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Task Encoding and Strategy Learning in Large Worlds Dagstuhl Workshop, 29 July 2010

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Task Encoding and Strategy Learning in Large WorldsDagstuhl Workshop, 29 July 2010

Subramanian Ramamoorthy

Institute of Perception, Action and Behaviour

School of Informatics

University of Edinburgh

What does this tell you about robust autonomy in large worlds?

- How does she represent the task in order to be able to deploy it in a wide
- variety of previously unseen and unmodelled environments?
- 2. How is this efficiently utilized for learning?

- Agent represents tasks/environments in terms of a hierarchy of abstractions, ranging from weak sufficient conditions (qualitative information) to detailed quantitative information
- Qualitative descriptions define an abstract problem that is useful for coarse reasoning about the large world
- Quantitative information can be dealt with locally or at a slower time scale
- Variety of learning methods can be combined to leverage their strengths

- An ideal abstraction is such that one can make many useful inferences at the abstract level, without recourse to quantitative details that are uncertain/unobserved/undefined
- Different from â€˜mereâ€™ clustering of states, etc.
- We want a decision making strategy to be fully defined at each level

- System consists of two subsystems â€“ pendulum and cart on finite track
- Only one actuator â€“ cart
- We want global asymptotic stability of 4-dim system
- The Game: Experimenter hits the pole with arbitrary velocity at any time, system picks controls
- What are the weak sufficient conditions defining this task?

Phase space of the pendulum

Adversary could push

system anywhere,

e.g., here

Can describe

global strategy

as a qualitative

transition graph

Larger disturbances

could truly change

quantitative details,

e.g., any number of

rotations around origin

The uncontrolled system

converges to this point

We want to reach and

stay here

Lemma (Spring â€“ Mass - Positive Damping):

Let a system be described by

where, and

Then it is asymptotically stable at (0,0).

Lemma (Spring â€“ Mass - Negative Damping):

Let a system be described by

where, and

Then it has an unstable fixed-point at (0,0), and no limit cycle.

The control law:

if Balance

else if Pump

else Spin

Constraints:

The switching strategy:

If then Balance

else if then Pump

else Spin

S. Ramamoorthy, B.J. Kuipers, Qualitative heterogeneous control of higher order systems, Hybrid Systems: Computation and Control (2003)

- No learning in this example but we can still learn things from it
- â€˜Symbolâ€™ ïƒ³ local system with well defined dynamical properties
- could also do this using automated formal methods [Shults+Kuipers,AIJ97]

- We can talk about task achievement for an entire family of dynamical systems
- Weak commitment to functional forms of f and g(also very large parameter intervals, etc.)
- Possibility for composition and interactive strategies at symbolic level

- Current work: how does one make general relational/logical statements about the behaviour of such models â€“ so that we can use â€˜reasoningâ€™ tools at abstract levels
- Could enable greedy learning of local models with interesting predicates

- No detailed models of dynamics
- Precisely specified footfalls
- Height/length variations
- Hard to represent & achieve with state of the art methods!

[Kuo, Science â€™05]

Define qualitative strategy

in low-dimensions (finite

horizon optimal control)

(X,U,W)

(X,W)

(X,U)

(X)

Lift resulting strategy to

the more complex c-space

(presently unknown!)

S. Ramamoorthy, B.J. Kuipers, Qualitative hybrid control of dynamic bipedal walking, Robotics: Science and Systems II, pp. 89-96 (2006)

- Random actions
- Imperfect gait
- Active learning

Known Analytically

Organize data in a k-NN graph

Where is manifoldin the graph?

- Manifold ïƒ³ Set of geodesic trajectories restricted to it
- If the manifold encodes task â€“ every geodesic must behave like template plan
- Diagram must commute!
- Minimize commutativity error

S. Ramamoorthy, B. Kuipers, Trajectory generation for dynamic bipedal walking through qualitative model based manifold learning, ICRA 08

- Consider high-dim data drawn from an unknown low-dim manifold
- We can approximate the tangent space:

- This can be learnt with a pair of optimization steps
- Simple example: 3-link arm
- The following error term defines the manifold:
- Another error minimization defines geodesic paths:

- The grey mesh is the Delaunay triangulation of the 100 data points
- shown for visualization of the desired manifold
- (from which curves in fig. c are drawn)

I. Havoutis, S. Ramamoorthy, Geodesic trajectory generation on learnt skill manifolds, ICRA 2010

Following the unconstrained geodesics, oblivious to obstacles

Constrained geodesic trajectory â€“ avoid obstacles, while staying within demonstrated class

I. Havoutis, S. Ramamoorthy, Constrained geodesic trajectory generation on approximately optimal skill manifolds, IROS 2010

- Let us make the abstract spaces concrete
- you are driving over a network of highways

- Two sources of uncertainty:
- Oncoming traffic (changing goals)
- Changing dynamics, navigability/costs

- In â€˜simpleâ€™/reasonably well understood worlds, acquire basis strategies
- e.g., imitation learning
- Could also be more bottom-up exploratory learning

- In a continually changing complex world, learn strategies in a game against a (fictitious) adversary

- Learn policy from expert:
- RL problem
- Reward as weighted combination of features

- 2-player zero-sum game
- select distribution over actions to maximise V(Ïˆ) â€“ V(Ï€E)
- nature varies R(s) through weights w

[Syed & Schapire 2008]

- Environment picks transition function, reward
- You pick mixture over basis strategies (finite horizon)
- Online regret minimization to compute strategies
- Composing elemental strategies in response to changing environment

B. Rosman, S. Ramamoorthy, A game theoretic procedure for learning hierarchically structured strategies, ICRA 2010.

- Many autonomous agent behaviours admit efficient descriptions in terms of consistent hierarchy of abstractions
- Challenges for learning:
- What unsupervised learning methods can we use to extract base concepts (how descriptive are these models)?
- Are there principled ways to refine these models over time?
- Efficient methods for online strategy learning: how best to define games at the abstract level so they are consistent with fully quantitative local problems?
- Life-long and social learning

- Complex manipulation problems in terms of hierarchies of abstractions
- At one level, one is only thinking of relational concepts: single hole
- At another level, one if faced with the full challenge of robotics â€“ grasping, etc.

- Ability to work with partial specifications and concepts
- Ability to refine representations as we have more and more experience

B. Rosman, S. Ramamoorthy, Learning spatial relationships

between objects (Under Review)