Iterative Hierarchical Optimization for Misspecified Problems (IHOMP)

Iterative Hierarchical Optimization for Misspecified Problems (IHOMP)

Abstract

For complex, high-dimensional Markov Decision Processes (MDPs), it may be necessary to represent the policy with function approximation. A problem is misspecified whenever, the representation cannot express any policy with acceptable performance. We introduce IHOMP : an approach for solving misspecified problems. IHOMP iteratively learns a set of context specialized options and combines these options to solve an otherwise misspecified problem. Our main contribution is proving that IHOMP enjoys theoretical convergence guarantees. In addition, we extend IHOMP to exploit Option Interruption (OI) enabling it to decide where the learned options can be reused. Our experiments demonstrate that IHOMP can find near-optimal solutions to otherwise misspecified problems and that OI can further improve the solutions.

1 Introduction

Reinforcement Learning (RL) algorithms can learn near-optimal solutions to well-defined problems. However, real-world problems rarely come in the form of a concrete problem description. A human has to translate the poorly defined target problem into a concrete problem description. A Misspecified Problem (MP) occurs when an optimal solution to the problem description is inadequate in the target problem. Unfortunately, creating a well-defined problem description is a challenging art. Furthermore, MPs can have serious consequences in many domains ranging from smart-grids \citepAbiri2013,Wu2010 and robotics \citepSmart2002 to inventory management systems \citepMann2014a. In this paper, we introduce a hierarchical approach that mitigates the consequences of problem misspecification.

RL problems are often described as Markov Decision Processes \citep[MDPs]Sutton1998. A solution to a MDP is a function that generates an action when presented with the current state, called a policy. The solution to a MDP is any policy that maximizes the long term sum of rewards. For problems with continuous, high dimensional state-spaces, explicitly representing the policy is infeasible, thus for the remainder of this paper we restrict our discussion to linearly parametrized policy representations \citepSutton1996,Roy2013.1

Why are problems misspecified? While a problem description can be misspecified for many reasons, one important case is due to the state representation. It is well established in the machine learning \citepLevi2004,Zhou2009 and RL \citepKonidaris2011 literature that “good” features can have a dramatic impact on performance. Finding “good” features to represent the state is a challenging domain specific problem that is generally considered outside of the scope of RL. Unfortunately, domain experts may not supply useful features either because they do not fully understand the target problem or the technicalities of reinforcement learning.

In addition, we may prefer a MP with a limited state representation for several reasons: (1) Regularization: We wish to have a limited feature representation to improve the generalization and avoid overfitting \citepSingh1995,Geramifard2012. (2) Memory and System constraints: Only a finite number of the features can be used due to computational constraints \citepRoy2013,Singh1995. In real-time systems, querying a feature may take too long. In physical systems, the sensor required to measure a desired feature may be prohibitively expensive. (3) Learning on Large Data: After learning on large amounts of data, augmenting a feature set with new features to get improved performance is non-trivial and often inefficient \citepGeramifard2012.

How can we mitigate misspecification? Learning a hierarchical policy can mitigate the problems associated with a MP and contrast this against a flat policy approach where a single, parameterized policy is used to solve the entire MDP.

To illustrate how learning a hierarchical policy can repair MPs, consider the S-shaped domain shown in Figure 1. To solve the task the agent must move from the bottom left corner to the goal region denoted by the letter ‘G’ in the top right. The state representation only permits policies that move in a straight line. So the problem is misspecified, and it is not solvable with a flat policy approach (Figure 1). However, if we break up the state-space, as shown in Figure 1, and learn one policy for each cell, the problem is solvable.

The partial policies shown in Figure 1 are an example of abstract actions, called options \citepSutton1999, macro-actions \citepHauskrecht1998,He2011, or skills \citepKonidaris2009. Learning useful options has been a topic of intense research \citepMcGovern2001,Moerman2009,Konidaris2009,Brunskill2014,Hauskrecht1998. However, previous approaches have proposed algorithms for learning options to learn or plan faster. In contrast, our objective is to learn options to repair a MP.

