Active Goal Recognition

Active Goal Recognition

Christopher Amato    Andrea Baisero
Khoury College of Computer Sciences
Northeastern University
Boston, MA 02115
Abstract

To coordinate with other systems, agents must be able to determine what the systems are currently doing and predict what they will be doing in the future—plan and goal recognition. There are many methods for plan and goal recognition, but they assume a passive observer that continually monitors the target system. Real-world domains, where information gathering has a cost (e.g., moving a camera or a robot, or time taken away from another task), will often require a more active observer. We propose to combine goal recognition with other observer tasks in order to obtain active goal recognition (AGR). We discuss this problem and provide a model and preliminary experimental results for one form of this composite problem. As expected, the results show that optimal behavior in AGR problems balance information gathering with other actions (e.g., task completion) such as to achieve all tasks jointly and efficiently. We hope that our formulation opens the door for extensive further research on this interesting and realistic problem.

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Introduction

AI methods are now being employed in a wide range of products and settings, from thermostats to robots for domestic, manufacturing and military applications. Such autonomous systems usually need to coordinate with other systems (e.g., sensors, robots, autonomous cars, people), which requires the ability to determine what the other systems are currently doing and predict what they will be doing in the future. This task is called plan and goal recognition.

Many methods already exist for plan and goal recognition (e.g., [21, 31, 6, 16, 29, 30, 11, 14]; see [37] for a recent survey). However, these methods assume that a passive observer continually observes the target (possibly missing some observation data), that is no cost to acquire observations, and that the observer has no other tasks to complete.

These assumptions fall short in real-world scenarios (e.g., assistive robotics at home or in public) where robots have their own tasks to carry out, and the recognition of others’ goals and plans must be incorporated into their overall behavior. For example, consider a team of robots assisting a disabled or elderly person: the robots would be tasked with activities such as fetching items and preparing meals, while also opening doors or otherwise escorting the person. As a result, the robots will need to balance completion of their own tasks with information gathering about the behavior of the other (target) agent.

Current goal recognition methods do not address this active goal recognition problem. While there has been some work on integrating plan recognition into human-machine interfaces [20, 12] or more generally addressing observation queries [26], it has been limited to utilizing query actions in service of recognition, rather than in an attempt to reason about recognition through general tasks.

In contrast, real-world domains will involve agents interacting with other agents in complex ways. For example, manufacturing and disaster response will likely consist of different people and robots conducting overlapping tasks in a shared space. Similarly, agents and robots of different make and capacity will be deployed in search and rescue operations in disaster areas, forming an ad hoc team that requires collaboration without prior coordination [17, 36].

Instead of considering a passive and fixed observer, we propose active goal recognition for combining the observer’s planning problem with goal recognition to balance information gathering with task completion for an observer agent and target agent. The active goal recognition problem is very general, considering costs for observing the target that could be based on costs of performing observation actions or missed opportunities from not completing other tasks. The observer and target could be deterministic or stochastic and operate in fully or partially observable domains (or any combination thereof). In our formulation, we assume there is a planning problem for the target as well as a planning problem for the observer that includes knowledge and reasoning about the target’s goal.

An example active goal recognition scenario is shown in Figure 1. Here, a single robot is tasked with cleaning the bathroom and retrieving food from the kitchen, while also opening the proper door for the person. The robot must complete its tasks while predicting when and where the human will leave. This domain is a realistic example of assistive agent domains; it has the goal recognition problem of predicting the humans’ goals as well as a planning problem for deciding what to do and when. This example is a simple illustration of the class of agent interaction problems that must be solved, and appropriate methods for active goal recognition will allow the autonomous agents to properly balance information gathering with task completion in these types of domains.

Figure 1: A depiction of active goal recognition, where the robot has to balance information gathering using limited sensors with task completion (e.g., gathering items or cleaning an area) to assist the human and complete tasks.

