Learning from Humans as an I-POMDP

04/01/2012 ∙ by Mark P. Woodward, et al. ∙ Harvard University 0

The interactive partially observable Markov decision process (I-POMDP) is a recently developed framework which extends the POMDP to the multi-agent setting by including agent models in the state space. This paper argues for formulating the problem of an agent learning interactively from a human teacher as an I-POMDP, where the agent programming to be learned is captured by random variables in the agent's state space, all signals from the human teacher are treated as observed random variables, and the human teacher, modeled as a distinct agent, is explicitly represented in the agent's state space. The main benefits of this approach are: i. a principled action selection mechanism, ii. a principled belief update mechanism, iii. support for the most common teacher signals, and iv. the anticipated production of complex beneficial interactions. The proposed formulation, its benefits, and several open questions are presented.

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1 Introduction

We propose formulating the problem of learning interactively from a human teacher as an interactive partially observable Markov decision process (I-POMDP). The I-POMDP is a formulation of the decision making problem faced by an agent in a stochastic, partially observable, multi-agent environment Gmytrasiewicz:2005 . In our formulation the human teacher is modeled as a distinct agent in the agent’s state space. The parameters of the agent programming to be learned are expressed as random variables also in the agent’s state space. Finally, all signals from the human teacher are represented as observed random variables.

Some of the benefits of this approach are a principled action selection mechanism, a principled belief update mechanism, the foreseen production of complex interactions with the human teacher, a single mechanism for interpreting all teacher signals including gestures, body posture, natural language, and direct modeling, and a single mechanism for selecting all agent actions including emotive displays, world manipulation, and spoken language.

In the following sections we present the proposed formulation, discuss its benefits, and finish with several open questions. But first we begin with a brief description of the envisioned domain for the our formulation.

2 Domain

The domain envisioned for this paper is an agent learning interactively from a non-technical human teacher. The interaction consists of signals generated by the teacher and the agent. Some examples of these signals are words, gestures, facial expressions, body posture, eye gaze, and rewards. We intend the proposed formulation to cover most forms of teaching including learning from demonstration and learning from reinforcements.

An important restriction of our proposal is that a teacher must be present for learning to take place. The teacher is explicitly modeled by the agent and learning is accomplished by interpreting signals resulting from the interaction with the teacher.111This does not preclude sub-systems from performing independent learning; for example, in a robotics domain, the agent might independently learn through experience that bumping into walls is disadvantageous. The main reason for this restriction is that the agent is seen as attempting to learn what the teacher wants it to learn, and, in general, the teacher needs to be present to convey what they want learned.

3 Proposed Framework

We briefly summarize the partially observable Markov decision process (POMDP) and its extension, the interactive partially observable Markov decision process (I-POMDP), before describing our formulation of learning from a human teacher as an I-POMDP.

3.1 Pomdp

A partially observable Markov decision process is a formulation of an agent’s decision process when operating in a sequential, stochastic, partially observable domain. Essentially, the agent knows only probabilistically how the world changes around it and only probabilistically how its sensor readings reflect the state of the world. Importantly, the optimal decision (a.k.a action) is the one that maximizes the agent’s expected utility. It is an expected utility because the agent knows only probabilistically the current state of the world, how it’s actions will affect the world, and what measurements it will receive assuming the world gets to a certain state. A POMDP is captured by the tuple . is the set of world states, is the set of actions the agent can perform,

is the motion model defining the probability of reaching any state

given an action executed from a state , is the set of measurements (a.k.a observations) the agent might receive, is the measurement model (a.k.a observation model) defining the probability of measuring given the world is in state , and finally is the utility function (a.k.a reward function) mapping states of the world or sometimes belief states to real numbers. It is some function of that the agent is trying to maximize when selecting the next action, for example the discounted sum of from to some horizon .

In the case of discrete states, the recursive Bayesian belief update for time step having taken action and received measurements is:222The continuous case is similar with integrals replacing summations and pdfs in place of pmfs for the beliefs .

(1)

where is the normalizing constant,

Action selection based on maximum expected utility is defined as

(2)

where

(3)

is the discount factor, and

(4)

For finite horizon solutions with horizon T, this recursion terminates with

(5)

where is the specified utility function. As an example of , in our initial robot learning experiments, we have used , Since entropy is a measure of uncertainty in the distribution, using negative entropy as the utility function means that the robot chooses actions that maximize certainty over the parameters being learned.

