sufficiently large T with examples.
For the stochastic multi-armed bandit (MAB) problem from a constrained model that generalizes the classical one, we show that an asymptotic optimality is achievable by a simple strategy extended from the $ε_t$-greedy strategy. We provide a finite-time lower bound on the probability of correct selection of an optimal near-feasible arm that holds for all time steps. Under some conditions, the bound approaches one as time $t$ goes to infinity. A particular example sequence of $\{ε_t\}$ having the asymptotic convergence rate in the order of $(1-\frac{1}{t})^4$ that holds from a sufficiently large …
We present a formal model of human decision-making in explore-exploit tasks using the context of multi-armed bandit problems, where the decision-maker must choose among multiple options with uncertain rewards. We address the standard multi-armed bandit problem, the multi-armed bandit problem with transition costs, and the multi-armed bandit problem on graphs. We focus on the case of Gaussian rewards in a setting where the decision-maker uses Bayesian inference to estimate the reward values. We model the decision-maker's prior knowledge with the Bayesian prior on the mean reward. We develop the…
This work explores adaptations of successful multi-armed bandits policies to the online contextual bandits scenario with binary rewards using binary classification algorithms such as logistic regression as black-box oracles. Some of these adaptations are achieved through bootstrapping or approximate bootstrapping, while others rely on other forms of randomness, resulting in more scalable approaches than previous works, and the ability to work with any type of classification algorithm. In particular, the Adaptive-Greedy algorithm shows a lot of promise, in many cases achieving better performanc…
Gittins indices provide an optimal solution to the classical multi-armed bandit problem. An obstacle to their use has been the common perception that their computation is very difficult. This paper demonstrates an accessible general methodology for the calculating Gittins indices for the multi-armed bandit with a detailed study on the cases of Bernoulli and Gaussian rewards. With accompanying easy-to-use open source software, this work removes computation as a barrier to using Gittins indices in these commonly found settings.
Over the past decade, contextual bandit algorithms have been gaining in popularity due to their effectiveness and flexibility in solving sequential decision problems---from online advertising and finance to clinical trial design and personalized medicine. At the same time, there are, as of yet, surprisingly few options that enable researchers and practitioners to simulate and compare the wealth of new and existing bandit algorithms in a standardized way. To help close this gap between analytical research and empirical evaluation the current paper introduces the object-oriented R package "conte…
We consider the stochastic linear (multi-armed) contextual bandit problem with the possibility of hidden simple multi-armed bandit structure in which the rewards are independent of the contextual information. Algorithms that are designed solely for one of the regimes are known to be sub-optimal for the alternate regime. We design a single computationally efficient algorithm that simultaneously obtains problem-dependent optimal regret rates in the simple multi-armed bandit regime and minimax optimal regret rates in the linear contextual bandit regime, without knowing a priori which of the two m…
Institute of Science and Technology Austria（奥地利科学技术研究所）
We consider the problem of minimizing the regret in stochastic multi-armed bandit, when the measure of goodness of an arm is not the mean return, but some general function of the mean and the variance.We characterize the conditions under which learning is possible and present examples for which no natural algorithm can achieve sublinear regret.
We study the problem of identifying the top $m$ arms in a multi-armed bandit game. Our proposed solution relies on a new algorithm based on successive rejects of the seemingly bad arms, and successive accepts of the good ones. This algorithmic contribution allows to tackle other multiple identifications settings that were previously out of reach. In particular we show that this idea of successive accepts and rejects applies to the multi-bandit best arm identification problem.
I present the first algorithm for stochastic finite-armed bandits that simultaneously enjoys order-optimal problem-dependent regret and worst-case regret. Besides the theoretical results, the new algorithm is simple, efficient and empirically superb. The approach is based on UCB, but with a carefully chosen confidence parameter that optimally balances the risk of failing confidence intervals against the cost of excessive optimism.
