Partially Observable Markov Decision Processes (POMDPs)
Partially Observable Markov Decision Processes (POMDPs)are useful to model many real-world problems, where the actual state is unknown to a decision making agent due to limited and noisy sensors. The agent has to consider the history of its past observations and actions to maintain a belief state as a distribution of possible states [Kaelbling et al.1998].
Solving POMDPs exactly is computationally intractable for domains with extremely large state spaces and long planning horizons due to the curse of dimensionality, where the space of possible belief states grows exponentially w.r.t. the number of states [Kaelbling et al.1998], and the curse of history, where the number of possible histories grows exponentially w.r.t. the horizon length [Pineau et al.2006].
Monte-Carlo planning algorithms are popular approaches to decision making in large POMDPs due to breaking both curses with statistical sampling and black box simulation [Silver and Veness2010, Somani et al.2013]. State-of-the-art algorithms like POMCP construct sparse closed-loop search trees over belief states and actions. Although these approaches are computationally efficient, the search trees can become arbitrarily large in highly complex domains. In such domains, performance could be limited by restricted memory resources, which is common in sensor networks or IoT settings with intelligent devices that have to make decisions with very limited resources and perception capabilities.
Open-loop planning is an alternative approach to efficient decision making, which only optimizes sequences of actions independently of the belief state space. Although open-loop planning generally converges to suboptimal solutions, it has been shown to be competitive against closed-loop planning in practice, when the problem is too large to provide sufficient computational resources [Weinstein and Littman2013, Perez Liebana et al.2015, Lecarpentier et al.2018]. However, open-loop approaches have been rarely used in POMDPs so far, despite their potential to efficient planning in very large domains [Yu et al.2005, Phan et al.2019].
In this paper, 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, which is matched to successive time steps of the decision process. The stack size is bounded by the planning horizon and can be automatically adapted on demand according to the underlying domain without any prior domain knowledge beyond a generative model.
We empirically test SYMBOL in four large 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.
A POMDP is defined by a tuple , where is a (finite) set of states, is the (finite) set of actions,
is the transition probability function,is the reward function, is a (finite) set of observations, is the observation probability function, and is a probabilitiy distribution over initial states [Kaelbling et al.1998]. It is always assumed that , , and at time step .
A history is a sequence of actions and observations. A belief state is a sufficient statistic for history
and defines a probability distribution over statesgiven .
is the space of all possible belief states. The belief state can be updated by Bayes theorem, where is a normalizing constant, is the last action, and is the history without and .
The goal is to find a policy , which maximizes the expectation of return for a horizon :
where is the discount factor.
can be evaluated with a value function , which is the expected return conditioned on belief states. An optimal policy has a value function , where for all and all .
2.2 Multi-armed Bandits
Multi-armed Bandits (MABs) are decision making problems with a single state . An agent has to repeatedly select an action in order to maximize its expected reward , where
is a random variable with an unknown distribution
. The agent has to balance between exploring actions to estimate their expected reward and exploiting its knowledge on all actions by selecting the action with the currently highest expected reward. This is theexploration-exploitation dilemma, where exploration can lead to actions with possibly higher rewards but requires time for trying them out, while exploitation can lead to fast convergence but possibly gets stuck in a local optimum. UCB1 and Thompson Sampling are possible approaches to solve MABs.
selects actions by maximizing the upper confidence bound of action values , where is the average reward of action , is an exploration constant, is the total number of action selections, and is the number of times action was selected. The second term is the exploration bonus, which becomes smaller with increasing [Auer et al.2002, Kocsis and Szepesvári2006].
is a Bayesian approach to balance between exploration and exploitation of actions [Thompson1933]. The unknown reward distribution of of each action is modeled by a parametrized likelihood function
with parameter vector. Given a prior distribution and a set of past observed rewards , the posterior distribution can be inferred by using Bayes rule . The expected reward of each action can be estimated by sampling to compute . The action with the highest expected reward is selected.