Proposed Algorithm: We introduce a meta-algorithm, Iterative Hierarchical Optimization for Misspecified Problems (IHOMP), that uses an RL algorithm as a “black box” to iteratively learn options that repair MPs. To force the options to specialize, IHOMP uses a partition of the state-space and trains one option for each class in the partition (Figure 1). Any arbitrary partitioning scheme can be used, however the partition impacts performance. During an iteration of IHOMP, an RL algorithm updates each option. The options may be initialized arbitrarily, but after the first iteration options with access to a goal region or non-zero rewards will learn how to exploit those rewards (e.g., Figure 1, Iteration 1). On further iterations, the newly acquired options propagate reward back to other regions of the state-space. Thus, options that previously had no reward signal exploit the rewards of other options that have received meaningful reward signals (e.g., Figure 1, Iterations 2 and 5). Although each option is only learned over a single partition class, it can be initialized in any state.

Figure 1: () An episodic MDP with S-shaped state-space and goal region . () Flat approach: A single policy failing to solve the entire task resulting in a misspecified model () Hierarchical approach: Using and combining simple policy representations to solve a task. () Learning options, denoted by black arrows. The domain is partitioned into five classes (sub-partitions) resulting in option-set . In iteration , all options except for (which has immediate access to the goal region) are arbitrary. In iteration , propagates reward back to . This process repeats until useful options are learned over the entire state-space.

Why partitions? If all options are trained on all data, then the options would not specialize defeating the purpose of learning multiple policies. Partitions are necessary to foster specialization. Natural partitionings arise in many different applications and are often easy to design. Consider navigation tasks (which we use in this paper for ease of visualization), which are ever-present in robotics \citepSmart2002, where partitions naturally lead an agent from one location to another in the state space. In addition, partitions are well suited to cyclical tasks; that is, tasks that have repeatable cycles (For example, a yearly cycle of 12 months). Here the state space can be easily partitioned based on time. Examples include inventory management systems \citepMann2014a as well as maintenance scheduling of generation units and transmission lines in smart grids \citepAbiri2013,Wu2010.

Automatically Learning partitions: The availability of a pre-defined partitioning of the state space is a strong assumption in some domains. We have developed a relaxation to this assumption that can enable partitions to be learned automatically using Regularized Option Interruption (ROI) \citepMann2014b, Sutton1999.

Contributions: Our main contributions are: (1) Introducing Iterative Hierarchical Optimization for Misspecified Problems (IHOMP), which learns options to repair and solve MPs. (2) Theorem 1 shows that IHOMP converges to a near-optimal solution relating the quality of the learned policy to the quality of the options learned by the “black box” RL algorithm. (3) Theorem 2 proves that Regularized Option Interruption (ROI) can be safely incorporated into IHOMP. (4) Experiments demonstrating that, given a misspecified problem, IHOMP can learn options to repair and solve the problem. Experiments showing IHOMP-ROI learning partitions and discovering reusable options. This divide-and-conquer approach may also enable us to scale and solve larger MDPs.

2 Background

Let be an MDP, where is a (possibly infinite) set of states, is a finite set of actions, is a mapping from state-action pairs to probability distributions over next states, maps each state-action pair to a reward in , and is the discount factor. A policy gives the probability of executing action from state .

Let be an MDP. The value function of a policy with respect to a state is where the expectation is taken with respect to the trajectory produced by following policy . The value function of a policy can also be written recursively as

(1)

which is known as the Bellman equation. The optimal Bellman equation can be written as Let . We say that a policy is -optimal if for all . The action-value function of a policy is defined by for a state and an action , and the optimal action-value function is denoted by . Throughout this paper, we will drop the dependence on when it is clear from the context.

3 Learning Options

An option is typically defined by a triple . However, we want to learn options that are both specialized to specific regions of the state-space but potentially reusable if they are useful in more general contexts. We focus on a special case of options, where an option is defined by a tuple , where is a parametric policy with parameter vector and indicates whether the option has finished () or not () given the current state .