The target of $\mathcal{X}$-armed bandit problem is to find the global maximum of an unknown stochastic function $f$, given a finite budget of $n$ evaluations. Recently, $\mathcal{X}$-armed bandits have been widely used in many situations. Many of these applications need to deal with large-scale data sets. To deal with these large-scale data sets, we study a distributed setting of $\mathcal{X}$-armed bandits, where $m$ players collaborate to find the maximum of the unknown function. We develop a novel anytime distributed $\mathcal{X}$-armed bandit algorithm. Compared with prior work on $\mathc…
In this paper we propose a multi-armed bandit inspired, pool based active learning algorithm for the problem of binary classification. By carefully constructing an analogy between active learning and multi-armed bandits, we utilize ideas such as lower confidence bounds, and self-concordant regularization from the multi-armed bandit literature to design our proposed algorithm. Our algorithm is a sequential algorithm, which in each round assigns a sampling distribution on the pool, samples one point from this distribution, and queries the oracle for the label of this sampled point. The design of…
University of Pennsylvania（宾夕法尼亚大学）；Michigan State University（密歇根州立大学）；Princeton University（普林斯顿大学）
Satisficing is a relaxation of maximizing and allows for less risky decision making in the face of uncertainty. We propose two sets of satisficing objectives for the multi-armed bandit problem, where the objective is to achieve reward-based decision-making performance above a given threshold. We show that these new problems are equivalent to various standard multi-armed bandit problems with maximizing objectives and use the equivalence to find bounds on performance. The different objectives can result in qualitatively different behavior; for example, agents explore their options continually in…
IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
In recent years, multi-armed bandit (MAB) framework has attracted a lot of attention in various applications, from recommender systems and information retrieval to healthcare and finance, due to its stellar performance combined with certain attractive properties, such as learning from less feedback. The multi-armed bandit field is currently flourishing, as novel problem settings and algorithms motivated by various practical applications are being introduced, building on top of the classical bandit problem. This article aims to provide a comprehensive review of top recent developments in multip…
Indiana University（印第安纳大学）；University of Alberta（阿尔伯塔大学）
Stochastic linear bandits are a natural and simple generalisation of finite-armed bandits with numerous practical applications. Current approaches focus on generalising existing techniques for finite-armed bandits, notably the optimism principle and Thompson sampling. While prior work has mostly been in the worst-case setting, we analyse the asymptotic instance-dependent regret and show matching upper and lower bounds on what is achievable. Surprisingly, our results show that no algorithm based on optimism or Thompson sampling will ever achieve the optimal rate, and indeed, can be arbitrarily …
This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.
Contextual multi-armed bandits are classical models in reinforcement learning for sequential decision-making associated with individual information. A widely-used policy for bandits is Thompson Sampling, where samples from a data-driven probabilistic belief about unknown parameters are used to select the control actions. For this computationally fast algorithm, performance analyses are available under full context-observations. However, little is known for problems that contexts are not fully observed. We propose a Thompson Sampling algorithm for partially observable contextual multi-armed ban…
The multi-armed bandit problem is a classical decision-making problem where an agent has to learn an optimal action balancing exploration and exploitation. Properly managing this trade-off requires a correct assessment of uncertainty; in multi-armed bandits, as in other machine learning applications, it is important to distinguish between stochasticity that is inherent to the system (aleatoric uncertainty) and stochasticity that derives from the limited knowledge of the agent (epistemic uncertainty). In this paper we consider the formalism of subjective logic, a concise and expressive framewor…
