We propose Stable Yet Memory Bounded Open-Loop (SYMBOL) planning, a general memory bounded approach to partially observable open-loop planning. SYMBOL maintains an adaptive stack of Thompson Sampling bandits, whose size is bounded by the planning horizon and can be automatically adapted according to the underlying domain without any prior domain knowledge beyond a generative model. We empirically test SYMBOL in four large POMDP benchmark problems to demonstrate its effectiveness and robustness w.r.t. the choice of hyperparameters and evaluate its adaptive memory consumption. We also compare its performance with other open-loop planning algorithms and POMCP.
Previous stack based approaches to open-loop planning maintain a fixed size stack of @math statistics over actions to sample open-loop plans with high expected return @cite_3 @cite_5 @cite_12 . While these approaches work well in practice, their convergence properties remain unclear because of the non-stationarity of the sampling statistics and the underlying state distributions due to the simultaneous adaptation of each statistic. Furthermore, the required number of statistics is highly domain dependent and hard to prespecify. SYMBOL maintains an of Thompson Sampling bandits, which automatically adjusts its size according to the underlying domain without prior domain knowledge. The creation and adaptation of each bandit depends on the bandits to preserve a stationary state and reward distribution for proper convergence of all bandits.
@cite_5 @cite_12 @cite_3
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We introduce Stacked Thompson Bandits (STB) for efficiently generating plans that are likely to satisfy a given bounded temporal logic requirement. STB uses a simulation for evaluation of plans, and takes a Bayesian approach to using the resulting information to guide its search. In particular, we show that stacking multiarmed bandits and using Thompson sampling to guide the action selection process for each bandit enables STB to generate plans that satisfy requirements with a high probability while only searching a fraction of the search space. The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP. We consider stochastic multi-armed bandit problems with complex actions over a set of basic arms, where the decision maker plays a complex action rather than a basic arm in each round. The reward of the complex action is some function of the basic arms' rewards, and the feedback observed may not necessarily be the reward perarm. For instance, when the complex actions are subsets of the arms, we may only observe the maximum reward over the chosen subset. Thus, feedback across complex actions may be coupled due to the nature of the reward function. We prove a frequentist regret bound for Thompson sampling in a very general setting involving parameter, action and observation spaces and a likelihood function over them. The bound holds for discretely-supported priors over the parameter space without additional structural properties such as closed-form posteriors, conjugate prior structure or independence across arms. The regret bound scales logarithmically with time but, more importantly, with an improved constant that non-trivially captures the coupling across complex actions due to the structure of the rewards. As applications, we derive improved regret bounds for classes of complex bandit problems involving selecting subsets of arms, including the first nontrivial regret bounds for nonlinear MAX reward feedback from subsets. Using particle filters for computing posterior distributions which lack an explicit closed-form, we present numerical results for the performance of Thompson sampling for subset-selection and job scheduling problems. Thompson Sampling has recently been shown to achieve the lower bound on regret in the Bernoulli Multi-Armed Bandit setting. This bandit problem assumes stationary distributions for the rewards. It is often unrealistic to model the real world as a stationary distribution. In this paper we derive and evaluate algorithms using Thompson Sampling for a Switching Multi-Armed Bandit Problem. We propose a Thompson Sampling strategy equipped with a Bayesian change point mechanism to tackle this problem. We develop algorithms for a variety of cases with constant switching rate: when switching occurs all arms change (Global Switching), switching occurs independently for each arm (PerArm Switching), when the switching rate is known and when it must be inferred from data. This leads to a family of algorithms we collectively term Change-Point Thompson Sampling (CTS). We show empirical results in 4 articial environments, and 2 derived from real world data: news click-through[Yahoo!, 2011] and foreign exchange data[Dukascopy, 2012], comparing them to some other bandit algorithms. In real world data CTS is the most eective.