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Register a new userOn Limited-Memory Subsampling Strategies for Bandits
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Added by
Yoan Russac
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Dorian Baudry
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There has been a recent surge of interest in nonparametric bandit algorithms based on subsampling. One drawback however of these approaches is the additional complexity required by random subsampling and the storage of the full history of rewards. Our first contribution is to show that a simple deterministic subsampling rule, proposed in the recent work of Baudry et al. (2020) under the name of last-block subsampling, is asymptotically optimal in one-parameter exponential families. In addition, we prove that these guarantees also hold when limiting the algorithm memory to a polylogarithmic function of the time horizon. These findings open up new perspectives, in particular for non-stationary scenarios in which the arm distributions evolve over time. We propose a variant of the algorithm in which only the most recent observations are used for subsampling, achieving optimal regret guarantees under the assumption of a known number of abrupt changes. Extensive numerical simulations highlight the merits of this approach, particularly when the changes are not only affecting the means of the rewards.
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Consider the following abstract coin tossing problem: Given a set of $n$ coins with unknown biases, find the most biased coin using a minimal number of coin tosses. This is a common abstraction of various exploration problems in theoretical computer science and machine learning and has been studied extensively over the years. In particular, algorithms with optimal sample complexity (number of coin tosses) have been known for this problem for quite some time. Motivated by applications to processing massive datasets, we study the space complexity of solving this problem with optimal number of coin tosses in the streaming model. In this model, the coins are arriving one by one and the algorithm is only allowed to store a limited number of coins at any point -- any coin not present in the memory is lost and can no longer be tossed or compared to arriving coins. Prior algorithms for the coin tossing problem with optimal sample complexity are based on iterative elimination of coins which inherently require storing all the coins, leading to memory-inefficient streaming algorithms. We remedy this state-of-affairs by presenting a series of improved streaming algorithms for this problem: we start with a simple algorithm which require storing only $O(log{n})$ coins and then iteratively refine it further and further, leading to algorithms with $O(loglog{(n)})$ memory, $O(log^*{(n)})$ memory, and finally a one that only stores a single extra coin in memory -- the same exact space needed to just store the best coin throughout the stream. Furthermore, we extend our algorithms to the problem of finding the $k$ most biased coins as well as other exploration problems such as finding top-$k$ elements using noisy comparisons or finding an $epsilon$-best arm in stochastic multi-armed bandits, and obtain efficient streaming algorithms for these problems.
We study a structured variant of the multi-armed bandit problem specified by a set of Bernoulli distributions $ u != !( u_{a,b})_{a in mathcal{A}, b in mathcal{B}}$ with means $(mu_{a,b})_{a in mathcal{A}, b in mathcal{B}}!in![0,1]^{mathcal{A}timesmathcal{B}}$ and by a given weight matrix $omega!=! (omega_{b,b})_{b,b in mathcal{B}}$, where $ mathcal{A}$ is a finite set of arms and $ mathcal{B} $ is a finite set of users. The weight matrix $omega$ is such that for any two users $b,b!in!mathcal{B}, text{max}_{ainmathcal{A}}|mu_{a,b} !-! mu_{a,b}| !leq! omega_{b,b} $. This formulation is flexible enough to capture various situations, from highly-structured scenarios ($omega!in!{0,1}^{mathcal{B}timesmathcal{B}}$) to fully unstructured setups ($omega!equiv! 1$).We consider two scenarios depending on whether the learner chooses only the actions to sample rewards from or both users and actions. We first derive problem-dependent lower bounds on the regret for this generic graph-structure that involves a structure dependent linear programming problem. Second, we adapt to this setting the Indexed Minimum Empirical Divergence (IMED) algorithm introduced by Honda and Takemura (2015), and introduce the IMED-GS$^star$ algorithm. Interestingly, IMED-GS$^star$ does not require computing the solution of the linear programming problem more than about $log(T)$ times after $T$ steps, while being provably asymptotically optimal. Also, unlike existing bandit strategies designed for other popular structures, IMED-GS$^star$ does not resort to an explicit forced exploration scheme and only makes use of local counts of empirical events. We finally provide numerical illustration of our results that confirm the performance of IMED-GS$^star$.
This paper focuses on building personalized player models solely from player behavior in the context of adaptive games. We present two main contributions: The first is a novel approach to player modeling based on multi-armed bandits (MABs). This approach addresses, at the same time and in a principled way, both the problem of collecting data to model the characteristics of interest for the current player and the problem of adapting the interactive experience based on this model. Second, we present an approach to evaluating and fine-tuning these algorithms prior to generating data in a user study. This is an important problem, because conducting user studies is an expensive and labor-intensive process; therefore, an ability to evaluate the algorithms beforehand can save a significant amount of resources. We evaluate our approach in the context of modeling players social comparison orientation (SCO) and present empirical results from both simulations and real players.
Current implementations of pseudo-Boolean (PB) solvers working on native PB constraints are based on the CDCL architecture which empowers highly efficient modern SAT solvers. In particular, such PB solvers not only implement a (cutting-planes-based) conflict analysis procedure, but also complementary strategies for components that are crucial for the efficiency of CDCL, namely branching heuristics, learned constraint deletion and restarts. However, these strategies are mostly reused by PB solvers without considering the particular form of the PB constraints they deal with. In this paper, we present and evaluate different ways of adapting CDCL strategies to take the specificities of PB constraints into account while preserving the behavior they have in the clausal setting. We implemented these strategies in two different solvers, namely Sat4j (for which we consider three configurations) and RoundingSat. Our experiments show that these dedicated strategies allow to improve, sometimes significantly, the performance of these solvers, both on decision and optimization problems.
We consider a multi-armed bandit problem specified by a set of Gaussian or Bernoulli distributions endowed with a unimodal structure. Although this problem has been addressed in the literature (Combes and Proutiere, 2014), the state-of-the-art algorithms for such structure make appear a forced-exploration mechanism. We introduce IMED-UB, the first forced-exploration free strategy that exploits the unimodal-structure, by adapting to this setting the Indexed Minimum Empirical Divergence (IMED) strategy introduced by Honda and Takemura (2015). This strategy is proven optimal. We then derive KLUCB-UB, a KLUCB version of IMED-UB, which is also proven optimal. Owing to our proof technique, we are further able to provide a concise finite-time analysis of both strategies in an unified way. Numerical experiments show that both IMED-UB and KLUCB-UB perform similarly in practice and outperform the state-of-the-art algorithms.
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