While the problem is very general, to make the discussion concrete, we describe a version of the problem with deterministic action outcomes and both agents acting in fully observable domains. We also assume the target remains partially observable to the observer unless the observer is in the proper states or chooses the proper actions. This research will be the first to consider active goal recognition. The goal is to extend the utility of goal recognition, making it a viable task while also interacting with other agents

In the following section, we provide background on goal recognition and planning. We then describe our proposed active goal recognition problem and representation. We also present initial results from transforming our deterministic and fully observable active goal recognition problems into a partially-observable Markov decision process (POMDP) and using an off-the-shelf solver to generate solutions. Experiments are run on two domains inspired by Figure 1. The results show that, as expected, it is important to balance information gathering and other costs (e.g., task completion). We discuss the limitations of this simple representation and solution methods as well as some proposed future work.

Background

We first discuss goal recognition and then discuss planning using the POMDP formulation.

Goal recognition

Plan and goal recognition have been widely studied in the planning community [37]. Early methods assumed the existence of a plan library and then attempted to determine the correct plan (and the correct goal) from observations of the plan execution [22, 24, 18]. Techniques for plan and goal recognition include Bayes nets [7], hidden Markov models [4] and grammar parsing [13]. In these methods, goal recognition is treated as the process of matching observations to elements of the given library. This process, while effective, does not easily admit planning or reasoning about which observations are needed, using familiar planning methods.

More recently, Ramírez and Geffner have reformulated plan and goal recognition as a planning problem, where it is assumed that the observed agent (which we will call the target) is acting to optimize its costs in a known domain [28, 29, 35]. In this case, a classical planner is used to solve a planning problem for each of the possible goals, while ensuring plans adhere to the observations and the known initial state. These approaches are typically more flexible than methods that use a plan library, as they dynamically generate plan hypotheses based on the observations, rather than from a fixed library.

More formally, a planning problem is defined as a tuple , where is a set of states, is the initial state, is a set of actions, and is a set of goals. A plan is a sequence of actions which changes the state from the initial one to (hopefully) one of the goal states. If actions have costs, then the optimal plan is that which reaches a goal while while minimizing the sum of action costs.

While the use of a planner in the recognition process is promising, the reliance on classical planning—characterized by deterministic actions and fully-observable states—is a limiting factor which only adds to the assumption of a passive observer. To remove these limitations, some extensions have been proposed. For example, for the case where the state for the target is still fully observable, but transitions are stochastic, goal recognition has been performed over Markov decision processes (MDPs) [1] and POMDPs [30]. Here, the goal recognition problem is reformulated as a planning problem, but instead of solving multiple classical planning problems, multiple MDPs are solved.

While the goal recognition methods perform well in a range of domains, they assume passive observation of the target. In many real-world domains, such as the robotic domains depicted in Figure 1, the observers will be mobile and have other tasks to complete besides watching the target. Active goal recognition requires a deeper integration of planning and goal recognition. This work will apply in the deterministic, fully observable case as well as stochastic, partially observable and multi-agent cases, making it a very general framework for interaction with other agents.

Partially Observable Planning

Our active goal recognition problem can be formulated as a planning problem with partial observability over the target agent. We will discuss this formulation in more detail in the next section, and first discuss general planning under uncertainty and partial observability with POMDPs.

A partially observable Markov decision process (POMDP) [19] represents a planning problem where an agent operates under uncertainty based on partial views of the world, and with the plan execution unfolding over time. At each time step, the agent receives some observation about the state of the system and then chooses an action which yields an immediate reward and causes the system to transition stochastically. Because the state is not directly observed, it is usually beneficial for the agent to remember the observation history in order to improve its estimate over the current state. The belief state (a probability distribution over the state) is a sufficient statistic of the observation history that can be updated at each step based on the previous belief state, the taken action and the consequent observation. The agent continues seeing observations and choosing actions until a given problem horizon has elapsed (or forever in the infinite-horizon case).

Formally, a POMDP is defined by tuple , where:

  • is a finite set of states with designated initial state distribution ;

  • is set of actions;

  • is a state transition probability function, , that specifies the probability of transitioning from state to when action is taken (i.e., );

  • is a reward function , the immediate reward for being in state and taking action ;

  • is a set of observations;

  • is an observation probability function , the probability of seeing observation given action was taken which results in state (i.e., );

  • and is the number of steps until termination, called the horizon.

A solution to a POMDP is a policy, . The policy can map observation histories to actions, , where is the set of observation histories, , up to time step, or, more concisely belief states to actions , where represents the set of distributions over states .