Given a set of options with size , the inter-option policy is defined by where is the state-space and is the index set over the options in . An inter-option policy selects which options to execute from the current state by returning the index of one of the options. By defining inter-option policies to select an index (rather than the options), we can use the same policy even as the set of options is adapting.

Figure 1 shows an arbitrary partitioning , consisting of sub-partitions , defined over the original MDP’s state space. Each is initialized with an arbitrary option and its corresponding Local-MDP . Local-MDP (see supplementary material for a full definition) is an episodic MDP that terminates once the agent escapes from and upon terminating receives a reward equal to the value of the state the agent would have transitioned to in the original MDP. Therefore, we construct a modified MDP called a Local-MDP and apply a planning or RL algorithm to solve it. The resulting solution (policy) is a specialized option.

Given a “good” set of options, planning can be significantly faster \citepSutton1999,Mann2014a. However, in many domains we may not be given a good set of options. Therefore it is necessary to learn and improve this set of options. In the next section, we introduce an algorithm for dynamically learning and improving options using iterative hierarchical optimization.

4 Iterative Hierarchical Optimization for Misspecified Problems (IHOMP)

Iterative Hierarchical Optimization for Misspecified Problems (IHOMP, Algorithm 1) takes the original MDP , a partition over the state-space and a number of iterations and returns a pair containing an inter-option policy and a set of options . The number of options is equal to the number of classes (sub-partitions) in the partition (line 1). The inter-option policy returned by IHOMP is defined (line 2) by where is the indicator function returning if its argument is true and otherwise and denotes the class in the partition . Thus simply returns the index of the option associated with the partition class containing the current state. On line 3, IHOMP initializes with arbitrary options (IHOMP can also be initialized with options that we believe might be useful to speed up learning).

0:  {MDP}, {Partitioning of }, {Iterations}
1:   {# of partitions.}
2:  
3:  Initialize with options. {1 option per partition.}
4:  for  do {Do iterations.}
5:     for  do {One update per option.}
6:        Policy Evaluation:
7:        Evaluate with to obtain
8:        Option Update:
9:        Construct Local-MDP from &
10:        Solve obtaining policy
11:        
12:        Replace in by
13:     end for
14:  end for
15:  return
Algorithm 1 Iterative Hierarchical Optimization for Misspecified Problems (IHOMP)

Next (lines 414), IHOMP performs iterations. In each iteration, IHOMP updates the options in (lines 513). Note that the value of a option depends on how it is combined with other options. If we allowed all options to change simultaneously, the options could not reliably propagate value off of each other. Therefore, IHOMP updates each option individually. Multiple iterations are needed so that the option set can converge (Figure 1).

The process of updating an option (lines 712) starts by evaluating with the current option-set (line 7). Any number of policy evaluation algorithms could be used here, such as TD with function approximation \citepSutton1998 or LSTD \citepBoyan2002, modified to be used with options. In our experiments, we used a straightforward variant of LSTD \citepSorg2010. Then we use the original MDP to construct a Local-MDP (line 9). Next, IHOMP uses a planning or RL algorithm to approximately solve the Local-MDP returning a parametrized policy (line 10). Any planning or RL algorithm for regular MDPs could fill this role provided that it produces a parametrized policy. However, in our experiments, we used a simple actor-critic PG algorithm, unless otherwise stated. Then a new option is created (line 11) where is the policy derived on line 10 and . The definition of means that the option will terminate only if it leaves the partition. Finally, we update the option set by replacing the option with (line 12). It is important to note that in IHOMP, updating an option is equivalent to solving a Local-MDP.

5 Analysis of IHOMP

We provide the first convergence guarantee for combining hierarchically and iteratively learning options in a continuous state MDP using IHOMP (Lemma 1 and Lemma 2, proven in the supplementary material). We use this guarantee as well as Lemma 2 to prove Theorem 1. This theorem enables us to analyze the quality of the inter-option policy returned by IHOMP. It turns out that the quality of the policy depends critically on the quality of the option learning algorithm. An important parameter for determining the quality of a policy returned by IHOMP is the misspecification error defined below.

