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We consider a stochastic contextual bandit problem where the dimension $d$ of the feature vectors is potentially large, however, only a sparse subset of features of cardinality $s_0 ll d$ affect the reward function. Essentially all existing algorithms for sparse bandits require a priori knowledge of the value of the sparsity index $s_0$. This knowledge is almost never available in practice, and misspecification of this parameter can lead to severe deterioration in the performance of existing methods. The main contribution of this paper is to propose an algorithm that does not require prior knowledge of the sparsity index $s_0$ and establish tight regret bounds on its performance under mild conditions. We also comprehensively evaluate our proposed algorithm numerically and show that it consistently outperforms existing methods, even when the correct sparsity index is revealed to them but is kept hidden from our algorithm.
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We study a bandit version of phase retrieval where the learner chooses actions $(A_t)_{t=1}^n$ in the $d$-dimensional unit ball and the expected reward is $langle A_t, theta_starrangle^2$ where $theta_star in mathbb R^d$ is an unknown parameter vector. We prove that the minimax cumulative regret in this problem is $smash{tilde Theta(d sqrt{n})}$, which improves on the best known bounds by a factor of $smash{sqrt{d}}$. We also show that the minimax simple regret is $smash{tilde Theta(d / sqrt{n})}$ and that this is only achievable by an adaptive algorithm. Our analysis shows that an apparently convincing heuristic for guessing lower bounds can be misleading and that uniform bounds on the information ratio for information-directed sampling are not sufficient for optimal regret.
The Oxford English Dictionary defines precision medicine as medical care designed to optimize efficiency or therapeutic benefit for particular groups of patients, especially by using genetic or molecular profiling. It is not an entirely new idea: physicians from ancient times have recognized that medical treatment needs to consider individual variations in patient characteristics. However, the modern precision medicine movement has been enabled by a confluence of events: scientific advances in fields such as genetics and pharmacology, technological advances in mobile devices and wearable sensors, and methodological advances in computing and data sciences. This chapter is about bandit algorithms: an area of data science of special relevance to precision medicine. With their roots in the seminal work of Bellman, Robbins, Lai and others, bandit algorithms have come to occupy a central place in modern data science ( Lattimore and Szepesvari, 2020). Bandit algorithms can be used in any situation where treatment decisions need to be made to optimize some health outcome. Since precision medicine focuses on the use of patient characteristics to guide treatment, contextual bandit algorithms are especially useful since they are designed to take such information into account. The role of bandit algorithms in areas of precision medicine such as mobile health and digital phenotyping has been reviewed before (Tewari and Murphy, 2017; Rabbi et al., 2019). Since these reviews were published, bandit algorithms have continued to find uses in mobile health and several new topics have emerged in the research on bandit algorithms. This chapter is written for quantitative researchers in fields such as statistics, machine learning, and operations research who might be interested in knowing more about the algorithmic and mathematical details of bandit algorithms that have been used in mobile health.
Motivated by a natural problem in online model selection with bandit information, we introduce and analyze a best arm identification problem in the rested bandit setting, wherein arm expected losses decrease with the number of times the arm has been played. The shape of the expected loss functions is similar across arms, and is assumed to be available up to unknown parameters that have to be learned on the fly. We define a novel notion of regret for this problem, where we compare to the policy that always plays the arm having the smallest expected loss at the end of the game. We analyze an arm elimination algorithm whose regret vanishes as the time horizon increases. The actual rate of convergence depends in a detailed way on the postulated functional form of the expected losses. Unlike known model selection efforts in the recent bandit literature, our algorithm exploits the specific structure of the problem to learn the unknown parameters of the expected loss function so as to identify the best arm as quickly as possible. We complement our analysis with a lower bound, indicating strengths and limitations of the proposed solution.
While neural networks are powerful function approximators, they suffer from catastrophic forgetting when the data distribution is not stationary. One particular formalism that studies learning under non-stationary distribution is provided by continual learning, where the non-stationarity is imposed by a sequence of distinct tasks. Most methods in this space assume, however, the knowledge of task boundaries, and focus on alleviating catastrophic forgetting. In this work, we depart from this view and move the focus towards faster remembering -- i.e measuring how quickly the network recovers performance rather than measuring the networks performance without any adaptation. We argue that in many settings this can be more effective and that it opens the door to combining meta-learning and continual learning techniques, leveraging their complementary advantages. We propose a framework specific for the scenario where no information about task boundaries or task identity is given. It relies on a separation of concerns into what task is being solved and how the task should be solved. This framework is implemented by differentiating task specific parameters from task agnostic parameters, where the latter are optimized in a continual meta learning fashion, without access to multiple tasks at the same time. We showcase this framework in a supervised learning scenario and discuss the implication of the proposed formalism.
In this paper we propose the first multi-armed bandit algorithm based on re-sampling that achieves asymptotically optimal regret simultaneously for different families of arms (namely Bernoulli, Gaussian and Poisson distributions). Unlike Thompson Sampling which requires to specify a different prior to be optimal in each case, our proposal RB-SDA does not need any distribution-dependent tuning. RB-SDA belongs to the family of Sub-sampling Duelling Algorithms (SDA) which combines the sub-sampling idea first used by the BESA [1] and SSMC [2] algorithms with different sub-sampling schemes. In particular, RB-SDA uses Random Block sampling. We perform an experimental study assessing the flexibility and robustness of this promising novel approach for exploration in bandit models.
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