The value of a policy, , from state is , which represents the expected value of the immediate rewards summed for each step of the problem, given the action prescribed by the policy. In the infinite-horizon case () the discount factor is included to obtain a finite sum. An optimal policy beginning at state is . The goal is to find an optimal policy beginning at some initial belief state, .

POMDPs have been extensively studied and many solution methods exist (e.g., for a small sample [19, 33, 23, 32, 34]). POMDPs are general models for representing problems with state uncertainty and (possibly) stochastic outcomes. Generic solvers have made great strides at solving large problems in recent years [34, 39] and specialized solvers can be developed that take advantage of special structure in problem classes.

Active Goal Recognition

A rich set of domains consist of a single agent conducting tasks that are difficult to complete (an example is shown in Figure 1). One author encounters these problems each day when he attempts to get three small children ready for school: there is no way to get himself ready while making sure they are dressed, have breakfast, brush their teeth and deal with the inevitable catastrophes that arise. They somehow manage to get out of the door, but it is often not an enjoyable experience for any involved (and they are sometimes missing important things like shoes or lunches). Having an autonomous agent to assist with some of these tasks would be extremely helpful. Of course, there are many other instances of similar problems such as having an autonomous wingmanin combat missions to protect and assist a human pilot [5], a robot assisting a disabled person to navigate an environment, retrieve objects and complete tasks (e.g., opening doors or preparing meals as in Figure 1) or a robot performing a search and rescue task with the help of another robot from a different manufacturer.

An important piece of solving this problem is active goal recognition—recognizing the goal of the target agent while the observer agent is completing other tasks. For example, in a room with multiple doors, robots would need to determine when a disabled person is going to go out a door and which one. Some work has been done in person tracking and intent recognition (e.g., [37, 8, 15]), but improving the methods and integrating them with decision-making remains an open problem.

One version of the active goal recognition problem is defined more formally below, but like the passive goal recognition problem of Ramírez and Geffner, we propose transforming the active goal recognition problem into a series of planning problems. In particular, like the passive problem, we can assume the domain of the target is known (the states, actions, initial state and possible goals), and given a set of observed states for the target, planning problems can be solved to reach the possible goals. The difference in the active case is that the observer agent’s observations of the target depend on the actions taken in its domain (which may be different than the target’s domain).

We now sketch an overview of our proposed approach and a preliminary model. The active goal recognition problem considers the planning problem of the observer and incorporates observation actions as well as the target’s goals into that problem. These observation actions can only be executed based on preconditions being true (such as being within visual range of the target). The goals of the observer are augmented to also include prediction of the target’s true goal (e.g., moving to the target’s goal location or a more generic prediction action). Therefore, the observer agent is trying to reach it’s own goal as well as correctly predict the chosen goal of the target (but observations of the target will be needed to perform this prediction well). This augmented planning problem of the observer, which includes knowledge of the target’s domain and observation actions is the active goal recognition problem. By solving the augmented planning problem for goals of the target, we can predict the possible future states of the target and act to both complete the observer’s planning problem and gain information about the target.

More formally, a sketch of a definition and model is:

Definition 1 (Active Goal Recognition (AGR)).

Given a planning problem for the observer agent, and a planning domain as well as possible goals for the target agent , construct a new planning problem .

Note that a domain is a planning problem without a (known) goal and the domains for the target and observer do not have to be the same, but they have to be known (e.g., maybe the agent just needs to predict the correct goal without acting in the same world as the target). Given (classical) planning representations of and , the planing problem, , can be constructed as a tuple , where

  • is a set of states for the observer and the target,

  • is the initial state in the observer’s planning problem,

  • is the initial state in the target,

  • is a set of actions for acting in the observer’s planning problem, , observing the target, , and deciding on the target’s goal,

  • is a set of goals in the observer’s planning problem, and

  • is a set of possible goals for the target.

