## Deep learning for control using augmented Hessian-free optimization

Traditionally, deep learning is applied to feed-forward tasks, like classification, where the output of the network doesn’t affect the input to the network. It is a decidedly harder problem when the output is recurrently connected such that network output affects the input. Because of this application of deep learning methods to control was largely unexplored until a few years ago. Recently, however, there’s been a lot of progress and research in this area. In this post I’m going to talk about an implementation of deep learning for control presented by Dr. Ilya Sutskever in his thesis Training Recurrent Neural Networks.

In his thesis, Dr. Sutskever uses augmented Hessian-free (AHF) optimization for learning. There are a bunch of papers and posts that go into details about AHF, here’s a good one by Andrew Gibiansky up on his blog, that I recommend you check out. I’m not going to really talk much here about what AHF is specifically, or how it differs from other methods, if you’re unfamiliar there are lots of places you can read up on it. Quickly, though, AHF is kind of a bag of tricks you can use with a fast method for estimating the curvature of the loss function with respect to the weights of a neural network, as well as the gradient, which allows it to make larger updates in directions where the loss function doesn’t change quickly. So rather than estimating the gradient and then doing a small update along each dimension, you can make the size of your update large in directions that change slowly and small along dimensions where things change quickly. And now that’s enough about that.

In this post I’m going to walk through using a Hessian-free optimization library (version 0.3.8) written by my compadre Dr. Daniel Rasmussen to train up a neural network to train up a 2-link arm, and talk about the various hellish gauntlets you need run to get something that works. Whooo! The first thing to do is install this Hessian-free library, linked above.

I’ll be working through code edited a bit for readability, to find the code in full you can check out the files up on my GitHub.

Build the network

Dr. Sutskever specified the structure of the network in his thesis to be 4 layers: 1) a linear input layer, 2) 100 Tanh nodes, 3) 100 Tanh nodes, 4) linear output layer. The network is connected up with the standard feedforward connections from 1 to 2 to 3 to 4, plus recurrent connections on 2 and 3 to themselves, plus a ‘skip’ connection from layer 1 to layer 3. Finally, the input to the network is the target state for the plant and the current state of the plant. So, lots of recursion! Here’s a picture:

The output layer connects in to the plant, and, for those unfamiliar with control theory terminology, ‘plant’ just means the system that you’re controlling. In this case an arm simulation.

Before we can go ahead and set up the network that we want to train, we also need to specify the loss function that we’re going to be using during training. The loss function in Ilya’s thesis is a standard one:

$L(\theta) = \sum\limits^{N-1}\limits_{t=0} \ell(\textbf{u}_t) + \ell_f(\textbf{x}_N),$
$\ell(\textbf{u}_t) = \alpha \frac{||\textbf{u}_t||^2}{2},$
$\ell_f(\textbf{x}_N) = \frac{||\textbf{x}^* - \textbf{x}_t||^2}{2}$

where $L(\theta)$ is the total cost of the trajectory generated with $\theta$, the set of network parameters, $\ell(\textbf{u})$ is the immediate state cost, $\ell_f(\textbf{x})$ is the final state cost, $\textbf{x}$ is the state of the arm, $\textbf{x}^*$ is the target state of the arm, $\textbf{u}$ is the control signal (torques) that drives the arm, and $\alpha$ is a gain value.

To code this up using the hessianfree library we do:

from hessianfree import RNNet
from hessianfree.nonlinearities import (Tanh, Linear, Plant)
from hessianfree.loss_funcs import SquaredError, SparseL2

l2gain = 10e-3 * dt # gain on control signal loss
rnn = RNNet(
# specify the number of nodes in each layer
shape=[num_states * 2, 96, 96, num_states, num_states],
# specify the function of the nodes in each layer
layers=[Linear(), Tanh(), Tanh(), Linear(), plant],
# specify the layers that have recurrent connections
rec_layers=[1,2],
# specify the connections between layers
conns={0:[1, 2], 1:[2], 2:[3], 3:[4]},
# specify the loss function
loss_type=[
# squared error between plant output and targets
SquaredError(),
# penalize magnitude of control signal (output of layer 3)
SparseL2(l2gain, layers=[3])],
use_GPU=True)


Note that if you want to run it on your GPU you’ll need PyCuda and sklearn installed. And a GPU.