Definition 1

Let be a partition over the target MDP’s state-space. The misspecification error is

(2)

where is the smallest , such that for all and is the policy returned by the option learning algorithm executed on .

The misspecification error quantifies the quality of the Local-MDP solutions returned by our option learning algorithm. If we used an exact solver to learn options, then . However, if we use an approximate solver, then will be non-zero and the quality will depend on the partition . Generally, using finer grain partitions will decrease . However, Theorem 1 reveals that adding too many options can also negatively impact the returned policy’s quality.

Theorem 1

Let . If we run IHOMP with partition for iterations, then the algorithm returns stitching policy such that

(3)

where is the number of partition classes in .

The proof of Theorem 1 is divided into three parts (a complete proof is given in the supplementary material). The main challenge is that updating one option can impact the value of other options. Our analysis starts by bounding the impact of updating one option. Note that represents a option set and represents a option set where we have updated the option (corresponding to the partition class ) in the set. In the first part, we show that the error between , the globally optimal value function, and , is a contraction when and is bound by otherwise (Lemma 1). In the second part, we apply an inductive argument to show that updating all options results in a contraction over the entire state space (Lemma 2). In the third part, we apply this contraction recursively, which proves Theorem 1.

This provides the first theoretical guarantees of convergence to a near optimal solution when combining hierarchically, and iteratively learning, a set of options in a continuous state MDP. Theorem 1 tells us that when the misspecification error is small, IHOMP returns a near-optimal inter-option policy. The first term on the right hand side of (3) is the approximation error. This is the loss we pay for the parametrized class of policies that we learn options over. Since represents the number of classes defined by the partition, we now have a formal way of analyzing the effect of the partitioning structure. In addition, complex options do not need to be designed by a domain expert; only the partitioning needs to be provided a-priori. The second term is the convergence error. It goes to as the number of iterations increases.

The guarantee provided by Theorem 1 may appear similar to \citep[Theorem 1]Hauskrecht1998. However, [Hauskrecht et al.(1998)Hauskrecht, Meuleau, Kaelbling, Dean, and Boutilier] derive options only at the beginning of the learning process and do not update them. On the other hand, IHOMP updates its option-set dynamically by propagating value throughout the state space during each iteration. Thus, IHOMP does not require prior knowledge of the optimal value function.

Theorem 1 does not explicitly present the effect of policy evaluation error, which occurs with any approximate policy evaluation technique. However, if the policy evaluation error is bounded by , then we can simply replace in (3) with . Again, smaller policy evaluation error leads to smaller approximation error.

6 Learning Partitions via Regularized Option Interruption

So far IHOMP has assumed a partition is given a-priori. However, it may be non-trivial to design a partition and, in many cases, the partition may be sub-optimal. To relax this assumption, we incorporate Regularized Option Interruption (ROI) \citepMann2014b into this work to enable IHOMP to automatically learn a near-optimal partition from an initially misspecified problem.

IHOMP keeps track of the action value function which represents the expected value of being in state and executing option , given the inter-option policy and option set . ROI uses this estimate of the action-value function to enable the agent to choose when to switch options according to the following termination rule:

(4)
Figure 2: Regularized Option Interruption (ROI): () The initial misspecified partition pre-defined by the user. () The actual optimal partitioning for the task. () The partition learned using ROI. If option is executing, the option will continue to execute at location and will terminate at location . At this point option will begin to execute.

Here corresponds to the termination probability of the option partition and . This rule is illustrated in Figure 2. A user has designed a partition resulting in a MP (Figure 2) compared to the optimal partition for this domain (Figure 2). IHOMP applies ROI to ‘modify’ the initial partition into the optimal one. By learning the optimal action-value function , IHOMP builds a near-optimal partition (Figure 2) that is implicitly stored within this action-value function. That is, if the agent is executing an option in partition class , and the value of continuing with option , , is less than for some regularization function (see the location in Figure 2), then switch to the new option partition (). Otherwise, continue executing the current option (see the location in Figure 2).