A solution to this planning problem is one that starts at the initial states of the observer and target and chooses actions that reach an augmented goal (with lowest cost or highest reward). The augmented goal, , is a combination of the observer’s and target’s goal: satisfying the conditions of the observer’s goal as well as predicting the target’s goal. Prediction is accomplished with, , a prediction action that chooses the goal of the target. may have preconditions requiring the observer to be in the target’s goal location (e.g., in navigation problems), thereby ensuring the observer predicts correctly (costs can also be used to penalize incorrect predictions). It may also be the case that is null and indirect observations are received through (e.g., the agent doesn’t choose to observe, but it happens passively in certain states). The planning problems for the target and observer could be deterministic or stochastic, fully observable or partially observable. The general AGR formulation makes no assumptions about these choices, but to make the problem concrete and simple, this paper focuses on the deterministic, fully observable case.

For example, consider the problem in Figure 1. Here, the robot and target (human) have similar, but slightly different domains. The target’s domain is assumed to be just a navigation problem to one of the two doors. Of course, the target is a human, so it may conduct other tasks along the way or take suboptimal paths. The robot’s domain consists of not just navigation, but actions for cleaning and picking up and dropping items (and the corresponding states). The goal for the robot in its original planning problem, , consist of the bathroom being clean and the food item being in its gripper. The observation actions are null in this case and it is assumed that when the robot is in line of sight of the human, it can observe the human’s location. The augmented goal consists of the original goal from as well as prediction of the target’s goal, which in this case requires navigation to the predicted door location. Costs could vary, but for simplicity, we could assume each action costs 1 until the goal is reached.

Unfortunately, this problem is no longer fully observable (because the state of the target is not known fully). As such, this problem can be thought of as a contingent planning problem or more generally as a POMDP. Next, we present a POMDP model, which we use for our experiments. The model and solutions serve as a proof of concept for the active goal recognition problem and future work can explore other methods (such as using contingent planning methods).

POMDP Representation

Given the planning problem for the observer agent and a planning domain and goals for the target agent, we construct the AGR POMDP as the tuple , where:

The state space

is the Cartesian product of observer states, target states and target goals, and states factorize as ;

The action space

is the union of actions available in the observer’s domain, together with observation and decision actions;

The transition model

factorizes into the marginal observer and target transition models, respectively and . Let indicate the action that the target makes in its own planning domain, then

(1)

where the indicator function which maps and . As a result, the target’s goal doesn’t change and the observer’s state only changes when taking a domain action (as opposed to an observation or decision action).

The reward function

can be represented directly from the costs from the observer’s task, observation costs and decision costs/rewards. That is, depending on the type of action the observer takes, it receives the corresponding cost (that may depend on its state and the state of the target):

(2)
The observation space

factorizes into an fully observable component for the observer’s own state and a noiseless, but partially observable target state component. Assume that the target state space factorizes in an observable component and a complementary unobservable component , and let indicate that no target-state observation is made, then , and an observation decomposes into . As such, the observer will be able to fully observe some target states and never observe others (as described next);

The observation model

is deterministic and target-state observations may be obtained explicitly when an observation action is made, or implicitly when the joint observer-target state is in a subset which warrants a target observation (e.g., when the agents are within line-of-sight). Therefore, the observer will perfectly observe its own state and may get a perfect observation of the target state if the observation action allows it or the observer is in a corresponding state:

(3)
The horizon

is set based on the planning problem.

Of course, POMDPs can represent more general forms of AGR problems that include stochasticity and consider more general observation, transition and reward models. Nevertheless, we believe this AGR POMDP model balances expressivity and problem structure.

Experiments

We implement two prototypical AGR domains (Figure 2): An abstract corridor domain and a more concrete map domain. In both domains, the observer must predict the goal of the target, while also performing other work. In the corridor domain, the observer and target do not share the same working environment, and target-state observations can only be obtained by making explicit observation actions. In contrast, observer and target in the map domain move in the same shared environment, and target-state observations can only be implicitly obtained as a function of the joint state rather than via an ad hoc action.

(a) Corridor Domain
(b) Map Domain
Figure 2: AGR domains used in our evaluation. Above, a corridor domain with size parameter . Below, a map domain with four task stations and an occluded work station.

Corridor Domain

In this domain, depicted in Figure 2(a), the target operates in corridor with positions to the right and positions to the left ( total positions). Each space contains a door which can only be opened remotely by the observer, and the target’s goal is to walk through a randomly selected goal door (known to the target only). At each time-step, the target will move towards the goal door, wait there if it is closed, or move to a terminal state if it has been opened. The observer is operating in a stateless multi-armed bandit-like domain, and its own non-cooperative task is to either work or remain idle.