An important thing to note as well is that in Dr. Sustkever’s thesis when we’re calculating the squared error of the distance from the arm state to the target, this is measured in joint angles. So it’s kind of a weird set up to be looking at the movement of the hand but have your cost function in joint-space instead of end-effector space, but it definitely simplifies training by making the cost more directly relatable to the control signal. So we need to calculate the joint angles of the arm that will have the hand at different targets around a circle. To do this we’ll take advantage of our inverse kinematics solver from way back when, and use the following code:

def gen_targets(arm, n_targets=8, sig_len=100):
#Generate target angles corresponding to target
#(x,y) coordinates around a circle
import scipy.optimize

x_bias = 0
if arm.DOF == 2:
y_bias = .35
dist = .075
elif arm.DOF == 3:
y_bias = .5
dist = .2

# set up the reaching trajectories around circle
targets_x = [dist * np.cos(theta) + x_bias \
for theta in np.linspace(0, np.pi*2, 65)][:-1]
targets_y = [dist * np.sin(theta) + y_bias \
for theta in np.linspace(0, np.pi*2, 65)][:-1]

joint_targets = []
for ii in range(len(targets_x)):
joint_targets.append(arm.inv_kinematics(xy=(targets_x[ii],
targets_y[ii])))
targs = np.asarray(joint_targets)

# repeat the targets over time
for ii in range(targs.shape[1]-1):
targets = np.concatenate(
(np.outer(targs[:, ii], np.ones(sig_len))[:, :, None],
np.outer(targs[:, ii+1], np.ones(sig_len))[:, :, None]), axis=-1)
targets = np.concatenate((targets, np.zeros(targets.shape)), axis=-1)
# only want to penalize the system for not being at the
# target at the final state, set everything before to np.nan
targets[:, :-1] = np.nan

return targets


And you can see in the last couple lines that to implement the distance to target as a final state cost penalty only we just set all of the targets before the final time step equal to np.nan. If we wanted to penalize distance to target throughout the whole trajectory we would just comment that line out.

Create the plant

You’ll notice in the code that defines our RNN I set the last layer of the network to be plant, but that that’s not defined anywhere. Let’s talk. There are a couple of things that we’re going to need to incorporate our plant into this network and be able to use any deep learning method to train it. We need to be able to:

1. Simulate the plant forward; i.e. pass in input and get back the resulting plant state at the next timestep.
2. Calculate the derivative of the plant state with respect to the input; i.e. how do small changes in the input affect the state.
3. Calculate the derivative of the plant state with respect to the previous state; i.e. how do small changes in the plant state affect the state at the next timestep.
4. Calculate the derivative of the plant output with respect to its state; i.e. how do small changes in the current position of the state affect the output of the plant.

So 1 is easy, we have the arm simulations that we want already, they’re up on my GitHub. Number 4 is actually trivial too, because the output of our plant is going to be the state itself, so the derivative of the output with respect to the state is just the identity matrix.

For 2 and 3, we’re going to need to calculate some derivatives. If you’ve read the last few posts you’ll note that I’m on a finite differences kick. So let’s get that going! Because no one wants to calculate derivatives!

Important note, the notation in these next couple pieces of code is going to be a bit different from my normal notation because they’re matching with the hessianfree library notation, which is coming from a reinforcement learning literature background instead of a control theory background. So, s is the state of the plant, and x is the input to the plant. I know, I know. All the same, make sure to keep that in mind.

# calculate ds0/dx0 with finite differences
d_input_FD = np.zeros((x.shape[0], # number of trials
x.shape[1], # number of inputs
self.state.shape[1])) # number of states
for ii in range(x.shape[1]):
# calculate state adding eps to x[ii]
self.reset_plant(self.prev_state)
inc_x = x.copy()
inc_x[:, ii] += self.eps
self.activation(inc_x)
state_inc = self.state.copy()
# calculate state subtracting eps from x[ii]
self.reset_plant(self.prev_state)
dec_x = x.copy()
dec_x[:, ii] -= self.eps
self.activation(dec_x)
state_dec = self.state.copy()

d_input_FD[:, :, ii] = \
(state_inc - state_dec) / (2 * self.eps)
d_input_FD = d_input_FD[..., None]


Alrighty. First we create a tensor to store the results. Why is it a tensor? Because we’re going to be doing a bunch of runs at once. So our state dimensions are actually trials x size_input. When we then take the partial derivative, we end up with trials many size_input x size_state matrices. Then we increase each of the parameters of the input slightly one at a time and store the results, decrease them all one at a time and store the results, and compute our approximation of the gradient.