We consider a novel multi-armed bandit framework where the rewards obtained by pulling the arms are functions of a common latent random variable. The correlation between arms due to the common random source can be used to design a generalized upper-confidence-bound (UCB) algorithm that identifies certain arms as $non-competitive$, and avoids exploring them. As a result, we reduce a $K$-armed bandit problem to a $C+1$-armed problem, where $C+1$ includes the best arm and $C$ $competitive$ arms. Our regret analysis shows that the competitive arms need to be pulled $\mathcal{O}(\log T)$ times, whi…
Centre Borelli, ENS Paris-Saclay（巴黎萨克雷高等师范学院博雷利中心）；Cdiscount（Cdiscount公司）；CREST, ENSAE Paris（巴黎经济与统计高等学校统计与经济研究中心）；Criteo AI Lab（Criteo人工智能实验室）
Continuously learning and leveraging the knowledge accumulated from prior tasks in order to improve future performance is a long standing machine learning problem. In this paper, we study the problem in the multi-armed bandit framework with the objective to minimize the total regret incurred over a series of tasks. While most bandit algorithms are designed to have a low worst-case regret, we examine here the average regret over bandit instances drawn from some prior distribution which may change over time. We specifically focus on confidence interval tuning of UCB algorithms. We propose a band…
Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）
In many online learning or multi-armed bandit problems, the taken actions or pulled arms are ordinal and required to be monotone over time. Examples include dynamic pricing, in which the firms use markup pricing policies to please early adopters and deter strategic waiting, and clinical trials, in which the dose allocation usually follows the dose escalation principle to prevent dose limiting toxicities. We consider the continuum-armed bandit problem when the arm sequence is required to be monotone. We show that when the unknown objective function is Lipschitz continuous, the regret is $O(T)$.…
Contextual multi-armed bandit problems arise frequently in important industrial applications. Existing solutions model the context either linearly, which enables uncertainty driven (principled) exploration, or non-linearly, by using epsilon-greedy exploration policies. Here we present a deep learning framework for contextual multi-armed bandits that is both non-linear and enables principled exploration at the same time. We tackle the exploration vs. exploitation trade-off through Thompson sampling by exploiting the connection between inference time dropout and sampling from the posterior over …
Bridging Adversarial and Nonstationary Multi-armed Bandit
Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）；The Hong Kong University of Science and Technology（香港科技大学）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
Princeton University（普林斯顿大学）；Università degli Studi di Milano（米兰大学）
Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the Thirties, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In …
Bosch Center for Artificial Intelligence（博世人工智能中心）；University of Southern Denmark（南丹麦大学）；Technische Universität Darmstadt（达姆施塔特工业大学）
We present a PAC-Bayesian analysis of lifelong learning. In the lifelong learning problem, a sequence of learning tasks is observed one-at-a-time, and the goal is to transfer information acquired from previous tasks to new learning tasks. We consider the case when each learning task is a multi-armed bandit problem. We derive lower bounds on the expected average reward that would be obtained if a given multi-armed bandit algorithm was run in a new task with a particular prior and for a set number of steps. We propose lifelong learning algorithms that use our new bounds as learning objectives. O…
Adaptive and sequential experiment design is a well-studied area in numerous domains. We survey and synthesize the work of the online statistical learning paradigm referred to as multi-armed bandits integrating the existing research as a resource for a certain class of online experiments. We first explore the traditional stochastic model of a multi-armed bandit, then explore a taxonomic scheme of complications to that model, for each complication relating it to a specific requirement or consideration of the experiment design context. Finally, at the end of the paper, we present a table of know…
This work explores the idea of a causal contextual multi-armed bandit approach to automated marketing, where we estimate and optimize the causal (incremental) effects. Focusing on causal effect leads to better return on investment (ROI) by targeting only the persuadable customers who wouldn't have taken the action organically. Our approach draws on strengths of causal inference, uplift modeling, and multi-armed bandits. It optimizes on causal treatment effects rather than pure outcome, and incorporates counterfactual generation within data collection. Following uplift modeling results, we opti…