The above action selection equations form a decision tree into the future, where the tree nodes are belief states, and the branches alternate between measurements and actions.

For further details on POMDPs see Kaelbling:1998 and Thrun:2005 .

3.2 I-Pomdp

An I-POMDP extends the POMDP to the multi-agent setting, for our case there will only be a single other agent which is the human teacher. An I-POMDP of an agent is captured by the tuple , which is the usual POMDP tuple, except includes a model of the other agents, which could themselves be I-POMDPs, and is extended to include all actions that all agents could take. If the teacher is modeled with an I-POMDP, we might also model the teachers model of the agent as another I-POMDP. In order to reach a solution, this recursive agent model nesting must eventually ground out with a non interactive model, such as a POMDP.

Belief updates and action selections are similar to the POMDP case, except that measurements, actions, and beliefs of the teacher, and of the teacher about the agent, etc. need to be incorporated. We leave these details to the referenced readings. The above POMDP equations give a good intuition for the types of computations required in an I-POMDP.

For further details on I-POMDPs see Gmytrasiewicz:2005 , it may also be helpful to read their earlier work on nested models as a primer Gmytrasiewicz:2000 .

3.3 Learning from Humans as an I-POMDP

In our proposed formulation, the agent programming is defined by random variables stored in the agent’s state space within . For example in a work-flow, these parameters could be the trigger conditions, the number of nodes, and the value of transitions. Another example could be the parameters defining the angle of a driver when engaging with a screw. Also in our proposal, all signals in the human robot interaction are expressed as measurements . Examples of signals were given in the Domain section above. Lastly, the human teacher is explicitly represented as an interactive agent in the agent’s state space. This means that the teacher’s belief about the state of the world, about the parameters of the programming, and about the belief of the agent are maintained and updated by the agent. Additionally, the agent, when selecting actions to maximize it’s expected utility, can take into account likely future actions of the teacher, and its own responses to those future actions, resulting in a potentially high utility state. Most of the interactions listed below in the Benefits section are a result of this recursively nested modeling.

On each time-step, the agent first updates its beliefs and then selects an action to perform. The beliefs being updated include the agent’s belief about the parameter’s of the programming and the physical state of the world, the agent’s belief about the action performed by the teacher on the previous time-step, the agent’s belief about the teacher’s belief about the parameters of the task and the physical state of the world, the agent’s belief about the teacher’s belief about the agent’s previous action, etc.

With the belief state updated, the agent then selects an action. One method to select this action would be, for each nested belief level, to search out to a time horizon over all possible actions and measurements which could be performed or received by both agents, evaluate the utility of the resulting belief state at the horizon and then work this utility back to the current actions by taking expectation at measurement branches and maximization at action branches. Optimal actions in lower nesting levels appear to the higher level agent as distributions over actions by the lower level agent.

The selected action would then be performed, and along with the actions of the teacher, would affect the state of the world, which would, on the next time-step, be perceived through measurements by the teacher and the agent. And the process would repeat.

3.4 A Note on Complexity

The computation required for the belief update and action selection mentioned in the previous time grows exponentially with depth of agent model nesting, the time-steps to the horizon, and the complexity of the state space. We believe that sampling techniques will likely be the the best solution to this exponential growth. Particle filters have been applied to exponential state growth Fox:1999 and to exponential nested agent models Doshi:2005

to good result, but we are unaware of effective techniques to handle exponential growth due to action and measurement branching in planning over future time-steps. Hopefully the exponential growth due to action branches can be moderated by sampling over actions based on heuristics learned over time. Similarly the exponential growth due to measurement branches might be moderated by sampling according to the likelihood of measurements.

Even with the proposed sampling techniques, computation will remain a major problem with the proposed formulation. We believe this is a necessary side effect of the complexity of the beliefs and plans being represented and that the benefits of this complexity, described in the next section, warrant wrestling with the exponential growth of computation.

3.5 A Note on Modeling

The presented formulation is model based, in that it requires a probabilistic motion model for the state space and probabilistic measurement models for the measurements observed by the agent. Additionally, the recursive agent models must ground out with a stochastic model of actions for either the teacher or agent. All of these models have parameters that need to be set, for example the attention span of a typical student, the rate of misspoken words by a human teacher, or the variance of a laser range scan. As the proposed technique matures, future agents will likely maintain these parameters by long term monitoring of their physical sensors and repeated interactions with humans. In the near term, these parameters can be set by controlled sensor calibration and data and from experiments with a human teaching another human.