2.3 Online Planning in POMDPs
Planning searches for an (near-)optimal policy given a model of the environment , which usually consists of explicit probability distributions of the POMDP. Unlike global planning, which searches the whole (belief) state space to find an optimal policy , local planning only focuses on finding a policy for the current (belief) state by taking possible future (belief) states into account [Weinstein and Littman2013]. Thus, local planning can be applied online at every time step at the current state to recommend the next action for execution. Local planning is usually restricted to a time or computation budget nb due to strict real-time constraints [Bubeck and Munos2010, Weinstein and Littman2013].
We focus on local Monte-Carlo planning, where is a generative model, which can be used as black box simulator [Silver and Veness2010]. Given and , the simulator provides a sample . Monte-Carlo planning algorithms can approximate and by iteratively simulating and evaluating actions without reasoning about explicit probability distributions of the POMDP.
Local planning can be closed- or open-loop. Closed-loop planning conditions the action selection on histories of actions and observations. Open-loop planning only conditions the action selection on previous sequences of actions (also called open-loop plans or simply plans) and summarized statistics about predecessor (belief) states [Bubeck and Munos2010, Perez Liebana et al.2015]. An example from [Phan et al.2019] is shown in Fig. 1. A closed-loop tree for is shown in Fig. (a)a, while Fig. (b)b shows the corresponding open-loop tree which summarizes the observation nodes of Fig. (a)a within the blue dotted ellipses into history distribution nodes. Open-loop planning can be further simplified by only regarding statistics about the expected return of actions at specific time steps (Fig. (c)c). In that case, a stack of statistics is used to sample and evaluate plans [Weinstein and Littman2013].
Partially Observable Monte-Carlo Planning (POMCP) is a closed-loop approach based on Monte-Carlo Tree Search (MCTS) [Silver and Veness2010]. POMCP uses a search tree of histories with o-nodes representing observations and a-nodes representing actions (Fig. (a)a). The tree is traversed by selecting a-nodes with a policy until a leaf o-node is reached, which is expanded, and its value is estimated with a rollout by using a policy . can be used to integrate domain knowledge into the planning process to focus the search on promising states [Silver and Veness2010]. The observed rewards are recursively accumulated (Eq. 1) to update the value estimate of each node in the simulated path. The original version of POMCP uses UCB1 for and converges to the optimal best-first tree given sufficient computation [Silver and Veness2010].
[Lecarpentier et al.2018] formulates an open-loop variant of MCTS using UCB1 as , called Open-Loop Upper Confidence bound for Trees (OLUCT), which could be easily extended to POMDPs by constructing a tree, which summarizes all o-nodes to history distribution nodes (Fig. (b)b).
Open-loop planning generally converges to suboptimal solutions in stochastic domains, since it ignores (belief) state values and optimizes the summarized values of each node (Fig. (b)b) instead [Lecarpentier et al.2018]. If the problem is too complex to provide sufficient computation budget nb or memory capacity, then open-loop approaches are competitive against closed-loop approaches, since they need to explore a much smaller search space to find an appropriate solution [Weinstein and Littman2013, Perez Liebana et al.2015, Lecarpentier et al.2018].
3 Related Work
Previous stack based approaches to open-loop planning maintain a fixed size stack of statistics over actions to sample open-loop plans with high expected return [Weinstein and Littman2013, Belzner and Gabor2017, Phan et al.2019]. 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 adaptive stack 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 convergence of all preceding bandits to preserve a stationary state and reward distribution for proper convergence of all bandits.
[Yu et al.2005] proposed an open-loop approach to decision making in POMDPs by using hierarchical planning. An open-loop plan is constructed at an abstract level, where uncertainty w.r.t. particular actions is ignored. A low-level planner controls the actual execution by explicitly dealing with uncertainty. SYMBOL is more general, since it performs planning directly on the original problem by using a generative model for black box optimization and does not require the POMDP to be transformed for hierarchical planning.
[Powley et al.2017] proposed a memory bounded version of MCTS with a fixed size state pool to add, discard, or reuse states depending on their visitation frequency. However, this approach cannot be easily adapted to tree-based open-loop approaches, because it requires (belief) states to be identifiable. SYMBOL does not require a pool to reuse states or nodes but maintains an adaptive stack of Thompson Sampling bandits. The bandits adapt according to the temporal dependencies between actions, while the size of the bandit stack is bounded by the planning horizon and automatically adapts itself according to the underlying domain.
