Accelerated Bayesian learning for decentralized two-armed bandit based decision making with applications to the Goore Game

The two-armed bandit problem is a classical optimization problem where a decision maker sequentially pulls one of two arms attached to a gambling machine, with each pull resulting in a random reward. The reward distributions are unknown, and thus, one must balance between exploiting existing knowledge about the arms, and obtaining new information. Bandit problems are particularly fascinating because a large class of real world problems, including routing, Quality of Service (QoS) control, game playing, and resource allocation, can be solved in a decentralized manner when modeled as a system of interacting gambling machines. Although computationally intractable in many cases, Bayesian methods provide a standard for optimal decision making. This paper proposes a novel scheme for decentralized decision making based on the Goore Game in which each decision maker is inherently Bayesian in nature, yet avoids computational intractability by relying simply on updating the hyper parameters of sibling conjugate priors, and on random sampling from these posteriors. We further report theoretical results on the variance of the random rewards experienced by each individual decision maker. Based on these theoretical results, each decision maker is able to accelerate its own learning by taking advantage of the increasingly more reliable feedback that is obtained as exploration gradually turns into exploitation in bandit problem based learning. Extensive experiments, involving QoS control in simulated wireless sensor networks, demonstrate that the accelerated learning allows us to combine the benefits of conservative learning, which is high accuracy, with the benefits of hurried learning, which is fast convergence. In this manner, our scheme outperforms recently proposed Goore Game solution schemes, where one has to trade off accuracy with speed. As an additional benefit, performance also becomes more stable. We thus believe that our methodology opens avenues for improved performance in a number of applications of bandit based decentralized decision making

A two-armed bandit based scheme for accelerated decentralized learning

The two-armed bandit problem is a classical optimization problem where a decision maker sequentially pulls one of two arms attached to a gambling machine, with each pull resulting in a random reward. The reward distributions are unknown, and thus, one must balance between exploiting existing knowledge about the arms, and obtaining new information. Bandit problems are particularly fascinating because a large class of real world problems, including routing, QoS control, game playing, and resource allocation, can be solved in a decentralized manner when modeled as a system of interacting gambling machines. Although computationally intractable in many cases, Bayesian methods provide a standard for optimal decision making. This paper proposes a novel scheme for decentralized decision making based on the Goore Game in which each decision maker is inherently Bayesian in nature, yet avoids computational intractability by relying simply on updating the hyper parameters of sibling conjugate priors, and on random sampling from these posteriors. We further report theoretical results on the variance of the random rewards experienced by each individual decision maker. Based on these theoretical results, each decision maker is able to accelerate its own learning by taking advantage of the increasingly more reliable feedback that is obtained as exploration gradually turns into exploitation in bandit problem based learning. Extensive experiments demonstrate that the accelerated learning allows us to combine the benefits of conservative learning, which is high accuracy, with the benefits of hurried learning, which is fast convergence. In this manner, our scheme outperforms recently proposed Goore Game solution schemes, where one has to trade off accuracy with speed. We thus believe that our methodology opens avenues for improved performance in a number of applications of bandit based decentralized decision making

Towards Thompson Sampling for Complex Bayesian Reasoning

Paper III, IV, and VI are not available as a part of the dissertation due to the copyright.Thompson Sampling (TS) is a state-of-art algorithm for bandit problems set in a Bayesian framework. Both the theoretical foundation and the empirical efficiency of TS is wellexplored for plain bandit problems. However, the Bayesian underpinning of TS means that TS could potentially be applied to other, more complex, problems as well, beyond the bandit problem, if suitable Bayesian structures can be found. The objective of this thesis is the development and analysis of TS-based schemes for more complex optimization problems, founded on Bayesian reasoning. We address several complex optimization problems where the previous state-of-art relies on a relatively myopic perspective on the problem. These includes stochastic searching on the line, the Goore game, the knapsack problem, travel time estimation, and equipartitioning. Instead of employing Bayesian reasoning to obtain a solution, they rely on carefully engineered rules. In all brevity, we recast each of these optimization problems in a Bayesian framework, introducing dedicated TS based solution schemes. For all of the addressed problems, the results show that besides being more effective, the TS based approaches we introduce are also capable of solving more adverse versions of the problems, such as dealing with stochastic liars.publishedVersio

On incorporating the paradigms of discretization and Bayesian estimation to create a new family of pursuit learning automata

There are currently two fundamental paradigms that have been used to enhance the convergence speed of Learning Automata (LA). The first involves the concept of utilizing the estimates of the reward probabilities, while the second involves discretizing the probability space in which the LA operates. This paper demonstrates how both of these can be simultaneously utilized, and in particular, by using the family of Bayesian estimates that have been proven to have distinct advantages over their maximum likelihood counterparts. The success of LA-based estimator algorithms over the classical, Linear Reward-Inaction (LRI)-like schemes, can be explained by their ability to pursue the actions with the highest reward probability estimates. Without access to reward probability estimates, it makes sense for schemes like the LRI to first make large exploring steps, and then to gradually turn exploration into exploitation by making progressively smaller learning steps. However, this behavior becomes counter-intuitive when pursuing actions based on their estimated reward probabilities. Learning should then ideally proceed in progressively larger steps, as the reward probability estimates turn more accurate. This paper introduces a new estimator algorithm, the Discretized Bayesian Pursuit Algorithm (DBPA), that achieves this by incorporating both the above paradigms. The DBPA is implemented by linearly discretizing the action probability space of the Bayesian Pursuit Algorithm (BPA) (Zhang et al. in IEA-AIE 2011, Springer, New York, pp. 608-620, 2011). The key innovation of this paper is that the linear discrete updating rules mitigate the counter-intuitive behavior of the corresponding linear continuous updating rules, by augmenting them with the reward probability estimates. Extensive experimental results show the superiority of DBPA over previous estimator algorithms. Indeed, the DBPA is probably the fastest reported LA to date. Apart from the rigorous experimental demonstration of the strength of the DBPA, the paper also briefly records the proofs of why the BPA and the DBPA are ε{lunate}-optimal in stationary environments