4 Benefits

In this section we describe many of the foreseen benefits of the proposed formulation. We begin with benefits to the learning from human research community due to the formulation being based on recognized principles. We then describe some complex interactions that should result from the formulation. Followed by a discussion of the extendability of the formulation to other interactive agent domains. We then finish with some miscellaneous benefits of the proposed formulation.

4.1 Principled Formulation

One of the main benefits of the proposed formulation is that it is based on recognized principles. Firstly, the belief over the state of the world, the parameters of the programming to be learned, and models of the teacher are all updated using Bayesian Inference, with the resulting implication that

all

signals by either agent be expressed as observed random variables. Secondly, the agent’s actions are selected in accordance with the principle of maximum expected utility. By formulating the problem of learning from humans based on the two principles of Bayesian Inference and maximum expected utility, techniques we develop can be useful to, and we can make use of techniques from, many fields which commonly base techniques on these principles, such as the broader reinforcement learning community, signal processing, and operations research.

4.2 Complex Interactions

As a product of the nesting of agent beliefs (beliefs about the teacher’s beliefs about the agent’s beliefs etc.), we expect to see a number of interesting exchanges. One type of exchange is where the agent has an inconsistency in its nested agent models and determines that acting to reduce this inconsistency will result in the highest expected utility.333The examples assume a utility function that favors faster learning, for example maximizing the discounted negative expected utility of the random variables defining the programming. The following are three examples of the agent taking this type of action to correct an inconsistency.

  • Interruption: The teacher is teaching about , and the agent interrupts to inform the teacher that they are clear on , but more uncertain on .

  • Clarification: The agent interrupts the teacher to clarify that a previous action came through as , and questions did they actually intend .

  • Correction: This is the inverse of clarification, the agent interrupts the teacher to communicate that the agent believes that the teacher believes that the agent was asking about , when it was actually asking about .

I-POMDPs have been demonstrated to generate similar exchanges in simple cooperative multi-agent settings Gmytrasiewicz:2001 .

4.3 Extendability

Another benefit of the proposed formulation is that it should easily extend to other useful agent settings. For example, in the proposed approach, as a sub-task of action selection, the agent is already determining optimal actions for the teacher, just one nesting level down, thus it should be straightforward to extend the formulation to the agent teaching a human or another agent. Similarly, the agent should be able to interactively learn from a non-human teaching agent. Additionally, since I-POMDPs were developed for the multi-agent setting, our formulation should extend to the common multi-agent settings of cooperation and competition, be it with human or non-human agents.

4.4 Miscellaneous Benefits

This section describes benefits that do not fall under the previous categories but are worth mentioning.

The first is that all actions fall under one umbrella. This means that we do not need separate action selection systems for physical movement, words, text, beeps, facial expressions, etc. As mentioned above, the agent will likely have built-in heuristics for determining the utility of specific actions to speed up action selection.444The proposed approach of a unified action selection mechanism is in stark contrast to the distributed behavior based robotics approach of Brooks:1991b which is likely a more accurate model of human action selection. We feel that having an identifiable, and thus easily adjustable, object function and action selection mechanism justify deviation from a more accurate distributed human action selection model which may be harder to develop.

Secondly, a number of action selections which may have needed explicit coding under other formulations happen without coding under the proposed formulation, because they fall out as the rational action selection. The following are several examples of behavior that should be exhibited without coding:

  • Facial expressions, such as raising eyebrows for a social robot, become the rational action, perhaps because conveying uncertainty will modify the teachers belief about the robots belief about the task in such a way that the teacher will provide further instruction resulting in a higher expected utility.

  • Similarly, periodically looking at the teacher might be rational behavior since it could maintain the teacher’s belief that the agent is still focused on learning, avoiding the potentially time consuming and consequently low utility producing actions by the teacher of checking if the agent is still paying attention.

  • Actions will be intuitively ordered, for example if a robot agent is going to ask the teacher about a series of objects and moving to each object takes time, the rational behavior would be to ask about objects in an order that minimizes travel time; The faster questions are asked and answered the higher the expected utility should be.

  • Often not acting will be the optimal action, since actions may interrupt the teacher, who is providing useful information, resulting in a lower expected utility than performing no action at all.

