Apprenticeship Learning via Inverse Reinforcement Learning

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Discriminative Training of Kalman FiltersP.
Abbeel, A. Coates, M. Montemerlo, A. Y. Ng, S.
Thrun

• Kalman filters estimate the state of a dynamical
  system from inputs and measurements.
• The Kalman filters parameters (e.g., state and
  observation variances) significantly affect its
  accuracy, and are difficult to choose.
• Current practice hand-engineer the parameters
  to get the best possible result..
• We propose to collect ground-truth data and then
  learn the Kalman filter parameters automatically
  from the data using discriminative training.

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Discriminative Training of Kalman FiltersP.
Abbeel, A. Coates, M. Montemerlo, A. Y. Ng, S.
Thrun
Ground truth
Hand-engineered
Learned
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Discriminative Training of Kalman Filters

• Pieter Abbeel, Adam Coates, Mike Montermerlo,
  Andrew Y. Ng, Sebastian Thrun
• Stanford University

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Motivation

• Extended Kalman filters (EKFs) estimate the state
  of a dynamical system from inputs and
  measurements.
• For fixed inputs and measurements, the estimated
  state sequence depends on
• Next-state function.
• Measurement function.
• Noise model.

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Motivation (2)

• Noise terms typically result from a number of
  different effects
• Mis-modeled system dynamics and measurement
  dynamics.
• The existence of hidden state in the environment
  not modeled by the EKF.
• The discretization of time.
• The algorithmic approximations of the EKF itself,
  such as the Taylor approximation commonly used
  for linearization.
• In Kalman filters noise is assumed independent
  over time, in practice noise is highly correlated.

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Motivation (3)

• Common practice careful hand-engineering of the
  Kalman filters parameters to optimize its
  performance, which can be very time consuming.
• Proposed solution automatic learning of the
  Kalman filters parameters from data where (part
  of) the state-sequence is observed (or very
  accurately measured).
• In this work we focus on learning the state and
  measurement variances, although the principles
  are more generally applicable.

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Example

• Problem estimate the variance of a GPS unit used
  to estimate the fixed position x of a robot.
• Standard model
• xmeasured xtrue ?
• ? N(0,?2) (Gaussian with mean 0, variance ?2)
• Assuming the noise is independent over time, we
  have after n measurements variance ?2/n.
• However if the noise is perfectly correlated, the
  true variance is ?2 gtgt ?2/n.
• Practical implication wrongly assuming
  independence leads to overconfidence in the GPS
  sensor. This matters not only for the variance,
  but also for the state estimate when information
  from multiple sensors is combined.

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Extended Kalman filter

• State transition equation
• xt f(xt-1,ut) ?
• ? N(0, R) (Gaussian with mean zero,
  covariance R)
• Measurement equation
• zt g(xt) ?
• ? N(0,Q) (Gaussian with mean zero, covariance
  Q)
• The extended Kalman filter linearizes the
  non-linear function f, g through their Taylor
  approximation
• f(xt-1,ut) ¼ f(?t-1,ut) Ft(xt-1-?t-1)
• g(xt) ¼ g(?t) Gt(xt - ?t)
• Here Ft and Gt are Jacobian matrices of f and g
  respectively, taken at the filter estimate ?.

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Extended Kalman Filter (2)

• For linear systems, the (standard) Kalman filter
  produces exact updates of expected state and
  covariance. The extended Kalman filter applies
  the same updates to the linearized system.
• Prediction update step
• Place holder
• Place holder
• Measurement update step
• Place holder
• Place holder
• Place holder

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Discriminative training

• Let y be a subset of the state variables x for
  which we obtain ground-truth data. E.g., the
  positions obtained with a very high-end GPS
  receiver that is not part of the robot system
  when deployed.
• We use h(.) to denote the projection from x to y
• y h(x).
• Discriminative training
• Given (u1T, z1T, y1T).
• Find the filter parameters that predict y1T most
  accurately.
• Note it is actually sufficient that y is a
  highly accurate estimate of x. See the paper for
  details.

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Three discriminative training criteria

• Minimizing the residual prediction error
• Maximizing the prediction likelihood
• Maximizing the measurement likelihood

(The last criterion does not require access to
y1T.)
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Evaluating the training criteria

• The extended Kalman filter computes
  p(xtz1t,u1t) N(?t, ?t) for all times t 2 1
  T.
• Residual prediction error and prediction
  likelihood can be evaluated directly from the
  filters output.
• Measurement likelihood

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Robot testbed
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The robots state, inputs and measurements

• State
• x,y position coordinates.
• ? heading.
• v velocity.
• b gyro bias.
• Control inputs
• r rotational velocity.
• a forward acceleration.
• Measurements
• Optical wheel encoders measure v.
• A (cheap) GPS unit measures x,y (1Hz, 3m
  accuracy).
• A magnetic compass measures ?.

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System Equations
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Experimental setup

• We collected two data sets (100 seconds each) by
  driving the vehicle around on a grass field. One
  data set is used to discriminatively learn the
  parameters, the other data set is used to
  evaluate the performance of the different
  algorithms.
• A Novatel RT2 differential GPS unit (10Hz, 2cm
  accuracy) was used to obtain ground truth
  position data. Note the hardware on which our
  algorithms are evaluated do not have the more
  accurate GPS.

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Experimental results (1)

• Evaluation metrics
• RMS error (on position)
• Prediction log-loss (on position)

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Experimental results (2)
Zoomed in on this area on next figure.
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Experimental results (3)
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Experimental Results (4)
Zoomed in on this area on next figure.
x (cheap) GPS
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Experimental Results (5)
x (cheap) GPS
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Related Work

• Conditional Random Fields (J. Lafferty, A.
  McCallum, F. Pereira, 2001). The optimization
  criterion (translated to our setting) is
• Issues for our setting
• The criterion does not use filter estimates. (It
  uses smoother estimates instead.)
• The criterion assumes the state is fully
  observed.

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Discussion and conclusion

• In practice Kalman filters often require
  time-consuming hand-engineering of the noise
  parameters to get optimal performance.
• We presented a family of algorithms that use a
  discriminative criterion to learn the noise
  parameters automatically from data.
• Advantages
• Eliminates hand-engineering step.
• More accurate state estimation than even a
  carefully hand-engineered filter.

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Discriminative
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Training of
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Kalman Filters
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• Pieter Abbeel, Adam Coates, Mike Montemerlo,
  Andrew Y. Ng and Sebastian Thrun
• Stanford University

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• Pieter Abbeel, Adam Coates, Mike Montemerlo,
  Andrew Y. Ng and Sebastian Thrun
• Stanford University

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Robot testbed