A day at the races : using best arm identification algorithms to reduce the cost of information retrieval user studies

Two major barriers to conducting user studies are the costs involved in recruiting participants and researcher time in performing studies. Typical solutions are to study convenience samples or design studies that can be deployed on crowd-sourcing platforms. Both solutions have benefits but also drawbacks. Even in cases where these approaches make sense, it is still reasonable to ask whether we are using our resources – participants’ and our time – efficiently and whether we can do better. Typically user studies compare randomly-assigned experimental conditions, such that a uniform number of opportunities are assigned to each condition. This sampling approach, as has been demonstrated in clinical trials, is sub-optimal. The goal of many Information Retrieval (IR) user studies is to determine which strategy (e.g., behaviour or system) performs the best. In such a setup, it is not wise to waste participant and researcher time and money on conditions that are obviously inferior. In this work we explore whether Best Arm Identification (BAI) algorithms provide a natural solution to this problem. BAI methods are a class of Multi-armed Bandits (MABs) where the only goal is to output a recommended arm and the algorithms are evaluated by the average payoff of the recommended arm. Using three datasets associated with previously published IR-related user studies and a series of simulations, we test the extent to which the cost required to run user studies can be reduced by employing BAI methods. Our results suggest that some BAI instances (racing algorithms) are promising devices to reduce the cost of user studies. One of the racing algorithms studied, Hoeffding, holds particular promise. This algorithm offered consistent savings across both the real and simulated data sets and only extremely rarely returned a result inconsistent with the result of the full trial. We believe the results can have an important impact on the way research is performed in this field. The results show that the conditions assigned to participants could be dynamically changed, automatically, to make efficient use of participant and experimenter time

A Learning Automaton-based Scheme for Scheduling Domestic Shiftable Loads in Smart Grids

In this paper, we consider the problem of scheduling shiftable loads, over multiple users, in smart electrical grids. We approach the problem, which is becoming increasingly pertinent in our present energy-thirsty society, using a novel distributed game-theoretic framework. In our specific instantiation, we consider the scenario when the power system has a local-area Smart Grid (SG) subnet comprising of a single power source and multiple customers. The objective of the exercise is to tacitly control the total power consumption of the customers’ shiftable loads so to approach the rigid power budget determined by the power source, but to simultaneously not exceed this threshold. As opposed to the “traditional” paradigm that utilizes a central controller to achieve the load scheduling, we seek to achieve this by pursuing a distributed approach that allows the users¹ to make individual decisions by invoking negotiations with other customers. The decisions are essentially of the sort where the individual users can choose whether they want to be supplied or not. From a modeling perspective, the distributed scheduling problem is formulated as a game, and in particular, a so-called “Potential” game. This game has at least one pure strategy Nash Equilibrium (NE), and we demonstrate that the NE point is a global optimal point. The solution that we propose, which utilize

A Learning Automaton-based Scheme for Scheduling Domestic Shiftable Loads in Smart Grids

In this paper, we consider the problem of scheduling shiftable loads, over multiple users, in smart electrical grids. We approach the problem, which is becoming increasingly pertinent in our present energy-thirsty society, using a novel distributed game-theoretic framework. In our specific instantiation, we consider the scenario when the power system has a local-area Smart Grid (SG) subnet comprising of a single power source and multiple customers. The objective of the exercise is to tacitly control the total power consumption of the customers’ shiftable loads so to approach the rigid power budget determined by the power source, but to simultaneously not exceed this threshold. As opposed to the “traditional” paradigm that utilizes a central controller to achieve the load scheduling, we seek to achieve this by pursuing a distributed approach that allows the users¹ to make individual decisions by invoking negotiations with other customers. The decisions are essentially of the sort where the individual users can choose whether they want to be supplied or not. From a modeling perspective, the distributed scheduling problem is formulated as a game, and in particular, a so-called “Potential” game. This game has at least one pure strategy Nash Equilibrium (NE), and we demonstrate that the NE point is a global optimal point. The solution that we propose, which utilizes the theory of Learning Automata (LA), permits the total supplied loads to approach the power budget of the subnet once the algorithm has converged to the NE point. The scheduling is achieved by attaching a LA to each customer. The paper discusses the applicability of three different LA schemes, and in particular the recently-introduced Bayesian Learning Automata (BLA). Numerical results, obtained from testing the schemes on numerous simulated datasets, demonstrate the speed and the accuracy of proposed algorithms in terms of their convergence to the game’s NE point.publishedVersionNivå