  • In a robot agent, the agent may perform pointing actions to direct the teacher’s attention. Both the decision to point and the duration of the point would be handled automatically by the system. The agent would compute the expected utility of various pointing durations and choose the duration with the maximum utility. A short point would allow the agent to move on but would leave the agent uncertain about the teacher’s attention, with a long point the agent would be certain about the teacher’s attention but will have pushed higher utility states further into the future. The principle of maximum expected utility would give the optimal duration to use.

5 Questions

In this section we first answer a couple questions posed by the organizers, before posing several open ended questions of our own.

5.1 Organizer Questions

Question:

How explicitly pedagogical is the human teacher? At one extreme, the human is a role-model not considering the agent at all; at the other, the human is carefully formulating a curriculum.

Answer:

Our approach is towards the pedagogical end of the spectrum. An intelligent agent, explicitly instructing the robot is required. Though the signals of instruction can be ambiguous and varied (demonstration, modeling, reinforcement, or natural language).

Question:

How dependent is learning on communication between human teacher and agent? At one extreme, the human and agent merely observe one another interacting with the environment; at the other there is a complex dialog between teacher and student.

Answer:

The proposed formulation assumes a dialog of signals between the teacher and the agent. That said, the formulation should extend to the learning by observing peer agents. Since other agents in the environment are assumed to be rational, observing their actions updates the belief about the random variables affecting their actions. Through observation, the belief about their objective function is also updated. If this objective function is deemed “similar" to the agent’s, then imitation is reasonably the rational behavior that would result.

Question:

How interdependent is learned information? At one extreme, learning can potentially happen in any order (e.g., mapping a state space); at the other, each new piece of knowledge must be formulated in terms of the previous one (e.g., Kirchhoff’s laws depend on current and voltage).

Answer:

Learned information is not inherently hierarchical in our approach. But, for example, it is likely rational for steps of a task to be taught in order, or reverse order, and since the robot assumes a rational teacher, the learning task would be much harder if the teacher choose to teach step one, then five, then two, as apposed to one, then two, then three, or three, then two, then one. What is rational depends on the zeroth level models of the teacher and robot. If these zeroth models have a certain flow to them, then the agent will learn best under that flow.

Also, making use of prior learned programming as heuristics for future learning is likely a good idea. For example, learning the task of clearing a table should be easier having learned to set the table; forks and spoons moved together before, so they will likely move together again.

Question:

What is the relative importance of how learning occurs vs. the end result of learning in the research? At one extreme, learning from humans is just a pragmatic way of configuring a real-world system; at the other, the system is only valued for the insight it provides on human learning.

Answer:

We view learning from humans as a pragmatic way of configuring a real-world system.

Notably, we view our approach as a poor approximation to human learning. Through introspection, we may think that our actions are chosen by maximizing our expected utility, but psychology studies have shown that that humans often do not select actions to maximize their expected utilityKahneman:1971 . And (Brooks 1991) makes a compelling argument that model based planning is not how biological agents select actionsBrooks:1991b .

As noted above, our approach is attractive, not because it resembles human learning, but because it is a principled approach to robot behavior during learning, in addition to the other benefits mentioned.

5.2 Open Questions

  • What is the right objective function? If pleasing the human, how do we measure this?

  • What are reasonable metrics for evaluating this formulation? e.g. time to teach, certainty/accuracy vs. time curves, ease of use.

  • At what granularity if any, does this become less “effective" than model free techniques such as Q-Learning? than model based techniques such as Value Iteration?

  • Are there any situations in which a signal from the human teacher should be considered part of the utility function?

  • How deep should the agent belief nesting be?

  • What should the ground agent model be?

  • What is the simplest student/teacher game to explore this formulation?

  • Since computation is an issue, how should we trade off accuracy of belief representation, depth of agent model nesting, and depth of search?

  • How can we make use of heuristics?

  • How can these heuristics be learned?

6 Conclusions

This paper proposed formulating the problem of learning from humans as an interactive partially observable Markov decision process (I-POMDP), where parameters defining the agent programming to be learned are captured as random variables in the agent’s state space and the teacher is explicitly modeled as a distinct agent also in the agent’s state space. POMDPs and I-POMDPs were briefly described before outlining the proposed formulation. We described several benefits of the formulation, namely principled action selection, principled belief updates, and the anticipated production complex interactions. We finished by posing several open questions regarding the formulation.

We believe that the benefits of the new formulation justify its computational complexity, and will help to advance research on agents learning from human teachers.

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