This leads to a new algorithm IHOMP-ROI (IHOMP with Regularized Option Interruption). The algorithm can be found in the supplementary material. The key difference between IHOMP and IHOMP-ROI is applying ROI during the policy evaluation step after each of the options have been updated. IHOMP-ROI automatically learns an improved partition between iterations. We show that ROI can be safely incorporated into IHOMP in Theorem 2. The theorem shows that incorporating ROI can only improve the policy produced by IHOMP. The full proof is given in the supplementary material.

Theorem 2

(IHOMP-ROI Approximate Convergence) Eq. (3) also holds for IHOMP-ROI.

7 Experiments and Results

We performed experiments on three well-known RL benchmarks: Mountain Car (MC), Puddle World (PW) \citepSutton1996 and the Pinball domain \citepKonidaris2009. We also perform experiments in a sub-domain of Minecraft 2. The MC and Minecraft domains have similar results to PW and therefore have been moved to the supplementary material. We use two variations for the Pinball domain, namely maze-world (moved to supplementary material), which we created, and pinball-world which is one of the standard pinball benchmark domains. Finally, we created a domain which we call the Two Rooms domain to demonstrate how IHOMP-ROI can improve partitions.

In each experiment, we defined a MP, where no flat policy is adequate, and in some of the tasks, cannot solve the task at all. These experiments simulate situations where the policy representation is constrained to avoid overfitting, manage system constraints, or coping with poorly designed features. In each case, IHOMP learns a significantly better policy compared to the non-hierarchical approach. In the Two Rooms domain, IHOMP-ROI improves the initial partition. Our experiments demonstrate potential to scale up to higher dimensional domains by hierarchically combining options over simple representations (A Video of IHOMP solving the Pinball tasks and the Minecraft sub-domain can be found in the supplementary material).

IHOMP is a meta-algorithm. We provide an algorithm for Policy Evaluation (PE) and Policy Learning (PL). For the MC and PW domains, we used SMDP-LSTD \citepSorg2010 for PE and a modified version of Regular-Gradient Actor-Critic (RG-AC) \citepBhatnagar2009 for PL (see supplementary material for details). In the Pinball domains, we used Nearest-Neighbor Function Approximation (NN-FA) for PE and UCB Random Policy Search (UCB-RPS) for PL. In the two rooms domain, we use a variation of LSTDQ with Option Interruption for PE and RG-AC for PL.

For the MC, PW, and Two Rooms domains, each intra-option policy is represented as a probability distribution over actions (independent of the state). We compare their performance to the original misspecified problem using a flat policy with the same representation. Grid-like partitions are generated for each task. Binary-grid features are used to estimate the value function. In the Pinball domains, each option is represented by polynomial features corresponding to each state dimension and a bias term. The value function is represented by a KD-Tree containing state-value pairs uniformly sampled in the domain. A value for a particular state is obtained by assigning the value of the nearest neighbor to that state that is contained within the KD-tree. These are example representations. In principal, any value function and policy representation that is representative of the domain can be utilized.

Figure 3: () Puddle World domain. () The average reward for IHOMPcompared to the solution obtained by a flat policy in the initially misspecified problem and an approximately optimal policy derived using Q-learning. () The average cost (negative reward) for each grid partition.

Puddle World: Puddle World is a continuous 2-dimensional world containing two puddles as shown in Figure 3. A successful agent (red ball) should navigate to the goal location (blue square), avoiding the puddles. The state space is the location of the agent. Initially, the agent is provided with a misspecified problem. That is, a flat policy that can only move in a single direction (thus it cannot avoid the puddles). Figure 3 compares this flat policy with IHOMP (for a grid partition (Four options)). The flat policy achieves low average reward. However, IHOMP turns the flat policy into options and hierarchically composes these options together, resulting in a richer solution space and a higher average reward as seen in Figure 3. This is comparable to the approximately optimal average reward attained by executing Approximate Value Iteration (AVI) for a huge number of iterations. In this experiment IHOMP is not initiated in the partition class containing the goal state but still achieves near-optimal convergence after only iterations.