Based on differential privacy (DP) framework, we introduce and unify privacy definitions for the multi-armed bandit algorithms. We represent the framework with a unified graphical model and use it to connect privacy definitions. We derive and contrast lower bounds on the regret of bandit algorithms satisfying these definitions. We leverage a unified proving technique to achieve all the lower bounds. We show that for all of them, the learner's regret is increased by a multiplicative factor dependent on the privacy level $ε$. We observe that the dependency is weaker when we do not require local …
We extend the classic multi-armed bandit (MAB) model to the setting of noncompliance, where the arm pull is a mere instrument and the treatment applied may differ from it, which gives rise to the instrument-armed bandit (IAB) problem. The IAB setting is relevant whenever the experimental units are human since free will, ethics, and the law may prohibit unrestricted or forced application of treatment. In particular, the setting is relevant in bandit models of dynamic clinical trials and other controlled trials on human interventions. Nonetheless, the setting has not been fully investigate in th…
In this paper, we study the multi-armed bandit problem in the batched setting where the employed policy must split data into a small number of batches. While the minimax regret for the two-armed stochastic bandits has been completely characterized in \cite{perchet2016batched}, the effect of the number of arms on the regret for the multi-armed case is still open. Moreover, the question whether adaptively chosen batch sizes will help to reduce the regret also remains underexplored. In this paper, we propose the BaSE (batched successive elimination) policy to achieve the rate-optimal regrets (wit…
Carnegie Mellon University（卡内基梅隆大学）；INRIA Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧分部）
Learning from prior tasks and transferring that experience to improve future performance is critical for building lifelong learning agents. Although results in supervised and reinforcement learning show that transfer may significantly improve the learning performance, most of the literature on transfer is focused on batch learning tasks. In this paper we study the problem of \textit{sequential transfer in online learning}, notably in the multi-armed bandit framework, where the objective is to minimize the cumulative regret over a sequence of tasks by incrementally transferring knowledge from p…
Multi-armed bandits are a quintessential machine learning problem requiring the balancing of exploration and exploitation. While there has been progress in developing algorithms with strong theoretical guarantees, there has been less focus on practical near-optimal finite-time performance. In this paper, we propose an algorithm for Bayesian multi-armed bandits that utilizes value-function-driven online planning techniques. Building on previous work on UCB and Gittins index, we introduce linearly-separable value functions that take both the expected return and the benefit of exploration into co…
We consider a multi-armed bandit framework where the rewards obtained by pulling different arms are correlated. We develop a unified approach to leverage these reward correlations and present fundamental generalizations of classic bandit algorithms to the correlated setting. We present a unified proof technique to analyze the proposed algorithms. Rigorous analysis of C-UCB (the correlated bandit version of Upper-confidence-bound) reveals that the algorithm ends up pulling certain sub-optimal arms, termed as non-competitive, only O(1) times, as opposed to the O(log T) pulls required by classic …
We define and analyze a multi-agent multi-armed bandit problem in which decision-making agents can observe the choices and rewards of their neighbors. Neighbors are defined by a network graph with heterogeneous and stochastic interconnections. These interactions are determined by the sociability of each agent, which corresponds to the probability that the agent observes its neighbors. We design an algorithm for each agent to maximize its own expected cumulative reward and prove performance bounds that depend on the sociability of the agents and the network structure. We use the bounds to predi…
We study a multi-armed bandit problem with covariates in a setting where there is a possible delay in observing the rewards. Under some mild assumptions on the probability distributions for the delays and using an appropriate randomization to select the arms, the proposed strategy is shown to be strongly consistent.
