Imitation Learning in Relational Domains Using Functional Gradient ...

**Imitation** **Learn ing**

**Functional** **Gradient** Boost**in**g

Sriraam Natarajan, Saket Joshi + , Prasad Tadepalli + , Kristian Kerst**in**g # , Jude Shavlik ∗

Wake Forest University School of Medic**in**e, + Oregon State University

# Fraunhofer IAIS, Germany, ∗ University of Wiscons**in**-Madison

It is common knowledge that both humans and animals learn new skills by observ**in**g others. This problem, which is

called imitation learn**in**g, can be formulated as learn**in**g a representation of a policy – a mapp**in**g from states to actions

– from examples of that policy. Our focus is on relational doma**in**s where states are naturally described by relations

among an **in**def**in**ite number of objects. Examples **in**clude real time strategy games such as Warcraft, regulation of

traffic lights, logistics, and a variety of plann**in**g doma**in**s. In this work, we employ two ideas. First, **in**stead of learn**in**g

a determ**in**istic policy to imitate the expert, we learn a stochastic policy where the probability of an action given a

state is represented by a sum of potential functions. Second, we leverage the recently developed functional-gradient

boost**in**g approach to learn a set of regression trees, each of which represents a potential function. The functional

gradient approach has been found to give state of the art results **in** many relational problems [6, 5]. Together, these

two ideas allow us to overcome the limited representational capacity of earlier approaches, while also giv**in**g us an

effective learn**in**g algorithm. Indeed, the functional-gradient approach to boost**in**g has already been found to yield

excellent results **in** imitation learn**in**g **in** robotics **in** propositional sett**in**g [7].

**Functional** **Gradient** Boost**in**g:Assume that the tra**in****in**g examples are of the form (xi, yi) for i = 1, ..., N and

yi ∈ {1, ..., K}. The goal is to fit a model P (y|x) ∝ eψ(y,x) . The standard method of supervised learn**in**g is based

on gradient-descent where the learn**in**g algorithm starts with **in**itial parameters θ0 and computes the gradient of the

likelihood function. Dietterich et al. used a more general approach to tra**in** the potential functions based on Friedman’s

[3] gradient-tree boost**in**g algorithm where the potential functions are represented by sums of regression trees that are

grown stage-wise.

In this work, the goal is to f**in**d a policy µ that is captured by the trajectories provided by the expert and represented

us**in**g features f j

i of the state and actions aji

of the expert, i.e., the goal is to determ**in**e a policy µ =P (ai|fi; ψ)

∀a, i where the features are relational. These features could def**in**e the objects **in** the doma**in** (squares **in** a gridworld,

players **in** robocup, blocks **in** blocksworld etc.), their relationships (type of objects, teammates **in** robocup etc.), or

temporal relationships (between current state and previous state) or some **in**formation about the world (traffic density

at a signal, distance to the goal etc.). We assume a functional parametrization over the policy and consider the

conditional distribution over actions ai given the features to be P (ai|fi; ψ) = eψ(ai;fi)

/ a ′ e

i

ψ(a′ i ;fi) , ∀ai ∈ A where

ψ(ai; fi) is the potential function of ai given the ground**in**g fi of the feature predicates at state si and the normalization

is over all the admissible actions **in** the current state. Instead of comput**in**g the functional gradient over the

potential function, the functional gradients are computed for each tra**in****in**g example (i.e., trajectories):∆m(a j j

i ; fi ) =

j

j

∇ψ j i log(P (aji

|fi ; ψ))|ψm−1 . These are po**in**t-wise gradients for examples 〈fi , aji

〉 on each state i **in** each trajectory

j conditioned on the potential from the previous iteration(shown as |ψm−1). Now this set of local gradients

form a set of tra**in****in**g examples for the gradient at stage m. The most important step **in** functional gradient boost**in**g is

the fitt**in**g of a regression function (typically a regression tree) hm on the tra**in****in**g examples [(f j

j

; f )] [3].

i

, aj

i

), ∆m(a j

i

Dietterich et al. [2] po**in**t out that although the fitted function hm is not exactly the same as the desired ∆m, it will

po**in**t **in** the same direction (assum**in**g that there are enough tra**in****in**g examples). So ascent **in** the direction of hm will

approximate the true functional gradient.

Proposition 0.1. The functional gradient w.r.t ψ(a j

∂ log P (a j j

|f i i ;ψ)

∂ψ(â j

i

where â j

i

;f j

i )

= I(aji

= âj

i

|f j

i

) − P (aj

i

j

|fi ; ψ)

i

; f j

i

) of the likelihood for each example 〈f j

i

is the action observed from the trajectory and I is the **in**dicator function that is 1 if aj

i = âi j

, aji

〉 is given by:

and 0 otherwise.