Let be the set of all positions . The observer-, target-, and goal- states consist respectively of a singleton , the set of positions augmented with a terminal state , and a the set of all positions . The action space consists of actions for the observer planning problem , an explicit observation action , and goal decision actions . The observable component of the target-state is the target’s position , while no direct observations can be made about its goal. There are no implicit observations, hence . The rewards are as follows:

A_idle:

Does nothing, obtains ;

A_work:

Performs work, obtains ;

A_obs:

Observes target position, obtains ;

A_open():

Represents the goal recognition action by attempting to open a door for the target, obtains if the correct door is chosen and the target is at the correct cell, otherwise .

Overall, the state-, action-, and observation- space sizes for this domain are , , and .

Map Domain

This domain (Figure 2(b)) consists of a room containing walls, task stations (representing the target task) and working stations (representing the observer task). The target task is to visit all task stations in any order except for the last one—the goal task station—which encodes the target goal and is randomly selected (known to the target only). At each time step, the target will move towards its closest next viable task station, until it reaches the goal station where it will wait for the observer’s help. The observer’s non-cooperative task is to go perform work at a work station.

Let be the set of all positions which do not contain walls, be those with a task station, and be those with a work station. The observer-states consist of all observer positions ; the target-states are the factorize as , where is the observable set of target positions enhanced with a terminal target state (which is reached after the observer helps correctly) and is the non-observable set of remaining task stations; while the target goal is one of the task stations . The action space consists of the actions available for the observer planning problem and a goal decision action (there are no explicit observation actions, i.e., ). The reward function is as follows:

A_idle:

Do nothing. Obtains ;

A_work:

Attempts to work. Obtains if the observer is at a working station, otherwise ;

A_{left,right,up,down}:

Moves in the specified direction unless there is a wall. Obtains ;

A_help:

Represents the goal recognition acion by attempting to help the target with the last task station. Obtains if the target and observer are both at the goal task station and all task stations have been cleared, otherwise .

Overall, the state-, action- and observation- space sizes for this domain are , , and .

Lower and Upper Bounds

Given an AGR domain, it is always viable (if suboptimal) to consider the policies and which focus exclusively on the goal recognition and observer task respectively (and ignore the other). This indicates that the value of the optimal policy in the AGR domain is bounded below by and , and inspires us to design two lower bound (LB) variants to the AGR domain. The LB-A variant is constructed by applying a very strong penalty () to the goal decision actions , thus inhibiting the observer’s willingness to spend resources on gathering information about the target’s task. Similarly, the LB-T variant is constructed by applying the same penalty to a subset of the observer’s own planning problem actions (in our domains, ), thus ensuring that the observer will not focus on its own task.

Furthermore, completing the goal recognition task is typically contingent on the access to meaningful observations which are usually associated with an acquisition cost. This suggests that, if it were possible to have these observations available at any time for no cost at all, the resulting optimal policy would perform at least as well as the optimal AGR policy (i.e., ). This inspires us to design an upper bound (UB) variant to the AGR domain. The UB variant is constructed by always giving target observations to the observer, regardless of current action or joint state.

Results

We compute policies for the corridor and map domains (and the respective LB and UB variants) using the SARSOP solver [23]. For each domain and variant thereof, we simulate 1000 episodes (to account for random goal assignments) and compute the set of returns for each episode, and the current state-beliefs for each step. SARSOP was chosen because it is one of the most scalable optimal POMDP solvers, but other solvers could also be used.

Controlled Goal-Belief Entropy Reduction

Let be the goal-belief (i.e., the marginal state-belief obtained summing over all non-goal components), and be the normalized goal-belief entropy:

(4)
(5)

Figure 3 shows the evolution of during the course of the sampled episodes, and a few observations can be made about it in each of the 4 variants: {enumerate*}[a)]

Given the static nature of the target goal in our domains, the mean is non-increasing in time;

because all observations are free in the UB variant, the respective goal entropy also shows the quickest average reduction;

because the observer is prompted to focus only on the target task in the LB-T variant, the respective goal entropy also decreases relatively fast—albeit not necessarily as fast as in the UB case;

because the observer is incentivized to parallelize the execution of both tasks in the AGR domain, the goal entropy decreases in average slower than in the UB and LB-T variants; and

because the observer is uninterested in the target task, the LB-A variant is the only one where is not guaranteed to converge towards zero.