4 Adaptive Thompson Sampling Stacks
4.1 Generalized Thompson Sampling
follows a Normal distributionwith unknown mean and precision , where
is the variance.
follows a Normal Gamma distributionwith , , and . The distribution over is a Gamma distribution and the conditional distribution over given is a Normal distribution .
Given a prior distribution and observations , the posterior distribution is defined by , where , , , and . is the mean of all values in and is the variance.
The posterior is inferred for each action to sample an estimate for the expected return. The action with the highest is selected. The complete formulation is given in Algorithm 1. A MAB stores , , and for each action . In UpdateBandit, the absolute difference between the old and the new mean value of is returned to evaluate the convergence of .
If sufficient domain knowledge for defining the prior is unavailable, the prior should be chosen such that all possibilities can be sampled (almost) uniformly [Bai et al.2014]. This can be achieved by choosing the priors such that the variance of the resulting Normal distribution becomes infinite ( and ). Since follows a Gamma distribution with expectation , and should be chosen such that . Given the hyperparameter space , , and , it is recommended to set and to center the Normal distribution. should be small enough and should be sufficiently large [Bai et al.2014].
Stable Yet Memory Bounded Open-Loop (SYMBOL) planning is a partially observable open-loop approach, which optimizes an adaptive stack of nMAB Thompson Sampling bandits (with ) to maximize the expected return.
Initially beginning with a single MAB , a simulation starts at state , which is sampled from an approximated belief state 111We use a particle filter for [Silver and Veness2010].. The first nMAB actions are sampled from all MABs in the current stack. The remaining actions are sampled from a rollout policy , which can be random or enhanced with domain knowledge [Silver and Veness2010]. The sampled plan is evaluated with to observe rewards , which are accumulated to returns according to Eq. 1. The first nMAB returns are used to update the MAB stack. A MAB is only updated or created, when all of its predecessors with converged. We assume that converged, if , where is the average of the last values of from previous updates to (Algorithm 1). The parameter copes with the non-stationarity of the return values to update each MAB, which is caused by the adaptation of the action selection.
The complete formulation of SYMBOL is given in Algorithm 2, where is the action-observation history, is the planning horizon, nb is the computation budget, is the convergence tolerance represented by the number of MAB updates to be considered, and is the convergence threshold.
Thompson Sampling bandits are able to converge, if their reward distributions are stationary [Agrawal and Goyal2013]. A reward distribution is stationary, if both the underlying state distribution and the successor policy are stationary. The former is ensured, if all preceding MABs converged, since their actions affect the underlying state distribution. The latter is ensured by keeping all successing MABs fixed (they are not updated unless the predecessors converged) and by using a stationary rollout policy .
Starting from the first MAB , we know that the state distribution is stationary. If the successor policy is stationary as well, will converge to the best action given [Agrawal and Goyal2013]. By induction, we can show the same for all successor MABs, given that all predecessor MABs converged. If significantly changed such that , all MABs with must remain fixed to ensure a stationary reward distribution to enable convergence of first. By adjusting the parameters and , the size and speed of convergence of the MAB stack can be controlled.
5.1 Evaluation Environments
The RockSample(n,k) problem simulates an agent moving in an grid, which contains rocks [Smith and Simmons2004]. Each rock can be or , but the true state of each rock is unknown. The agent has to sample good rocks, while avoiding to sample bad rocks. It has a noisy sensor, which produces an observation for a particular rock. The probability of sensing the correct state of the rock decreases exponentially with the agent’s distance to that rock. Sampling gives a reward of , if the rock is good, and otherwise. If a good rock was sampled, it becomes bad. Moving past the east edge of the grid gives a reward of and the episode terminates. We set .
In Battleship five ships of size 1, 2, 3, 4, and 5 respectively are randomly placed into a grid, where the agent has to sink all ships without knowing their actual positions [Silver and Veness2010]. Each cell hitting a ship gives a reward of . There is a reward of per time step and a terminal reward of for hitting all ships. We set .