University of Colorado, Boulder（科罗拉多大学博尔德分校）；University of Virginia（弗吉尼亚大学）；Clemson University（克莱姆森大学）
We study incentivized exploration for the multi-armed bandit (MAB) problem where the players receive compensation for exploring arms other than the greedy choice and may provide biased feedback on reward. We seek to understand the impact of this drifted reward feedback by analyzing the performance of three instantiations of the incentivized MAB algorithm: UCB, $\varepsilon$-Greedy, and Thompson Sampling. Our results show that they all achieve $\mathcal{O}(\log T)$ regret and compensation under the drifted reward, and are therefore effective in incentivizing exploration. Numerical examples are …
University of Pennsylvania（宾夕法尼亚大学）；Massachusetts Institute of Technology（麻省理工学院）
We present a new type of acquisition functions for online decision making in multi-armed and contextual bandit problems with extreme payoffs. Specifically, we model the payoff function as a Gaussian process and formulate a novel type of upper confidence bound (UCB) acquisition function that guides exploration towards the bandits that are deemed most relevant according to the variability of the observed rewards. This is achieved by computing a tractable likelihood ratio that quantifies the importance of the output relative to the inputs and essentially acts as an \textit{attention mechanism} th…
We analyze the regret of combinatorial Thompson sampling (CTS) for the combinatorial multi-armed bandit with probabilistically triggered arms under the semi-bandit feedback setting. We assume that the learner has access to an exact optimization oracle but does not know the expected base arm outcomes beforehand. When the expected reward function is Lipschitz continuous in the expected base arm outcomes, we derive $O(\sum_{i =1}^m \log T / (p_i Δ_i))$ regret bound for CTS, where $m$ denotes the number of base arms, $p_i$ denotes the minimum non-zero triggering probability of base arm $i$ and $Δ_…
School of Information Science and Technology University of Science and Technology of China（中国科学技术大学信息科学技术学院）
We consider a variant of the classic multi-armed bandit problem where the expected reward of each arm is a function of an unknown parameter. The arms are divided into different groups, each of which has a common parameter. Therefore, when the player selects an arm at each time slot, information of other arms in the same group is also revealed. This regional bandit model naturally bridges the non-informative bandit setting where the player can only learn the chosen arm, and the global bandit model where sampling one arms reveals information of all arms. We propose an efficient algorithm, UCB-g,…
University of Cambridge（剑桥大学）；Alan Turing Institute（艾伦·图灵研究所）
Drawing a sample from a discrete distribution is one of the building components for Monte Carlo methods. Like other sampling algorithms, discrete sampling suffers from the high computational burden in large-scale inference problems. We study the problem of sampling a discrete random variable with a high degree of dependency that is typical in large-scale Bayesian inference and graphical models, and propose an efficient approximate solution with a subsampling approach. We make a novel connection between the discrete sampling and Multi-Armed Bandits problems with a finite reward population and p…
We consider a stochastic bandit problem with countably many arms that belong to a finite set of types, each characterized by a unique mean reward. In addition, there is a fixed distribution over types which sets the proportion of each type in the population of arms. The decision maker is oblivious to the type of any arm and to the aforementioned distribution over types, but perfectly knows the total number of types occurring in the population of arms. We propose a fully adaptive online learning algorithm that achieves O(log n) distribution-dependent expected cumulative regret after any number …
We define and analyze a multi-agent multi-armed bandit problem in which decision-making agents can observe the choices and rewards of their neighbors under a linear observation cost. Neighbors are defined by a network graph that encodes the inherent observation constraints of the system. We define a cost associated with observations such that at every instance an agent makes an observation it receives a constant observation regret. We design a sampling algorithm and an observation protocol for each agent to maximize its own expected cumulative reward through minimizing expected cumulative samp…
Indiana University - Purdue University Indianapolis（印第安纳大学-普渡大学印第安纳波利斯分校）
We propose a novel framework for integrating Hawkes processes with multi-armed bandit algorithms to solve spatio-temporal event forecasting and detection problems when data may be undersampled or spatially biased. In particular, we introduce an upper confidence bound algorithm using Bayesian spatial Hawkes process estimation for balancing the tradeoff between exploiting geographic regions where data has been collected and exploring geographic regions where data is unobserved. We first validate our model using simulated data and then apply it to the problem of disaster search and rescue using c…
