## Pytorch Trajectory Optimization Part 4: Cleaner code, 50Hz

Cleaned up the code more and refactored some things.

Added backtracking. It will backtrack on the dx until the function is actually decreasing.

Prototyped the online part with shifts. Seems to work well with a fixed penalty parameter rho~100. Runs at ~50Hz with pretty good performance at 4 optimization steps per time step. Faster or slower depending on the number of newton steps per time step we allow ourselves.  Still to see if the thing will control an actual cartpole.

The majority of time is spent just backwards calculating the hessian still (~50%).

I’ve tried a couple different schemes (direct projection of the delLy terms or using y = torch.eye). None particularly seem to help.

The line search is also fairly significant (~20% of the time) but it really helps with both stability and actually decreasing the number of hessian steps, so it is an overall win. Surprisingly during the line search, projecting out the batch to 0 doesn’t matter much. How could this possibly make sense?

What I should do is pack this into a class that accepts new state observations and initializes with the warm start. Not clear if I should force the 4 newton steps on you or let you call them yourself. I think if you use too few it is pretty unstable (1 doesn’t seem to work well. 2 might be ok and gets you up to 80Hz maybe.)

The various metaparameters should be put into the __init__. The stopping cutoff  1e-7, Starting rho (~0.1), rho increase (x10) , backtrack alpha decrease factor (0.5 right now), the online rho (100). Hopefully none of these matter two much. I have noticed going too small with cutoff leading to endless loops.

Could swapping the ordering of time step vs variable number maybe help?

For inequality constraints like the track length and forces, exponential barriers seems like a more stable option compared to log barriers. Log barriers at least require me to check if they are going NaN.

I attempted the pure Lagrangian version where lambda is just another variable. It wasn’t working that great.

## Pytorch Trajectory Optimization 3: Plugging in the Hessian

I plugged in the hessian extraction code for using newton steps.

When I profiled it using the oh so convenient https://github.com/rkern/line_profiler I found almost all of my time was spent in the delLy.backwards step. For each hessian I needed to run this B (the band width) times and each time cost ~0.6ms. For the entire run to converge took about 70 iterations and 1000 runs of this backwards step, which came out to 0.6 seconds. It is insane, but actually even calculating the band of the hessian costs considerably more time than inverting it.

So to speed this up, I did a bizarre thing. I replicated the entire system B times. Then I can get the entire hessian band in a single call to backwards. remarkably, although B ~ 15 this only slowed backwards down by 3x. This is huge savings actually, while obviously inefficient. The entire program has gone down from 1.1s to 0.38s, roughly a 3x improvement. All in all, this puts us at 70/0.38 ~ 185 Hz for a newton step. Is that good enough? I could trim some more fat. The Fast MPC paper http://web.stanford.edu/~boyd/papers/fast_mpc.html says we need about ~5 iterations to tune up a solution, this would mean running at 40Hz. I think that might be ok.

Since the hessian is hermitian it is possible to use roughly half the calls (~B/2), but then actually extracting the hessian is not so simple. I haven’t figured out a way to comfortably do such a thing yet. I think I could figure out the first column and then subtract (roughly some kind of gaussian elimination procedure).

It has helped stability to regularize everything with a surprising amount of weight in the cost. I guess since I anticipate all values being in the range of -10,10, maybe this makes sense.

Now I need to try not using this augmented Lagrangian method and just switching to a straight newton step.

Edit: Ooh. Adding a simple backtracking line search really improves stability.

## PyTorch Trajectory Optimization Part 2: Work in Progress

I actually have been plotting the trajectories, which is insane that I wasn’t already doing in part 1. There was clearly funky behavior.

Alternating the Lagrange multiplier steps and the state variable steps seems to have helped with convergence. Adding a cost to the dynamical residuals seems to have helped clean them up also.

I should attempt some kind of analysis rather than making shit up. Assuming quadratic costs (and dynamics), the problem is tractable. The training procedure is basically a dynamical system.

Changed the code a bit to use more variables. Actually trying the cart pole problem now. The results seem plausible. A noisy but balanced dynamical residual around zero. And the force appears to flip it’s direction as the pole crosses the horizontal.

Polyak’s step length

The idea is that if you know the optimal value you’re trying to achieve, that gives you a scale of gradient to work with. Not as good as a hessian maybe, but it’s somethin’. If you use a gradient step of $x + (f-f*)\frac{\nabla f}{|\nabla f|^2}$ it at least has the same units as x and not f/x. In some simple models of f, this might be exactly the step size you’d need. If you know you’re far away from optimal, you should be taking some big step sizes.

The Polyak step length has not been useful so far. Interesting idea though.

Problems:

1. The step size is ad hoc.
2. Lagrange multiplier technique does not seem to work
3. Takes a lot of steps
4. diverges
5. seems to not be getting an actual solution
6. Takes a lot of iterations

On the table: Better integration scheme. Hermite collocation?

Be more careful with scaling, what are the units?

mutiplier smoothing. Temporal derivative of lagrange multiplier in cost?

alternate more complete solving steps

huber on theta position cost. Square too harsh? Punishes swing away too much?

more bullshit optimization strats as alternatives to grad descent

weight sooner more than later. Care more about earlier times since want to do model predictive control

Just solve eq of motion don’t worry about control as simpler problem

Pole up balancing

logarithm squeezing method – nope

The lambda * x model of lagrange mulitplier. Leads to oscillation

Damping term?

This learning rate is more like a discretization time step than a decay parameter. Well the product of both actually.

Heat equation model. Kind of relaxing everything into place

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