PocMan is a partially observable version of PacMan [Silver and Veness2010]. The agent navigates in a maze and has to eat randomly distributed food pellets and power pills. There are four ghosts moving randomly in the maze. If the agent is within the visible range of a ghost, it is getting chased by the ghost and dies, if it touches the ghost, terminating the episode with a reward of . Eating a power pill enables the agent to eat ghosts for 15 time steps. In that case, the ghosts will run away, if the agent is under the effect of a power pill. At each time step a reward of is given. Eating food pellets gives a reward of and eating a ghost gives . The agent can only perceive ghosts, if they are in its direct line of sight in each cardinal direction or within a hearing range. Also, the agent can only sense walls and food pellets, which are adjacent to it. We set .
We implemented different partially observable planning algorithms to compare with SYMBOL 222Code available at https://github.com/thomyphan/planning. All algorithms with a rollout phase use a policy which randomly selects actions from a set of legal actions , depending on the currently simulated state . Since open-loop planning can encounter different states at the same node or time step (Fig. 1), the set of legal actions may vary for each state . Thus, we mask out currently illegal actions, regardless of whether they have high action values.
We use the POMCP implementation from [Silver and Veness2010]. selects actions from with UCB1. In each simulation step, there is at most one expansion step, where new nodes are added to the search tree. Thus, the tree size should increase linearly w.r.t. nb in large POMDPs.
POOLUCT and POOLTS
are implemented as open-loop versions of POMCP (Fig. (b)b), where actions are selected from using UCB1 (POOLUCT) or Thompson Sampling (POOLTS) as node selection policy . Similarly to POMCP, the search tree size should increase linearly w.r.t. nb, but with less nodes, since open-loop trees store summarized information about history distributions (Fig. (b)b).
uses an adaptive stack of nMAB Thompson Sampling bandits according to Algorithm 2. Starting at , all MABs apply Thompson Sampling to , depending on the currently simulated state . If , then is used. Given a horizon of , SYMBOL always maintains MABs. Although nMAB depends on , , and nb, it never exceeds (Algorithm 2).
Partially Observable Stacked Thompson Sampling (POSTS)
uses a fixed size stack of Thompson Sampling bandits [Phan et al.2019]. Similarly to SYMBOL, all MABs apply Thompson Sampling to , depending on the currently simulated state . Unlike SYMBOL, all MABs are updated simultaneously according to regardless of the convergence of the preceding MABs as suggested in [Weinstein and Littman2013, Belzner and Gabor2017, Phan et al.2019].
We ran each approach on RockSample, Battleship, and PocMan with different settings for 100 times or at most 12 hours of total computation. We evaluated the performance of each approach with the undiscounted return (), because we focus on the actual effectiveness instead of the quality of optimization [Bai et al.2014]. For POMCP and POOLUCT we set the UCB1 exploration constant to the reward range of each domain as proposed in [Silver and Veness2010].
Since we assume no additional domain knowledge, we focus on uninformative priors with , , and [Bai et al.2014]. With this setting, controls the degree of initial exploration during the planning phase, thus we only vary for POOLTS, POSTS, and SYMBOL. To preserve readability of the figures, we only provide the best configuration of POOLTS and POSTS for comparison.
5.3.1 Hyperparameter Sensitivity
We evaluated the sensitivity of SYMBOL w.r.t. the convergence threshold . Fig. 2a-d show the performance of SYMBOL with compared to all other approaches described in Section 5.2. All SYMBOL variants are able to keep up with their tree-based counterpart POOLTS, while outperforming POOLUCT. SYMBOL is able to keep up with POMCP in RockSample(11,11) and Battleship. POMCP outperforms all open-loop approaches in PocMan. SYMBOL scales better in performance with increasing nb than POSTS, which seems to converge prematurely after . Except in PocMan, SYMBOL scales slightly better with increasing nb when , probably due to more stable convergence of the MABs. Fig. 2e-h show the average stack sizes nMAB of SYMBOL for 333Using budgets between 1024 and 16384 led to similar plots, thus we stick to as suggested in [Phan et al.2019]., , and different . In RockSample, , when , but it does not grow any further. In Battleship, , but the stack size slightly increases w.r.t . In PocMan, nMAB quickly increases w.r.t . does not have any significant impact on nMAB.