Next we’ll do the same for calculating the derivative of the state with respect to the previous state.

# calculate ds1/ds0
d_state_FD = np.zeros((x.shape[0], # number of trials
self.state.shape[1], # number of states
self.state.shape[1])) # number of states
for ii in range(self.state.shape[1]):
# calculate state adding eps to self.state[ii]
state = np.copy(self.prev_state)
state[:, ii] += self.eps
self.reset_plant(state)
self.activation(x)
state_inc = self.state.copy()
# calculate state subtracting eps from self.state[ii]
state = np.copy(self.prev_state)
state[:, ii] -= self.eps
self.reset_plant(state)
self.activation(x)
state_dec = self.state.copy()

d_state_FD[:, :, ii] = \
(state_inc - state_dec) / (2 * self.eps)
d_state_FD = d_state_FD[..., None]


Great! We’re getting closer to having everything we need. Another thing we need is a wrapper for running our arm simulation. It’s going to look like this:

def activation(self, x):
state = []
# iterate through and simulate the plant forward
# for each trial
for ii in range(x.shape[0]):
self.arm.reset(q=self.state[ii, :self.arm.DOF],
dq=self.state[ii, self.arm.DOF:])
self.arm.apply_torque(u[ii])
state.append(np.hstack([self.arm.q, self.arm.dq]))
state = np.asarray(state)

self.state = self.squashing(state)


This is definitely not the fastest code to run. Much more ideally we would put the state and input into vectors and do a single set of computations for each call to activation rather than having that for loop in there. Unfortunately, though, we’re not assuming that we have access to the dynamics equations / will be able to pass in vector states and inputs.

Squashing
Looking at the above code that seems pretty clear what’s going on, except you might notice that last line calling self.squashing. What’s going on there?

The squashing function looks like this:

def squashing(self, x):
index_below = np.where(x < -2*np.pi)
x[index_below] = np.tanh(x[index_below]+2*np.pi) - 2*np.pi
index_above = np.where(x > 2*np.pi)
x[index_above] = np.tanh(x[index_above]-2*np.pi) + 2*np.pi
return x


All that’s happening here is that we’re taking our input, and doing nothing to it as long as it doesn’t start to get too positive or too negative. If it does then we just taper it off and prevent it from going off to infinity. So running a 1D vector through this function we get:

This ends up being a pretty important piece of the code here. Basically it prevents wild changes to the weights during learning to result in the system breaking down. So the state of the plant can’t go off to infinity and cause an error to be thrown, stopping our simulation. But because the target state is well within the bounds of where the squashing function does nothing, post-training we’ll still be able to use the resulting network to control a system that doesn’t have this fail safe built in. Think of this function as training wheels that catch you only if you start falling.

With that, we no have pretty much all of the parts necessary to begin training our network!

Training the network

We’re going to be training this network on the centre-out reaching task, where you start at a centre point and reach to a bunch of target locations around a circle. I’m just going to be re-implementing the task as it was done in Dr. Sutskever’s thesis, so we’ll have 64 targets around the circle, and train using a 2-link arm. Here’s the code that we’ll use to actually run the training:

for ii in range(last_trial+1, num_batches):
# train a bunch of batches using the same input every time
# to allow the network a chance to minimize things with
# stable input (speeds up training)
err = rnn.run_batches(plant, targets=None,
max_epochs=batch_size,
optimizer=HessianFree(CG_iter=96, init_damping=100))

# save the weights to file, track trial and error
# err = rnn.error(inputs)
err = rnn.best_error
name = 'weights/rnn_weights-trial%04i-err%.3f'%(ii, err)
np.savez_compressed(name, rnn.W)


Training your own network

A quick aside: if you want to run this code yourself, get a real good computer, have an arm simulation ready, the hessianfree Python library installed, and download and run this train_hf.py file. (Note: I used version 0.3.8 of the hessianfree library, which you can install using pip install hessianfree==0.3.8) This will start training and save the weights into a weights/ folder, so make sure that that exists in the same folder as train_hf.py. If you want to view the results of the training at any point run the plot_error.py file, which will load in the most recent version of the weights and plot the error so far. If you want to generate an animated plot like I have below run gen_animation_plots.py and then the last command from my post on generating animated gifs.

