Differentiable Bayes Filters Lecture 9 Announcement - Feedback - - PowerPoint PPT Presentation

Differentiable Bayes Filters

Lecture 9

Announcement

• Feedback for Project proposal latest on Wednesday night
• Kevin Zakka started course notes (see Piazza) - bonus points for contributing

What will you take home today?

Recap Bayesian Filters Kalman Filter Extended Kalman Filter Particle Filter Differentiable Filters Backpropagation through a Kalman Filter Backpropagation through a Particle Filter

Recap Bayes Filters

xt−1 xt xt+1 zt+1 zt−1 zt ut ut−1 ut+1

Kalman Filter

xt = f(xt−1, ut−1, ωt) zt = h(xt, νt)

xt = f(xt−1, ut−1, ωt) = Axt−1 + But + ωt zt = h(xt, νt) = Hxt + νt

ω ∼ N(0, Qt) ν ∼ N(0, Rt) x ∼ N(x, P)

Kalman Filter

Prediction Step: Update Step: ˆ xt = Axt−1 + But (1) ˆ Pt = APt−1AT + Qt (2) it = zt − Hˆ xt (3) Kt = ˆ PtHT Hˆ PtHT + Rt (4) xt = ˆ xt + Ktit (5) Pt = (In − KtH)ˆ Pt (6)

Extended Kalman Filter

Prediction Step: Update Step: ˆ xt = Axt−1 + But (1) ˆ Pt = APt−1AT + Qt (2) it = zt − Hˆ xt (3) Kt = ˆ PtHT Hˆ PtHT + Rt (4) xt = ˆ xt + Ktit (5) Pt = (In − KtH)ˆ Pt (6)

A|xt H|xt

Particle Filter

p(xt|z1:t, u1:t, x0) Xt = {x0

t, x1 t, · · · , xN t }

Particle Filter

Xt = f(Xt−1, ut, ωt) X0 p(zt|xi

t)

xi : wi

t = wi t−1p(zt|xi t)

When to use what? How to choose hyperparameters?

Priors and Hyperparameters

A lot of hardcoded knowledge!

1.

State Representation

2.

Models

• Forward Model
• State to next state
• Action to next state
• Measurement Model

3.

Probabilistic Properties

• Process Noise
• Measurement Noise

Differentiable filters

Can we learn models and hyperparameters from data? Approach: Embed algorithmic structure of Bayesian Filtering into a recurrent neural network.

• prevents overfitting through regularization
• Avoids manual tuning and modeling

A note on variables

In Robotics and Control: In Machine Learning: In Artificial Intelligence:

BackpropKF : Learning Discriminative Deterministic State

• Estimators. Haarnoja et al. NeurIPS 2016
• Differentiable version of the Kalman Filter
• Uses Images as observations; learns a sensors that outputs state directly

Differentiable Kalman Filter - Structure

Differentiable Kalman Filter - Structure

Differentiable Kalman Filter – Loss Function

L(l0...T , µ0...T , Σ0...T , w) = λ1

T

X

t=0

1 2((lt µt)T Σ−1

t (lt µt) + log(|Σt|)) + λ2 T

X

t=0

k (lt µt) k2 +λ3 k w k2

Differentiable Kalman Filter – Experiments and Baselines

Differentiable Kalman Filter – Experiments and Baselines

• Kitti – Visual Odometry Datatset
• 22 stereo sequences with LIDAR
• 11 sequences with ground truth

(GPS/IMU data)

• 11 sequences without ground truth (for

evaluation)

Differentiable Kalman Filter – Experiments and Baselines

Results reproduced by Claire Chen

Differentiable Particle Filters: End-to-End Learning with Algorithmic Priors. Jonschkowski et al. RSS 2018.

Differentiable Particle Filters: End-to-End Learning with Algorithmic Priors. Jonschkowski et al. RSS 2018.

Differentiable Particle Filters: End-to-End Learning with Algorithmic Priors. Jonschkowski et al. RSS 2018.

∀xi

t−1 ∈ Xt−1

sample ni ∼ N(0, 1) ωi

t =

p Qini xi

t = f(xi t−1, ut, ωi t)

Differentiable Particle Filters: End-to-End Learning with Algorithmic Priors. Jonschkowski et al. RSS 2018.

Particle Filter Networks with Application to Visual

• Localization. Karkus et al. CORL 2018.

Differentiable Particle Filter – Loss Function

Differentiable Particle Filter – Experiments and Baselines

Differentiable Particle Filter – Experiments and Baselines

Differentiable Particle Filter – Experiments and Baselines