We also experimented with the convergence tolerance but did not observe significantly different results than shown in Fig. 2. nMAB tends to decrease with increasing , which is due to the amount of time required to consider a MAB as converged. When , then SYMBOL was less stable in all domains (except in Battleship), leading to high variance in performance when nb is large.
5.3.2 Performance-Memory Tradeoff
We evaluated the performance-memory tradeoff of all approaches by introducing a memory capacity nMEM, where the computation is interrupted, when the number of nodes exceeds nMEM. For POMCP, we count the number of o-nodes and a-nodes (Fig. (a)a). For POOLTS and POOLUCT, we count the number of history distribution nodes (Fig. (b)b). For SYMBOL and POSTS, we count nMAB. POSTS always uses a planning horizon of to satisfy the memory bound. The results are shown in Fig. 3 for , , for POOLTS, POSTS, and SYMBOL, , and .
In RockSample and Battleship, POMCP is outperformed by SYMBOL and POOLTS. SYMBOL always performs best in these domains, when . POMCP performs best in PocMan by outperforming SYMBOL, when and POOLTS keeps up with SYMBOL, when . SYMBOL always outperforms POSTS, while using a lower maximum number of MABs. POSTS is only able to keep up with the best SYMBOL setting in RockSample(11,11) after creating 100 MABs, while SYMBOL only uses about 20 MABs for planning. POOLUCT performs worst except in Battleship, improving less and slowest with increasing nMEM.
We presented SYMBOL, a general memory bounded approach to partially observable open-loop planning with an adaptive stack of Thompson Sampling bandits.
Our experiments show that SYMBOL is a good alternative to tree-based planning in POMDPs. SYMBOL is competitive against tree-based open-loop planning like POOLUCT and POOLTS and is able to keep up with POMCP in domains with large action spaces and low stochasticity like RockSample or Battleship. SYMBOL is robust w.r.t. the choice of the hyperparameters and in terms of performance, with strongly affecting the memory consumption in domains with high stochasticity as shown for PocMan in Fig. (h)h. should be sufficiently small to ensure stable convergence of the MABs, although more computation budget will be required to build up adequate MAB stacks, if is too small. Fig. 2e-h indicate that appropriate MAB stack sizes are highly domain dependent and cannot be generally specified beforehand without expensive parameter tuning. Thus, adaptive and robust approaches like SYMBOL seem to be promising for general and efficient decision making in POMDPs.
When restricting the memory capacity, SYMBOL clearly outperforms all tree-based approaches, while requiring significantly less MABs. Although being bounded by at most, SYMBOL always created much less MABs in all domains, resulting in extremely memory efficient planning (Fig. 3). POOLTS requires thousands of nodes to keep up with SYMBOL, while POMCP is only able to outperform SYMBOL in PocMan after creating more than 100 nodes, which still consumes much more memory than SYMBOL (Fig. (d)d).
SYMBOL is able to outperform the fixed size stack approach POSTS, showing the effectiveness of the adaptive stack concept, where proper convergence is ensured by the convergence threshold and the convergence tolerance .
While state-of-the-art approaches to efficient online planning [Silver and Veness2010, Somani et al.2013, Bai et al.2014] heavily rely on sufficient memory resources in highly complex domains, SYMBOL is a memory bounded alternative, which maintains an adaptive stack of MABs. SYMBOL is able to automatically adapt its stack according to the underlying domain without any prior domain knowledge.
In the future, we plan to integrate SYMBOL into hierarchical planning to optimize macro-actions for certain subgoals.
- [Agrawal and Goyal2013] Shipra Agrawal and Navin Goyal. Further Optimal Regret Bounds for Thompson Sampling. In Artificial Intelligence and Statistics, pages 99–107, 2013.
- [Auer et al.2002] Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-Time Analysis of the Multiarmed Bandit Problem. Machine learning, 47(2-3):235–256, 2002.
- [Bai et al.2013] Aijun Bai, Feng Wu, and Xiaoping Chen. Bayesian Mixture Modelling and Inference based Thompson Sampling in Monte-Carlo Tree Search. In Advances in Neural Information Processing Systems, pages 1646–1654, 2013.