Another means of seeing the results of your trained up network is to use the controller I’ve implemented in my controls benchmarking suite, which looks for a set of saved weights in the controllers/weights folder, and will load it in and use it to generate command signals for the arm by running it with

python run.py arm2_python ahf reach --dt=1e-2


where you replace arm2_python with whatever arm model you trained your model on. Note the --dt=1e-2 flag, that is important because the model was trained with a .01 timestep and things get a bit weird if you suddenly change the dynamics on the controller.

OK let’s look at some results!

Results

OK full discretion, these results are not optimizing the cost function we discussed above. They’re implementing a simpler cost function that only looks at the final state, i.e. it doesn’t penalize the magnitude of the control signal. I did this because Dr. Sutskever says in his thesis he was able to optimize with just the final state cost using much smaller networks. I originally looked at neurons with 96 neurons in each layer, and it just took forgoddamnedever to run. So after running for 4 weeks (not joking) and needing to make some more changes I dropped the number of neurons and simplified the task.

The results below are from running a network with 32 neurons in each layer controlling this 2-link arm, and took another 4-5 weeks to train up.

Hey that looks good! Not bad, augmented Hessian-free learning, not bad. It had pretty consistent (if slow) decline in the error rate, with a few crazy bumps from which it quickly recovered. Also take note that each training iteration is actually 32 runs, so it’s not 12,50-ish runs it’s closer to 400,000 training runs that it took to get here.

One biggish thing that was a pain was that it turns out that I only trained the neural network for reaching in the one direction, and when you only train it to reach one way it doesn’t generalize to reaching back to the starting point (which, fair enough). But, I didn’t realize this until I was took the trained network and ran it in the benchmarking code, at which point I was not keen to redo all of the training it took to get the neural network to the level of accuracy it was at under a more complicated training set. The downside of this is that even though I’ve implemented a controller that takes in the trained network and uses it to control the arm, to do the reaching task I have to just do a hard reset after the arm reaches the target, because it can’t reach back to the center, like all the other controllers. All the same, here’s an animation of the trained up AHF controller reaching to 8 targets (it was trained on all 64 above though):

Things don’t always go so smoothly, though. Here’s results from another training run that took around 2-3 weeks, and uses a different 2-link arm model (translated from Matlab code written by Dr. Emo Todorov):

What I found frustrating about this was that if you look at the error over time then this arm is doing as well or better than the previous arm at a lot of points. But the corresponding trajectories look terrible, like something you would see in a horror movie based around getting good machine learning results. This of course comes down to how I specified the cost function, and when I looked at the trajectories plotted over time the velocity of the arm is right at zero at the final time step, which it is not quiiiitte the case for the first controller. So this second network has found a workaround to minimize the cost function I specified in a way I did not intend. To prevent this, doing something like weighting the distance to target heavier than non-zero velocity would probably work. Or possibly just rerunning the training with a different random starting point you could get out a better controller, I don’t have a great feel for how important the random initialization is, but I’m hoping that it’s not all too important and its effects go to zero with enough training. Also, it should be noted I’ve run the first network for 12,500 iterations and the second for less than 6,000, so I’ll keep letting them run and maybe it will come around. The first one looked pretty messy too until about 4,000 iterations in.

Training regimes

Frustratingly, the way that you train deep networks is very important. So, very much like the naive deep learning network trainer that I am, I tried the first thing that pretty much anyone would try:

• run the network,
• update the weights,
• repeat.

This is what I’ve done in the results above. And it worked well enough in that case.

If you remember back to the iLQR I made a little while ago, I was able to change the cost function to be

$L(\theta) = \sum\limits^{N-1}\limits_{t=0} \ell(\textbf{u}_t) + \ell_f(\textbf{x}_N),$
$\ell(\textbf{u}_t, \textbf{x}_t) = \alpha \frac{||\textbf{u}_t||^2}{2} + \frac{||\textbf{x}^* - \textbf{x}_t||^2}{2},$
$\ell_f(\textbf{x}_N) = \frac{||\textbf{x}^* - \textbf{x}_t||^2}{2}$

(i.e. to include a penalty for distance to target throughout the trajectory and not just at the final time step) which resulted in straighter trajectories when controlling the 2-link arm. So I thought I would try this here as well. Sadly (incredibly sadly), this was fairly fruitless. The network didn’t really learn or improve much at all.