The expression is very similar to the one derived **in** [2]. The key feature of the above expression is that the functional

gradient at each state of the trajectory is dependent on the observed action â. If the example is positive (i.e., it is an

1

i

action executed by the expert), the gradient (I −P ) is positive **in**dicat**in**g that the policy should **in**crease the probability

of choos**in**g the action. On the contrary if the example is a negative example (i.e., for all other actions), the gradient is

negative imply**in**g that it will push the probability of choos**in**g the action towards 0. Follow**in**g prior work [4, 6, 5], we

use **Relational** Regression Trees (RRTs)[1] to fit the gradient function at every feature **in** the tra**in****in**g example (s**in**ce

we are **in** the relational sett**in**g). These trees upgrade the attribute-value representation used with**in** classical regression

trees. The key idea **in** this work is to represent the distribution over each action as a set of RRTs on the features. These

trees are learned such that at each iteration the new set of RRTs aim to maximize the likelihood of the distributions

w.r.t ψ. Hence, when comput**in**g P (a(X)|f(X)) for a particular value of state variable X (say x), each branch **in** each

tree is considered to determ**in**e the branches that are satisfied for that particular ground**in**g (x) and their correspond**in**g

regression values are added to the potential ψ. For example, X could be a certa**in** block **in** the blocksworld, a player **in**

Robocup or a square **in** the gridworld.

Experiments: For our experiments we chose four doma**in**s. These doma**in**s range from blocksworld which has a long

tradition **in** plann**in**g, to a grid world doma**in** that has a rich relational hierarchical structure to a multi-agent system

for traffic control to Robocup doma**in** that has cont**in**uous features. We compare the performance of our tree-boost**in**g

for relational imitation learn**in**g TBRIL system to a classical RRT learner, TILDE [1] and to propositional functional

gradient boost**in**g (PFGB) and empirically show the superiority of our method. In the blocksworld doma**in**, the goal

is to reach a target configuration where there is no tower of more than three blocks. In the traffic signal doma**in**, the

goal is to ensure smooth flow of traffic by controll**in**g the signals (8 different configurations each for 4 signals). In

the gridworld doma**in**, the goal is to navigate the grid by open**in**g doors and collect resources or destroy enemies. In

the Robocup doma**in**, the task is to breakaway from an opposition player **in** a 2-on-1 game. For each doma**in**, we

obta**in**ed traces of an expert’s policy and created tra**in****in**g and test sets. Our performance metric is the area under the

precision-recall (PR) curve **in** the test set aga**in**st the number of trajectories from the expert. The results are presented

**in** Figure 1. As can be seen, TBRIL system outperforms both TILDE and PFGB systems. PFGB showed very poor

performance **in** the blocksworld and the traffic signal doma**in** and hence we do not present the results here.

Figure 1: AUC-PR vs. number of trajectories (a) Blocksworld (b) Gridworld (c) Traffic Signal (d) Robocup

To our knowledge this is the first adaptation of a Statistical **Relational** technique to the problem of learn**in**g relational

policies from an expert. In this sense, our work can be understood as mak**in**g a contribution **in** both ways: apply**in**g

gradient boost**in**g for relational imitation learn**in**g and identify**in**g one more potential application of SRL.

Acknowledgements: SN acknowledges the support of Translational Sciences Institute, Wake Forest University School of Medic**in**e. SJ is

funded by the CI Fellowship (2010), under NSF subaward No. CIF-B-211. PT gratefully acknowledges the support of NSF under grant IIS-0964705.

KK is supported by Fraunhofer ATTRACT fellowship STREAM and by the EC under contract number FP7-248258-First-MM. JS acknowledges

the support of Defense Advanced Research Projects Agency under DARPA grant FA8650-06-C-7606.

References

[1] H. Blockeel. Top-down **in**duction of first order logical decision trees. AI Commun., 12(1-2), 1999.

[2] T.G. Dietterich, A. Ashenfelter, and Y. Bulatov. Tra**in****in**g conditional random fields via gradient tree boost**in**g. In ICML, 2004.

[3] J.H. Friedman. Greedy function approximation: A gradient boost**in**g mach**in**e. Annals of Statistics, 29, 2001.

[4] B. Gutmann and K. Kerst**in**g. TildeCRF: Conditional random fields for logical sequences. In ECML, 2006.

[5] K. Kerst**in**g and K. Driessens. Non–parametric policy gradients: A unified treatment of propositional and relational doma**in**s. In ICML, 2008.

[6] S. Natarajan, T. Khot, K. Kerst**in**g, B. Guttmann, and J. Shavlik. Boost**in**g **Relational** Dependency networks. In ILP, 2010.

[7] N. Ratliff, D. Silver, and A. Bagnell. **Learn ing** to search:

25–53, 2009.

Topic: **Learn ing** algorithms

Poster/Presentation

2