- [Bai et al.2014] Aijun Bai, Feng Wu, Zongzhang Zhang, and Xiaoping Chen. Thompson Sampling based Monte-Carlo Planning in POMDPs. In Proceedings of the Twenty-Fourth International Conference on Automated Planning and Scheduling, pages 29–37. AAAI Press, 2014.
- [Belzner and Gabor2017] Lenz Belzner and Thomas Gabor. Stacked Thompson Bandits. In Proceedings of the 3rd International Workshop on Software Engineering for Smart Cyber-Physical Systems, pages 18–21. IEEE Press, 2017.
- [Bubeck and Munos2010] Sébastien Bubeck and Rémi Munos. Open Loop Optimistic Planning. In COLT, pages 477–489, 2010.
- [Chapelle and Li2011] Olivier Chapelle and Lihong Li. An Empirical Evaluation of Thompson Sampling. In Advances in neural information processing systems, pages 2249–2257, 2011.
- [Kaelbling et al.1998] Leslie Pack Kaelbling, Michael L Littman, and Anthony R Cassandra. Planning and Acting in Partially Observable Stochastic Domains. Artificial intelligence, 101(1):99–134, 1998.
- [Kaufmann et al.2012] Emilie Kaufmann, Nathaniel Korda, and Rémi Munos. Thompson Sampling: An Asymptotically Optimal Finite-Time Analysis. In International Conference on Algorithmic Learning Theory, pages 199–213. Springer, 2012.
- [Kocsis and Szepesvári2006] Levente Kocsis and Csaba Szepesvári. Bandit based Monte-Carlo Planning. In ECML, volume 6, pages 282–293. Springer, 2006.
- [Lecarpentier et al.2018] Erwan Lecarpentier, Guillaume Infantes, Charles Lesire, and Emmanuel Rachelson. Open Loop Execution of Tree-Search Algorithms. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, pages 2362–2368. IJCAI Organization, 7 2018.
[Perez Liebana et al.2015]
Diego Perez Liebana, Jens Dieskau, Martin Hunermund, Sanaz Mostaghim, and Simon
Open Loop Search for General Video Game Playing.
Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, pages 337–344. ACM, 2015.
- [Phan et al.2019] Thomy Phan, Lenz Belzner, Marie Kiermeier, Markus Friedrich, Kyrill Schmid, and Claudia Linnhoff-Popien. Memory Bounded Open-Loop Planning in Large POMDPs Using Thompson Sampling. 33th AAAI Conference on Artificial Intelligence, 2019.
- [Pineau et al.2006] Joelle Pineau, Geoffrey Gordon, and Sebastian Thrun. Anytime Point-based Approximations for Large POMDPs. Journal of Artificial Intelligence Research, 27:335–380, 2006.
- [Powley et al.2017] Edward Powley, Peter Cowling, and Daniel Whitehouse. Memory Bounded Monte Carlo Tree Search. AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, 2017.
- [Silver and Veness2010] David Silver and Joel Veness. Monte-Carlo Planning in Large POMDPs. In Advances in neural information processing systems, pages 2164–2172, 2010.
- [Silver et al.2017] David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the Game of Go without Human Knowledge. Nature, 550(7676):354–359, 2017.
- [Smith and Simmons2004] Trey Smith and Reid Simmons. Heuristic Search Value Iteration for POMDPs. In Proceedings of the 20th conference on Uncertainty in artificial intelligence, pages 520–527. AUAI Press, 2004.
- [Somani et al.2013] Adhiraj Somani, Nan Ye, David Hsu, and Wee Sun Lee. DESPOT: Online POMDP Planning with Regularization. In Advances in neural information processing systems, pages 1772–1780, 2013.
- [Thompson1933] William R Thompson. On the Likelihood that One Unknown Probability exceeds Another in View of the Evidence of Two Samples. Biometrika, 25(3/4):285–294, 1933.
- [Weinstein and Littman2013] Ari Weinstein and Michael L Littman. Open-loop Planning in Large-Scale Stochastic Domains. In Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence, pages 1436–1442. AAAI Press, 2013.
- [Yu et al.2005] C Yu, Jason Chuang, Brian Gerkey, G Gordon, and Andrew Ng. Open-Loop Plans in Multi-Robot POMDPs. Technical report, Stanford CS Dept, 2005.