After much consideration and quandary on my part, I talked with Dr. Dan and he suggested that I try another method:

• run the network,
• record the input,
• hold the input constant for a few batches of weight updating,
• repeat.

This method gave much better results. BUT WHY? I hear you ask! Good question. Let me give giving explanation a go.

Essentially, it’s because the cost function is more complex now. In the first training method, the output from the plant is fed back into the network as input at every time step. When the cost function was simpler this was OK, but now we’re getting very different input to train on at every iteration. So the system is being pulled in different directions back and forth at every iteration. In the second training regime, the same input is given several times in a row, which let’s the system follow the same gradient for a few training iterations before things change again. In my head I picture this as giving the algorithm a couple seconds to catch its breath dunking it back underwater.

This is a method that’s been used in a bunch of places recently. One of the more high-profile instances is in the results published from DeepMind on deep RL for control and for playing Go. And indeed, it also works well here.

To implement this training regime, we set up the following code:

for ii in range(last_trial+1, num_batches):

# run the plant forward once
rnn.forward(input=plant, params=rnn.W)

# get the input and targets from above rollout
inputs = plant.get_vecs()[0].astype(np.float32)
targets = np.asarray(plant.get_vecs()[1], dtype=np.float32)

# train a bunch of batches using the same input every time
# to allow the network a chance to minimize things with
# stable input (speeds up training)
err = rnn.run_batches(inputs, targets, max_epochs=batch_size,
optimizer=HessianFree(CG_iter=96, init_damping=100))

# save the weights to file, track trial and error
# err = rnn.error(inputs)
err = rnn.best_error
name = 'weights/rnn_weights-trial%04i-err%.3f'%(ii, err)
np.savez_compressed(name, rnn.W)


So you can see that we do one rollout with the weights, then go in and get the inputs and targets that were used in that rollout, and start training the network while holding those constant for batch_size epochs (training sessions). From a little bit of messing around I’ve found batch_size=32 to be a pretty good number. So then it runs 32 training iterations where it’s updating the weights, and then saves those weights (because we want a loooottttt of check-points) and then restarts the loop.

Embarrassingly, I’ve lost my simulation results from this trial, somehow…so I don’t have any nice plots to back up the above, unfortunately. But since this is just a blog post I figured I would at least talk about it a little bit, since people might still find it useful if they’re just getting into the field like me. and just update this post whenever I re-run them. If I rerun them.

What I do have, however, are results where this method doesn’t work! I tried this with the simpler cost function, that only looks at the final state distance from the target, and it did not go so well. Let’s look at that one!

My guess here is basically that the system has gotten to a point where it’s narrowed things down in the parameter space and now when you run 32 batches it’s overshooting. It needs feedback about its updates after every update at this point. That’s my guess, at least. So it could be the case that for more complex cost functions you’d want to train it while holding the input constant for a while, and then when the error starts to plateau switch to updating the input after every parameter update.

Conclusions

All in all, AHF for training neural networks in control is pretty awesome. There are of course still some major hold-backs, mostly related to how long it takes to train up a network, and having to guess at effective training regimes and network structures etc. But! It was able to train up a relatively small neural network to move an arm model from a center point to 64 targets around a circle, with no knowledge of the system under control at all. In Dr. Sutskever’s thesis he goes on to use the same set up under more complicated circumstances, such as when there’s a feedback delay, or a delay on the outgoing control signal, and unexpected noise etc, so it is able to learn under a number of different, fairly complex situations. Which is pretty slick.

Related to the insane training time required, I very easily could be missing some basic thing that would help speed things up. If you, reader, get ambitious and run the code on your own machine and find out useful methods for speeding up the training please let me know! Personally, my plan is to next investigate guided policy search, which seems like it’s found a way around this crazy training time.

## Online Goal Babbling – motor learning paper review

Recently I was linked to an article about learning how to control a highly complex arm from scratch: How infants tell us how to control the Bionic Handling Assistant. The work seemed very interesting so I went and pulled one of the papers linked in the article, Online Goal Babbling for rapid bootstrapping of inverse models in high dimensions.