Technion, Israel Institute of Technology（以色列理工学院）；Nvidia Research（英伟达研究院）
The Combinatorial Multi-Armed Bandit problem is a sequential decision-making problem in which an agent selects a set of arms on each round, observes feedback for each of these arms and aims to maximize a known reward function of the arms it chose. While previous work proved regret upper bounds in this setting for general reward functions, only a few works provided matching lower bounds, all for specific reward functions. In this work, we prove regret lower bounds for combinatorial bandits that hold under mild assumptions for all smooth reward functions. We derive both problem-dependent and pro…
We investigate the optimality of perturbation based algorithms in the stochastic and adversarial multi-armed bandit problems. For the stochastic case, we provide a unified regret analysis for both sub-Weibull and bounded perturbations when rewards are sub-Gaussian. Our bounds are instance optimal for sub-Weibull perturbations with parameter 2 that also have a matching lower tail bound, and all bounded support perturbations where there is sufficient probability mass at the extremes of the support. For the adversarial setting, we prove rigorous barriers against two natural solution approaches us…
The paper addresses the Multiplayer Multi-Armed Bandit (MMAB) problem, where $M$ decision makers or players collaborate to maximize their cumulative reward. When several players select the same arm, a collision occurs and no reward is collected on this arm. Players involved in a collision are informed about this collision. We present DPE (Decentralized Parsimonious Exploration), a decentralized algorithm that achieves the same regret as that obtained by an optimal centralized algorithm. Our algorithm has better regret guarantees than the state-of-the-art algorithm SIC-MMAB \cite{boursier2019}.…
Functional multi-armed bandit and the best function identification problems
Lomonosov Moscow State University（莫斯科罗蒙诺索夫莫斯科国立大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Skoltech（斯克里普切尔技术学院）；Avito；Moscow, Russia（莫斯科，俄罗斯）
LMU Munich（慕尼黑大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）
We consider a resource-aware variant of the classical multi-armed bandit problem: In each round, the learner selects an arm and determines a resource limit. It then observes a corresponding (random) reward, provided the (random) amount of consumed resources remains below the limit. Otherwise, the observation is censored, i.e., no reward is obtained. For this problem setting, we introduce a measure of regret, which incorporates the actual amount of allocated resources of each learning round as well as the optimality of realizable rewards. Thus, to minimize regret, the learner needs to set a res…
Northeastern University（东北大学）；CNRS & CRIStAL, UMR 9189, Université de Lille（法国国家科学研究中心与里尔大学晶体实验室（UMR 9189））；Inria Lille, SequeL team（法国国家信息与自动化研究所里尔分所序列团队）
We consider the problem of near-optimal arm identification in the fixed confidence setting of the infinitely armed bandit problem when nothing is known about the arm reservoir distribution. We (1) introduce a PAC-like framework within which to derive and cast results; (2) derive a sample complexity lower bound for near-optimal arm identification; (3) propose an algorithm that identifies a nearly-optimal arm with high probability and derive an upper bound on its sample complexity which is within a log factor of our lower bound; and (4) discuss whether our log^2(1/delta) dependence is inescapabl…
Technion – Institute of Technology（以色列理工学院）；Meta；Nvidia Research（英伟达研究院）
We consider a stochastic multi-armed bandit setting where reward must be actively queried for it to be observed. We provide tight lower and upper problem-dependent guarantees on both the regret and the number of queries. Interestingly, we prove that there is a fundamental difference between problems with a unique and multiple optimal arms, unlike in the standard multi-armed bandit problem. We also present a new, simple, UCB-style sampling concept, and show that it naturally adapts to the number of optimal arms and achieves tight regret and querying bounds.
ENS Paris-Saclay（巴黎萨克雷高等师范学院）；Criteo AI Lab（Criteo人工智能实验室）
Motivated by cognitive radio networks, we consider the stochastic multiplayer multi-armed bandit problem, where several players pull arms simultaneously and collisions occur if one of them is pulled by several players at the same stage. We present a decentralized algorithm that achieves the same performance as a centralized one, contradicting the existing lower bounds for that problem. This is possible by "hacking" the standard model by constructing a communication protocol between players that deliberately enforces collisions, allowing them to share their information at a negligible cost. Thi…
We study the stochastic multi-armed bandit problem with non-equivalent multiple plays where, at each step, an agent chooses not only a set of arms, but also their order, which influences reward distribution. In several problem formulations with different assumptions, we provide lower bounds for regret with standard asymptotics $O(\log{t})$ but novel coefficients and provide optimal algorithms, thus proving that these bounds cannot be improved.
