## Variational Method of the Quantum Simple Harmonic Oscillator using PyTorch

A fun method (and useful!) for solving the ground state of the Schrodinger equation is to minimize the energy integral $dx \psi^\dagger H \psi$ while keeping the total probability 1. Pytorch is a big ole optimization library, so let’s give it a go.

I’ve tried two versions, using a stock neural network with relus and making it a bit easier by giving a gaussian with variable width and shift.

We can mimic the probability constraint by dividing by to total normalization $\int dx \psi^\dagger \psi$. A Lagrange multiplier or penalty method may allows us to access higher wavefunctions.

SGD seems to do a better job getting a rounder gaussian, while Adam is less finicky but makes a harsh triangular wavefunction.

The ground state solution of $-\frac{d^2\psi}{dx^2} + x^2\psi=E\psi$ is $e^{-x^2/2}$, with an energy of 1/2 (unless I botched up my head math). We may not get it, because we’re not sampling a very good total domain. Whatever, for further investigation.

Very intriguing is that pytorch has a determinant in it, I believe. That opens up the possibility of doing a Hartree-Fock style variational solution.

Here is my garbage

Edit: Hmm I didn’t compensate for the fact I was using randn sampling. That was a goof. I started using unifrom sampling, which doesn’t need compensation

## Shit Compiling to Categories using Type level Programming in Haskell

So I’veÂ  been trying to try and approximate Conal Elliott’s compiling to categories in bog standard Haskell and I think I’ve got an interesting chunk.

His approach in using a GHC plugin is better for a couple reasons. One really important thing is that he gets to piggy back on GHC optimizations for the lambdas. I have only implemented a very bad evaluation strategy. Perhaps we could recognize shared subexpressions and stuff, but it is more work. I seem somewhat restricted in what I can do and sometimes type inference needs some help. Not great. However, GHC plugins definitely bring their own difficulties.

What I’ve got I think still has roots in my thinking from this previous post

There are a couple insights that power this implementation

1. A fully polymorphic tuple to tuple converter function is uniquely defined by it’s type. For example, swap :: (a,b) -> (b,a), id:: a -> a, fst :: (a,b) ->a, snd::(a,b)->b are all unique functions due to the restrictions of polymorphism. Thus typelevel programming can build the implementation from the type.
2. Getting a typeclass definition to notice polymorphism is hard though. I haven’t figured out how to do it, if it is even possible. We get around it by explicitly newtyping every pattern match on a polymorphic variable like so \(V x, V y) -> (y,x). Two extra characters per variable. Not too shabby.
3. You can abuse the type unification system as a substitution mechanism. Similar to HOAS, you can make a lambda interpreter at the type level that uses polymorphism as way of generating labels for variables. This is probably related to Oleg Kiselyov’s type level lambda calculus, but his kind of confuses meÂ http://okmij.org/ftp/Computation/lambda-calc.html#haskell-type-level
4. You can inject a categorical literal morphism using a wrapping type to be extracted later using an apply operator and type App f a. An infix () makes it feel better.

So here is the rough structure of what the typelevel programming is doing

You can do a depth first traversal of the input tuple structure, when you hit V, unify the interior type with a Nat labelled Leaf. At the same time, you can build up the actual value of the structure so that you can apply the lambda to it and get the output which will hold a tree that has morphism literals you want.

Then inspect the output type of the lambda which now has Nat labels, and traverse the labelled tree again to find the appropriate sequence of fst and snd to extract what you need. If you run into an application App, you can pull the literal out now and continue traversing down.

At the moment I’ve only tested with the (->) Category, which is a bit like demonstrating a teleporter by deconstructing a guy and putting back in the same place, but I think it will work with others. We’ll see. I see no reason why not.

At the moment, I’m having trouble getting GHC to not freakout unless you explicitly state what category you’re working in, or explicitly state that you’re polymorphic in the Category k.

Some future work thoughts: My typelevel code is complete awful spaghetti written like I’m a spastic trapped animal. It needs some pruning. I think all those uses of Proxy can be cleaned up by using TypeApplications. I need to implement some more categories. Should I conform to the Constrained-Categories package? Could I use some kind of hash consing to find shared structures? Could Generic or Generic1 help us autoplace V or locate polymorphism? Maybe a little Template Haskell spice to inject V?

Here’s the most relevant bits, with my WIP git repositoryÂ here

## Deducing Row Sparse Matrices from Matrix Vector Product Samples

So as part of betting the banded hessian out of pytorch, I’ve been trying to think about what it is I am doing and I’ve come up with an alternative way of talking about it.

If every row of a matrix has <N non zero entries, you can back out that matrix from N matrix vector samples of it. You have many choices for the possible sampling vectors. Random works well.

The unknown in this case is the row of the matrix, represented in green. We put a known set of inputs into it and get outputs. Each row of the output, represented in red, can tell use the matrix row. We have to invert the matrix that consists of all the elements that the nonzero elements of that row touches represented in blue. That black T is a transpose. To turn those row vectors into column vectors, we have to transpose everything.

For a banded matrix, we have a shifting sample matrix that we have to invert, one for each row.

A cute trick we can use to simplify the edge cases where we run off the ends is to extend the sample matrix with some random trash. That will actually put the entries in the appropriate place and will keep the don’t cares close to zero also.

In my previous post I used a stack of identity matrices. These are nice because a shifted identity matrix is a circular permutation of the entries, which is very easy to undo. That was what the loop that used numpy.roll was doing. Even better, it is easy to at least somewhat vectorize the operation and you can produce those sampling vectors using some clever use of broadcasting of an identity matrix.

An alternative formulation that is a little tighter. I want the previous version because the samples isn’t actually always random. Often they won’t really be under our control.