Diving into that title, online means that we’re using information from every movement as it’s gathered to improve our control, as opposed to ‘batch’ where learning only occurs every so-many trials. Bootstrapping is the process of bringing a system up to a functionally useful level. High dimensions then refers to the complexity of the system being controlled, where every component that requires a control signal is another dimension. Humans, for example, require extremely high dimensional control signals. Inverse models refer to a type of internal model, which ‘describe relations between motor commands and their consequences’. Forward models predict the results of a movement, and inverse models allow suggest a motor command that can be used to achieve a desired consequence, such as ‘the ball falls on the floor’ or ‘my hand touches the red balloon’.

Alright, so that’s the title, let’s dive into the paper proper.

Online goal babbling

The idea behind this research is to let the system learn how to control itself by exploring the environment. The reason why you would want to do this is so that you don’t have to analytically specify how the system should move. Analytic solutions require accurate models of the system dynamics, and calculating these quickly become horrendously complex. To the point that the equations of motion for a relatively simple 3-link arm moving are pages upon pages upon pages. On top of this, because your solution is only is as good as your model and the model you just took all that time to calculate isn’t adaptive, if your system dynamics change at all through wear and tear or an environment change, you’re in big trouble. You can use feedback control to help compensate for the errors introduced, but you can only respond as fast as you can receive and process sensory signals, which is often too long in practical applications.

So this moves us to adaptive feedforward control, based on learning an inverse model of the system dynamics. Importantly, what this paper describes is a kinematic inverse controller, not a kinetic inverse controller; meaning that given a desired target position for the end effector (hand) of an arm it provides a sequence of joint angle sets that will lead to the end effector reaching the target, as opposed to providing a sequence of joint angle torques to drive the system to the target. At some point, a control system will have to generate force commands, rather than just specify what position it wants the system to be in. But, knowing what joint angles and trajectory the joint angles should follow is an important problem, because systems that are of interest to control tend to exhibit a fair amount of configuration redundancy between ‘task-space’ and the actual system state-space. Task-space being something like the 3-dimensional $(x,y,z)$ position of the end-effector, which we are interested in controlling, and the actual system state-space being something like the joint-angles or muscle-lengths. Configuration redundancy is the problem of more than one possible set of joint-angles putting the end-effector in the desired location. Often the number of potential solutions is incredibly large, think about the number of possible ways you can reach to an object. So how do you learn an appropriate trajectory for your joint angles during a reaching movement? That is the problem being addressed here.

To learn an inverse model from scratch, the system needs to explore. How should it explore? Moving randomly eventually leads to an exhaustive search, but this is a poor choice in complex systems because it takes a large amount of time, increasing exponentially with the degrees of freedom (i.e. number of joint angles) of the system. So let’s look to babies, they’re responsible for learning how to control a very complex system, how the heck do they learn to move?

‘Motor babbling’ is a term that was coined to describe the seemingly random way babies moved about. It was suggested that they just flail about without purpose until they gain some understanding of how their bodies work, at which point they can start moving with purpose. But! Baby movement was shown way back in the 80’s to in fact not be just random, but instead to be highly goal directed. And when they find a solution, they stick with it, they don’t worry about it being the best one as long as it gets the job done. Only later are movements tweaked to be more efficient. As mentioned above, in systems as complicated as the human body the task-space (i.e. position of the hand) is much smaller than the motor space (i.e. length of all muscles), and there are a bunch of different solutions to a given task. With all these different potential solutions to a given problem, an exhaustive search isn’t even close to being necessary. Also, if babies aren’t just randomly exploring space to figure things out, they don’t have to flip a switch somewhere in their brain that says “ok stop messing around and let’s move with purpose”.

This paper provides a technique for stable online inverse model learning that can be used from initial bootstrapping, and shows how online learning can even be highly beneficial for bootstrapping. So let’s dive into the Online Goal Babbling learning method.

The inverse model function

Let’s denote the inverse model function we’re learning as $g()$, joint angles as $q$, and end effector positions as $x$. Then to denote giving a desired end effector position and getting out a set of joint angles we write $g(x^*) = q$, where $x^*$ represents a target in end effector space.

