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Register a new userA Smoothed Analysis of the Greedy Algorithm for the Linear Contextual Bandit Problem
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Jamie Morgenstern
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Bandit learning is characterized by the tension between long-term exploration and short-term exploitation. However, as has recently been noted, in settings in which the choices of the learning algorithm correspond to important decisions about individual people (such as criminal recidivism prediction, lending, and sequential drug trials), exploration corresponds to explicitly sacrificing the well-being of one individual for the potential future benefit of others. This raises a fairness concern. In such settings, one might like to run a greedy algorithm, which always makes the (myopically) optimal decision for the individuals at hand - but doing this can result in a catastrophic failure to learn. In this paper, we consider the linear contextual bandit problem and revisit the performance of the greedy algorithm. We give a smoothed analysis, showing that even when contexts may be chosen by an adversary, small perturbations of the adversarys choices suffice for the algorithm to achieve no regret, perhaps (depending on the specifics of the setting) with a constant amount of initial training data. This suggests that generically (i.e. in slightly perturbed environments), exploration and exploitation need not be in conflict in the linear setting.
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We investigate the sparse linear contextual bandit problem where the parameter $theta$ is sparse. To relieve the sampling inefficiency, we utilize the perturbed adversary where the context is generated adversarilly but with small random non-adaptive perturbations. We prove that the simple online Lasso supports sparse linear contextual bandit with regret bound $mathcal{O}(sqrt{kTlog d})$ even when $d gg T$ where $k$ and $d$ are the number of effective and ambient dimension, respectively. Compared to the recent work from Sivakumar et al. (2020), our analysis does not rely on the precondition processing, adaptive perturbation (the adaptive perturbation violates the i.i.d perturbation setting) or truncation on the error set. Moreover, the special structures in our results explicitly characterize how the perturbation affects exploration length, guide the design of perturbation together with the fundamental performance limit of perturbation method. Numerical experiments are provided to complement the theoretical analysis.
Online learning algorithms, widely used to power search and content optimization on the web, must balance exploration and exploitation, potentially sacrificing the experience of current users in order to gain information that will lead to better decisions in the future. While necessary in the worst case, explicit exploration has a number of disadvantages compared to the greedy algorithm that always exploits by choosing an action that currently looks optimal. We ask under what conditions inherent diversity in the data makes explicit exploration unnecessary. We build on a recent line of work on the smoothed analysis of the greedy algorithm in the linear contextual bandits model. We improve on prior results to show that a greedy approach almost matches the best possible Bayesian regret rate of any other algorithm on the same problem instance whenever the diversity conditions hold, and that this regret is at most $tilde O(T^{1/3})$.
Bandit learning algorithms typically involve the balance of exploration and exploitation. However, in many practical applications, worst-case scenarios needing systematic exploration are seldom encountered. In this work, we consider a smoothed setting for structured linear contextual bandits where the adversarial contexts are perturbed by Gaussian noise and the unknown parameter $theta^*$ has structure, e.g., sparsity, group sparsity, low rank, etc. We propose simple greedy algorithms for both the single- and multi-parameter (i.e., different parameter for each context) settings and provide a unified regret analysis for $theta^*$ with any assumed structure. The regret bounds are expressed in terms of geometric quantities such as Gaussian widths associated with the structure of $theta^*$. We also obtain sharper regret bounds compared to earlier work for the unstructured $theta^*$ setting as a consequence of our improved analysis. We show there is implicit exploration in the smoothed setting where a simple greedy algorithm works.
In this paper, we consider the contextual variant of the MNL-Bandit problem. More specifically, we consider a dynamic set optimization problem, where in every round a decision maker offers a subset (assortment) of products to a consumer, and observes their response. Consumers purchase products so as to maximize their utility. We assume that the products are described by a set of attributes and the mean utility of a product is linear in the values of these attributes. We model consumer choice behavior by means of the widely used Multinomial Logit (MNL) model, and consider the decision makers problem of dynamically learning the model parameters, while optimizing cumulative revenue over the selling horizon $T$. Though this problem has attracted considerable attention in recent times, many existing methods often involve solving an intractable non-convex optimization problem and their theoretical performance guarantees depend on a problem dependent parameter which could be prohibitively large. In particular, existing algorithms for this problem have regret bounded by $O(sqrt{kappa d T})$, where $kappa$ is a problem dependent constant that can have exponential dependency on the number of attributes. In this paper, we propose an optimistic algorithm and show that the regret is bounded by $O(sqrt{dT} + kappa)$, significantly improving the performance over existing methods. Further, we propose a convex relaxation of the optimization step which allows for tractable decision-making while retaining the favourable regret guarantee.
Balancing exploration and exploitation (EE) is a fundamental problem in contex-tual bandit. One powerful principle for EE trade-off isOptimism in Face of Uncer-tainty(OFU), in which the agent takes the action according to an upper confidencebound (UCB) of reward. OFU has achieved (near-)optimal regret bound for lin-ear/kernel contextual bandits. However, it is in general unknown how to deriveefficient and effective EE trade-off methods for non-linearcomplex tasks, suchas contextual bandit with deep neural network as the reward function. In thispaper, we propose a novel OFU algorithm namedregularized OFU(ROFU). InROFU, we measure the uncertainty of the reward by a differentiable function andcompute the upper confidence bound by solving a regularized optimization prob-lem. We prove that, for multi-armed bandit, kernel contextual bandit and neuraltangent kernel bandit, ROFU achieves (near-)optimal regret bounds with certainuncertainty measure, which theoretically justifies its effectiveness on EE trade-off.Importantly, ROFU admits a very efficient implementation with gradient-basedoptimizer, which easily extends to general deep neural network models beyondneural tangent kernel, in sharp contrast with previous OFU methods. The em-pirical evaluation demonstrates that ROFU works extremelywell for contextualbandits under various settings.
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