Intuitively, the results show that the observer in the AGR domain tends to delay information acquisition until it becomes useful (in expectation) to act upon that information, yet not too much as to perform substantially suboptimally w.r.t. the goal recognition task (i.e., the entropy decrease is noticeably slower than in the UB and LB-T variants, but it still catches up when it becomes possible to obtain complete certainty and guess the task correctly). As shown next, time is employed optimally by exploiting the observation delays to perform its own task.

(a) Corridor Domain
(b) Map Domain
Figure 3: Normalized goal-belief entropy as a function of time. Dots represent mean entropies for a given time step; vertical lines represent the min-max range of entropies for that time step; and the horizontal or slanted lines indicate the change of entropy values across adjacent time-steps (the more opaque the line, the more frequent the transition).

Sample Returns

Table 1 shows empirical statistics on the sample returns obtained for each domain and bound variant. As expected, the AGR method performs better than both LBs, and worse than the UB. In the previous section we have shown that the certainty over the target goal in an AGR grows slowly compared to the UB. Here we see that this does not translate into much of a loss in terms of expected returns, since the average AGR and UB performances are within 1–2 standard deviations from each other. The LB-A performance is deterministic because the optimal strategy does not depend on the target state or goal.

We propose that such empirical statistics contain meaningful information about the AGR design. For instance, if the performance in an AGR domain and respective UB is comparable, this should indicate that the observer’s goal recognition and own domain tasks can be efficiently parallelized—either because observations are cheap, or because few observations at key moments are sufficient, or because there are actions which move both tasks forward at the same time. On the other hand, if the AGR performance more strongly resembles that of a LB, it may indicate that the rewards associated with the observer’s goal recognition and own domain tasks are too unbalanced, that one of the two tasks is disregarded, and that a design revision of the AGR problem may be necessary.

Corridor Domain Map Domain
mean st.d. mean st.d.
UB
AGR
LB-A
LB-T
Table 1: Empirical mean and standard deviations for the sample returns obtained running the policy obtained by SARSOP in each AGR domain and LB/UB variant.

Optimal Policies

In the corridor domain, the observer spends the first few time steps performing work rather than observing the target, due to the the fact that it is relatively unlikely than observations at the beginning of an episode are particularly informative about the target’s goal. A few time steps into the episode, it performs the first observation, and then either opens the door, if the target is found to be waiting, or goes back to work to check again after another few time steps. Comparatively, the LB-T behavior performs a similar routine, except that it remains idle during the dead times rather than being able to obtain more rewards by working. In the ideal UB case, due to the target positions always being observed, the observer is able to open the correct door as soon as the target stops at a certain position, and uses the spare time optimally by performing work.

In the map domain, the observer proceeds directly towards the nearby working station and performs work until enough time has passed for the agent to be approaching its goal task station. At that point, the observer searches for the target, helps, and then moves back to the work station. The UB variant performs a similar routine, except that it can (naturally) exploit the additional information about the target’s path to time its movement more optimally: While the A_help actions in the AGR case are performed at time steps 13–16 (depending on the goal task station), they are performed at time steps 12–14 in the UB case. This difference is small enough that the observer decides to use its available time working rather than moving around trying to get more precise information about the target’s path and task. As in the corridor domain, the behavior in the LB-T case is similar to that of the AGR domain, with the exception that the observer stays idle while waiting, rather than profiting by working.

In both domains, the optimal LB-A behavior is straightforward—to work as much as possible, while completing disregarding observations (explicit or implicit) about the target task.

Related Work

Beyond the work already discussed, there has been recent research on incorporating agent interaction into goal recognition problems. For insstance, goal recognition has been formulated as a POMDP where the agent has uncertainty about the target’s goals and chooses actions to assist them [11]. Also, the case has been considered where the observer can assist the target to complete its task by solving a planning problem [12]. In both cases, the observers sole goal is to watch the target and assist them. In contrast, we consider the more general case of active observers that are also completing other tasks, balancing information gathering and task completion. We also consider multi-agent versions of the problem, unlike previous work.