We’re going to initialize the inverse model function by having every end effector position return some default resting state of the arm, or home position, that we’ve defined, $q_{home}$. Additionally, we’re going to throw on some exploratory noise to the answer provided by the inverse model, so that the joint angles to move to at time $t$ are defined as $q_t = g(x^*_t, \theta_t) + E_t(x_t^*)$, where $E_t(x^*_t)$ is the exploratory noise. When the system then moves to $q_t$ the end effector position, $x_t$, is observed and the parameters of the inverse model, $\theta$, are updated immediately.

To generate a smooth path from the starting position to the target, the system divides the distance up into a number of sub-targets (25 in the paper) for the system to hit along the way. This is an important point, as it’s how the inverse model function is used to create a path that represents the system moving; a series of targets are provided and the inverse model is queried “what should the joint configuration be for this position?”

As mentioned before, it is possible to have he end effector in the same position, but the joints in a different configuration. Learning across these examples is dangerous and can lead to instability in the system as neighbouring targets could be learned with very dissimilar joint configurations, preventing smooth movement through joint-space. To prevent this, observed information is weighted by a term $w_t$ as it is taken in, based on deviation from the resting position of the arm ($q_{home}$) and efficiency of end effector movement. What this leads to is a consistent solution to be converged upon throughout movement space, causing the inverse model to generate smooth, comfortable (i.e. not near joint limits, but near the resting state of the arm) movements.

Additions to movement commands

To recenter movement exploration every so often, and prevent the system from exploring heavily in some obscure joint-space, every time a new target to move to is selected there is some probability (.1 in the paper) that the system will return to $q_{home}$. To return home the system traces out a straight trajectory in joint-space, not worrying about how it is moving through end effector space. This return probability reinforces learning how to move well near $q_{home}$, and acts as a ‘developmentally plausible stabilizer that helps to stay on known areas of the sensorimotor space.’

Also, we mentioned adding exploratory noise. How is that noise generated? The noise is calculated through a small, randomly chosen linear function that varies slowly over time: $E_t(x^*_t) = A_t \cdot x^* + c_t$, where the entries to the matrix $A_0$ and vector $b_0$ are chosen independently from a normal distribution with zero mean and variance $\sigma^2$. To move, a set of small values is chosen from a normal distribution with variance significantly smaller than $\sigma^2$, and added to elements of $A_t$ and $c_t$. A normalization term is also used to keep the overall deviation stable around the standard deviation $\sigma$. And that’s how we get our slowly changing linear random noise function.

Online learning

To learn the inverse model function, we’re going to use the technique of creating a bunch of linear models that are accurate in a small area of state space, and weighting their contribution to the output based on how close we are to their ‘region of validity’. The specific method used in the paper is a local-linear map, from (H.Ritter, Learning with the self-organizing map 1991). We define our linear models as $g^{(k)}(x) = M^{(k)} \cdot x + b^{(k)}$, intuited as following the standard $mx + b$ definition of a line, but moving into multiple dimensions. $M^{(k)}$ is our linear transformation of the input for model $k$, and $b^{(k)}$ is the offset.

If an input is received that is outside of the region of validity for all of the local linear models, then another one is added to improve the approximation of the function at that point. New local models are initiated with the Jacobian matrix $J(x) = \frac{\partial g(x)}{\partial x}$ of the inverse estimate. In other words, we look at how the approximation of the function is changing as we move in this $x$ direction in state space, or estimate the derivative of the function, and use that to set the initial value of our new linear model.

To update our inverse model function, we fit it to the current example $(x_t, q_t)$ by reducing the weighted square error: $err = w_t \cdot ||q_t - g(x_t)||^2$. With this error, a standard gradient descent method is used to update the slopes and offsets of the dimensions of the linear models:

$M^{(k)}_{t+1} = M^{(k)}_t - \eta \frac{\partial err}{\partial M^{(k)}_t}, \;\;\;\;\; b^{(k)}_{t+1} = b^{(k)}_{t} - \eta \frac{\partial err}{/partial b^{(k)}_{t}}$,

where $\eta$ is the learning rate.

Results

So here are some figures of the results of applying the algorithm, and they’re pretty impressive. First we see learning the inverse model for a 5-link arm:

And one of the really neat things about this is the scalability of this algorithm. Here’s applying OGB to a 50-link arm:

And here are some results looking at varying the learning rate (x-axis) and the effects on (a) time until 10% error reached, (b) final performance error, and (c) final average distance from $q_{home}$, the resting state configuration.