Figure 3 compares the performance of different partitions where a grid represents the flat policy of the initially misspecified problem. The option learning error is significantly smaller for all the partitions greater than , resulting in lower cost. On the other hand, according to Theorem 1, adding more options increases the cost. A trade off therefore exists between and . In practice, tends to dominate . In addition to the trade off, the importance of the partition design is evident when analyzing the cost of the and grids. In this scenario, the partition design is better suited to Puddle World than the partition, resulting in lower cost.

Pinball: We tested IHOMP on the challenging pinball-world task (Figure 4) [Konidaris and Barto(2009)]. The agent is initially provided with a -feature flat policy . This results in a misspecified problem as the agent is unable to solve the task using this limited representation as shown by the average reward in Figure 4. Using IHOMP with a grid, options were learned. IHOMP clearly outperforms the flat policy as shown in Figure 4. It is less than optimal but still manages to sufficiently perform the task (see value function, Figure 4). The drop in performance is due to a complicated obstacle setup, non-linear dynamics and partition design. Nevertheless, this shows that IHOMP can produce a reasonable solution with a limited representation.

Figure 4: () Pinball domain. () Average reward for IHOMP. IHOMP converges after a single iteration as we start IHOMP in the partition containing the goal. () The learned value function.

Improving Partitions: Providing a ‘good’ option partitioning a-priori is a strong assumption. It may be non-trivial to design the partitioning especially in continuous, high-dimensional domains. A sub-optimal partitioning may still mitigate misspecification, but it will not result in a near-optimal solution. To relax this assumption, we have incorporated Regularized Option Interruption into IHOMP to produce the IHOMP-ROI Algorithm. This algorithm learns the options and improves the partition, effectively determining where the options should be executed in the state space.

We tested IHOMP-ROI on the two rooms domain shown in Figure 5. The agent (red ball) needs to navigate to the goal region (blue square). The policy parameterization is limited to a distribution over actions (moving in a single direction). This limited representation results in a MP as the agent is unable to traverse between the two rooms. If we use IHOMP with a sub-optimal partitioning containing two options as shown by the red and green cells in Figure 5, the problem is still misspecified. Here, the agent leaves the red cell and immediately gets trapped behind the wall whilst in the green cell. Using IHOMP-ROI, as shown in Figure 5, the agent learns both the options and a partition such that the agent can navigate to the goal. The green region in the bottom left corner comes about as a function approximation error but does not prevent the agent from reaching the goal. If reader looks carefully in Figure 5, they will notice something unexpected. The optimal partitioning learned for the Two Rooms domain includes executing the red option in region B. This is intuitive given the parameterizations learned for each of the options. The red option has a dominant right action whereas the green option has a dominant upward action. When the agent in region B, it makes more sense to execute the red option to reach the goal. Thus, IHOMP-ROI provides an effective way to, not only learn an optimal partition, but to also discover where options should be reused.

Figure 5: () Two Rooms domain () Two Rooms domain with a sub-optimal partition after performing IHOMP () The learned partition after running IHOMP-ROI () The average reward from executing IHOMP-ROI compared to IHOMP without ROI and the flat policy.

8 Discussion

We introduced IHOMP  a RL planning algorithm for iteratively learning options and an inter-option policy \citepSutton1999 to repair a MP. We provide theoretical results for IHOMP that directly relate the quality of the final inter-option policy to the misspecification error. IHOMP is the first algorithm that provides theoretical convergence guarantees while iteratively learning a set of options in a continuous state space. In addition, we have developed IHOMP-ROI which makes use of regularized option interruption \citepSutton1999, Mann2014b to learn an improved partition to solve an initially misspecified problem. IHOMP-ROI is also able to discover regions in the state space where the options should be reused. In high-dimensional domains, partitions can be learned from expert demonstrations \citepAbbeel2005 and intra-option policies can be represented as Deep Q-Networks \citepMnih2015. Option reuse can be especially useful for transfer learning \citepTessler2016 and multi-agent settings \citepGarant2015.

Footnotes

  1. Our results are generalizable and complementary to non-linear parametric policy representations.
  2. https://minecraft.net/en/

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