There has also been a large amount of work on multi-agent planning (e.g., [10, 9, 38]).Such methods have, at times, modeled teams of people and other agents, but they assume the people and other agents are also controllable and do not incorporate goal recognition. Ad hoc teamwork is similar, but assumes a set of agents comes together to jointly complete a task rather than an agent or team of agents assisting one or more agents or otherwise completing tasks in their presence  [17, 36].

Overall, as described above, research exists for goal recognition, but the work assumes passive and not active observers. This is an important gap in the literature. The problem in this paper represents a novel and the proposed methods allow an agent to reason about and coordinate with another agent in complex domains.

Discussion

In this paper, we used a general POMDP representation for the active goal recognition problem as well as an off-the-shelf solver. This was sufficient for the problems discussed, but both the model and solution methods could be substantially improved. POMDPs are general problem representations, but the problem representation can serve as inspiration to begin adding the appropriate assumptions and structure to the problem and developing solution methods that exploit this structure—finding the proper subclass and corresponding solution methods.

In terms of the model, our AGR problem has a great deal of structure. The simple version assumes deterministic outcomes and agents that operate in fully observable domains. Furthermore, the goal of the target is assumed to be unknown, but does not change. As such, the belief can be factored into a number of components, with the observer’s own state fully observable, but the target’s is information partially observable. Furthermore, and as seen by the entropy analysis above, information is never lost as the belief over the target’s goal continues to improve as more observations are seen. Stochastic and partially observable versions of AGR will also have this factored belief structure. This structure will allow specialized solution methods that are much more efficient than off-the-shelf methods such as the one used above.

In terms of solution methods, many options are possible. Offline POMDP solvers (such as SARSOP [23]) could be extended to take advantage of the special structure, but more promising may be extending online POMDP solvers [32, 34]. Online solvers interleave planning and execution by planning for a single action only, executing that action, observing the outcome and then planning again. Online solvers are typically more scalable and more robust to changes. These online methods typically work by using a forward tree search (e.g., Monte Carlo tree search, MCTS [34]), which limits search to reachable beliefs rather than searching the entire state space. Therefore, online methods should be able to scale to very large state spaces, while also taking advantage of the factored states, actions and observations. In our experiments SARSAP could no longer solve versions of our domain with more than 6000 states. We expect online methods (such as those based on MCTS) to scale to problems that are orders of magnitude larger.

Other methods would also be able to take advantage of the special structure in AGR. For instance, many methods exist for contingent planning in partially observable domains (e.g., [25, 27, 2]). These methods may be able to be directly applied to some versions of the AGR problem and variants could be extended to choose (for instance) the most likely target goal and replan when unexpected observations are seen (e.g., like [3]). Because of the information gathering structure of the belief, simpler (e.g., greedy) information-theoretic and decision-theoretic methods could also be used that directly use the probability distribution over goals to explicitly reason about information gathering and task completion. The information-theoretic case could consider planning actions that reduce the entropy of the distribution over goals, while decision-theoretic methods could be greedy, one-step methods that consider both the change in entropy and the action costs. Future work on modeling and solving AGR problems as well as comparing these solutions will show the strengths and weaknesses of the approaches and provide a range of methods that fit with different types of domains.

Conclusions

Goal recognition is typically described as a passive task—an agent’s only focus. We generalize this setting and provide an active goal recognition (AGR) formulation in which the agent has other supplementary tasks to execute alongside the original goal recognition task. This AGR problem represents a realistic model of interaction between an agent and another system—ranging from a person to an autonomous car from a different manufacturer—where both interactive and individual tasks come together. Our POMDP representation is one way of modeling such type of problems, but others are possible. We provide a tractable instance as well as preliminary results showing the usefulness of this problem statement. Empirical results show that, rather than observing the target continuously, the optimal strategy in a well crafted AGR domain involves the selection of key moments when observations should be made such as to obtain important information about the target goal as soon as if becomes relevant (rather than as soon as possible), and to have more available time to continue performing other tasks. These results represent a proof of concept that AGR methods must balance information gathering with the completion of other tasks, accounting for the respective costs and benefits. We expect that further research efforts will be able to expand upon our initial results, and introduce new exciting models and methods for representing and solving this newly introduced class of problems.

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