This all still doesn’t address the hermitian problem. The constraint that A is hermitian hypothetically allows you to take about half the samples. But the constraints make it tougher to solve. I haven’t come up with anything all that much better than vectorizing the matrix and building a matrix out of the samples in the appropriate spots.

I think such a matrix will be banded if A is banded, so that’s something at least.

## 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.Â  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 cutoffÂ Â 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 convenientÂ https://github.com/rkern/line_profilerÂ 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 paperÂ 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.

## Cartpole Camera System – OpenCV + PS EYE + IR

We tried using colored tape before. It was okay after manual tuning, but kind of sucked. Commerical motion tracking systems use IR cameras and retroreflectors.

We bought some retroreflective tape and put it on the pole. http://a.co/0A9Otmr

We removed our PS EYE IR filter. The PS EYE is really cheap (~7\$) and has a high framerate mode (100+ fps). People have been using it for a while for computer vision projects.

http://wiki.lofarolabs.com/index.php/Removing_the_IR_Filter_from_the_PS_Eye_Camera

We followed the instructions, but did not add the floppy disk and sanded down the base of the lens to bring the image back into focus.

We bought an IR LED ring light which fit over the camera with the plastic cover removed and rubber banded it in place.

http://a.co/2sGUY08

If you snip the photoresistor it is always on, since the photoresistor is high resistance in the dark. We used a spare 12V power supply that we soldered a connector on for.

We had also bought an IR pass filter on amazon, but it does not appear to help.

Useful utilties: qv4l2, v4l2-ctl and v4l2-utils. You can change lots of stuff.

qv4l2 -d 1 is very useful for experiementation

Useful options to  v4l2-ctl : -d selects camera, -p sets framerate -l gives a list of changeable options. You have to turn off the automatic stuff before it becomes changeable. Counterintuitively auto-exposure seems to have 1 as off.

There has been a recent update to opencv to let the v4l2 buffer size be changed. We’re hoping this will really help with our latency issues

A useful blog. We use v4l2-ctl for controlling the exposure programmatically

http://www.jayrambhia.com/blog/capture-v4l2

Oooh. The contour method + rotated rectangle is working really well for matching the retroreflective tape.

https://docs.opencv.org/3.3.1/dd/d49/tutorial_py_contour_features.html

You need to reduce the video size to 320×240 if you want to go to the highest framerate of 187fps

In regards to the frame delay problem from before, it’s not clear that we’re really seeing it? We are attempting both the screen timestamp technique and also comparing to our rotary encoder. In the screen timestamp technique, it is not so clear that what we measure there is latency, and if it is, it includes the latency of the monitor itself, which is irrelevant.

## Extracting a Banded Hessian in PyTorch

So pytorch does have some capability towards higher derivatives, with the caveat that you have to dot the gradients to turn them back into scalars before continuing. What this means is that you can sample a single application of theÂ  Hessian (the matrix of second derivatives) at a time.

One could sample out every column of the hessian for example. Performance-wise I don’t know how bad this might be.

For a banded hessian, which will occur in a trajectory optimization problem (the bandedness being a reflection of the finite difference scheme), you don’t need that many samples. This feels more palatable. You only need to sample the hessian roughly the bandwidth number of times, which may be quite small. Plus, then you can invert that banded hessian very quickly using special purpose banded matrix solvers, which are also quite fast. I’m hoping that once I plug this into the trajectory optimization, I can use a Newton method (or SQP?) which will perform better than straight gradient descent.

If you pulled just a single column using [1,0,0,0,0,0..] for example, that would be wasteful, since there are so many known zeros in the banded matrix. Instead something like [1,0,0,1,0,0,1,0,0..] will not have any zeros in the result. This gets us every 3rd row of the matrix. Then we can sample with shifted versions likeÂ [0,1,0,0,1,0,0,1,0,0..]. until we have all the rows somewhere. Then there is some index shuffling to put the thing into a sane ordering, especially so that we can useÂ https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solveh_banded.html which requires the banded matrix to be given in a particular form.

An alternative approach might be to use an fft with some phase twiddling. Also it feels like since the Hessian is hermitian we ought to be able to use about half the samples, since half are redundant, but I haven’t figured out a clean way to do this yet. I think that perhaps sampling with random vectors and then solving for the coefficients would work, but again how to organize the code for such a thing?

Here’s a snippet simulatng extracting the band matrix from matrix products.

and here is the full pytorch implementation including a linear banded solve.

Output:

## 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

http://stanford.edu/class/ee364b/lectures/subgrad_method_notes.pdf

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

______________________________

Made some big adjustments

Switched to using pytorch optimizers. Adam seems to work the best. Maybe 5x as fast convergence as my gradient descent. Adagrad and Adadelta aren’t quite as good. Should still try momentum. Have to reset the initial conditions after every iteration. A better way? Maybe pass x0 in to calc_loss separately?

Switched over to using the method of multipliersÂ http://www.cs.cmu.edu/~pradeepr/convexopt/Lecture_Slides/Augmented-lagrangian.pdf

The idea is to increase the quadratic constraint cost slowly over time, while adjusting a Lagrange mutiplier term to compensate also. Seems to be working better. The scheduling of the increase is still fairly ad hoc.

The left is residuals of obeying the equations of motion, the middle is the force and trajectories themselves and the right is cost vs iteration time. Not entirely clear that a residual of 0.04 is sufficient. Integrated over time this could be an overly optimistic error of 0.2 ish I’d guess. That is on the edge of making me uncomfortable. Increase rho more? Also that force schedule seems funky and overly complex. Still, improvement from before. Feels like we’re cookin’ with gas