The goal is to choose the most interesting article to users, or in bandit terms, to maximize the total number of clicks on the recommended articles, i. IS methods estimate posterior densities or expectations
in probabilistic models that are analytically intractable.

Proceedings of the 16th Conference on Uncertainty in
Artificial IntelligenceFinite-Time Analysis of Multi-armed Bandits Problems with
Kullback-Leibler Divergences. e. Analytical expressions for the parameter posteriors p(θt|H1:t) are available only for few reward functions (e.

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, Bernoulli and linear contextual Gaussian), but not for many other useful cases, such as logistic rewards. The basic idea of IS is to draw, from an alternative distribution, samples which are subsequently weighted to guarantee estimation accuracy (and often reduced variance). The increasing uncertainty of the parameter posterior encourages exploration of arms that have not been played recently, but may have evolved into new parameter spaces with exploitable rewards distributions. We use a dataset333Available at R6A – Yahoo! Front Page Today Module User Click Log Dataset.

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As in importance sampling, the expectation of a function can be approximated as a weighted averageFor a finite set of particles, the algorithm performance is dependent on the choice of the proposal distributionThe optimal proposal distribution is given as the target distributionHowever, the transition prior is often used as importance function, since it is easier to draw particles (or samples) and perform subsequent importance weight calculations:Sequential Importance Resampling (SIR) filters with transition prior as importance function are commonly known as bootstrap filter and condensation algorithm. Monte Carlo (MC) methods are a family of numerical techniques based on repeated random sampling, which have been shown to be flexible enough for both numerical integration and drawing samples from probability distributions of interest. This is known as the Sequential Importance Resampling (SIR) method (24), which we rely on for our proposed framework in Section 3. e.

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We propose instead to compute the quantile function of interest by means of the SIR approximation to the parameter posterior, where one can evaluate the expected reward at each round t based on the available posterior samples, i.  5 that the regret of SIR-based methods, for the contextual linear-Gaussian case with known parameters, is equivalent to the optimal case (i. The reward function is stochastic, parameterized by the intrinsic properties of each arm (i. We consider this setting of special interest because the induced expected rewards change over time and so, the decision on the optimal arm swaps accordingly. , those within the exponential family of distributions (28). , Gaussian distributed bandits, whose mean reward is linear in the contextual features), which are well-studied MAB models (45, 10, 2, 3, 27, 28).

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A single step of sequential importance resampling check this site out as follows:The term Sampling Importance Resampling is also sometimes used when referring to SIR filters. , in environments that evolve over time. Specifically, we consider the SIR method for (dynamic) bandits, where the world (might) evolve over time, i. In a Bayesian approach to the MAB problem, prior knowledge on the model and parameters is incorporated into the algorithm. We evaluate the proposed SIR-based methods for bandits with dynamics as in Eqn.

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Efficient particle-based online smoothing in general hidden Markov
models: the PaRIS algorithm. 3), one needs to resort to approximations, e. The implementation of the proposed method is available in this public repository. , θa,t∼N(θa,t|Laθa,t−1,Ca); while for the unknown parameter case, the marginalized transition density
222Details of the derivation are provided in
Appendix A . We have presented a (sequential) importance sampling-based framework for the MAB problem, where we combine sequential Monte Carlo inference with state-of-the-art Bayesian MAB policies. g.

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We note that particle stream degeneracy is an important issue when the smoothing distribution is of interest, i. The rewards For this reward distribution, the parameter conjugate prior distribution is the Beta distributionAfter observing actions a1:t click here for more info rewards y1:t, the parameter posterior follows an updated Beta distributionwith sequential updatesor, alternatively, batch updatesThe expected reward for each arm followsand the quantile function is based on the Beta distributionFor bandits with continuous rewards, the Gaussian distribution is often applicable, where contextual dependencies can Go Here be included. We here propose to leverage Monte Carlo
estimation and, in particular, the flexibility of (sequential) importance
sampling (IS) to allow for accurate estimation of the statistics of interest
within the MAB problem. .

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