Efficient Policy GradientOptimization/Learning of Feedback

Controllers

Chris Atkeson

Punchlines

• Optimize and learn policies. Switch from “value iteration” to “policy

iteration”.• This is a big switch from optimizing and

learning value functions.• Use gradient-based policy optimization.

Motivations

• Efficiently design nonlinear policies• Make policy-gradient reinforcement

learning practical.

Model-Based Policy Optimization• Simulate policy u = π(x,p) from some initial

states x0 to find policy cost.• Use favorite local or global optimizer to

optimize simulated policy cost.• If gradients are used, they are typically

numerically estimated.• Δp = -ε ∑x0w(x0)Vp 1st order gradient

• Δp = -(∑x0w(x0)Vpp)-1 ∑x0w(x0)Vp 2nd order

Can we make model-based policy gradient more efficient?

Analytic Gradients• Deterministic policy: u = π(x,p) • Policy Iteration (Bellman Equation): Vk-1(x,p) = L(x,π(x,p)) + V(f(x,π(x,p)),p)• Linear models: f(x,u) = f0 + fxΔx + fuΔu

L(x,u) = L0 + LxΔx + LuΔu

π(x,p) = π0 + πxΔx + πpΔp

V(x,p) = V0 + VxΔx + VpΔp• Policy Gradient: Vx

k-1 = Lx + Luπx + Vx(fx + fuπx)

Vpk-1 = (Lu + Vxfu)πp + Vp

Handling Constraints

• Lagrange multiplier approach, with constraint violation value function.

Vpp: Second Order Models

Regularization

LQBR: Linear (dynamics) Quadratic (cost) Bilinear (policy) Regulator

Timing Test

Antecedents

• Optimizing control “parameters” in DDP: Dyer and McReynolds 1970.

• Optimal output feedback design (1960s-1970s)

• Multiple model adaptive control (MMAC)• Policy gradient reinforcement learning• Adaptive critics, Werbos: HDP, DHP, GDHP,

ADHDP, ADDHP

When Will LQBR Work?

• Initial stabilizing policy is known (“output stabilizable”)

• Luu is positive definite.

• Lxx is positive semi-definite and (sqrt(Lxx),Fx) is detectable.

• Measurement matrix C has full row rank.

Locally Linear Policies

Local Policies

GOAL

Cost Of One Gradient Calculation

Continuous Time

Other Issues

• Model Following• Stochastic Plants• Receding Horizon Control/MPC• Adaptive RHC/MPC• Combine with Dynamic Programming• Dynamic Policies -> Learn State Estimator

Optimize Policies

• Policy Iteration, with gradient-based policy improvement step.

• Analytic gradients are easy.• Non-overlapping sub-policies make second

order gradient calculations fast.• Big problem: How choose policy structure?