Tag Archives: allocentric

Reinforcement Learning part 4: Combining egocentric and allocentric

Alright, in my last post on RL we looked at egocentric learning and what that is. We also saw that both ego and allocentric learning have their strong points, and they compliment each other. So, some combination of these two approaches seems like a good idea. In this post we’re going to look at combining the output from these two kinds of learners and see the strengths of different weightings. Per usual, all the code from this post is up on my Github.

Combining allocentric and egocentric approaches

There are a few different ways that come to mind that we could combine these two learning styles. In each of these cases the overall setup is that we’re going to run two learning systems, one that is allocentric and one that is egocentric, combine their action value mappings for the current state in some way, and then decide which action to take based on their combined output. So here are several possible ways to combine the outputs:

  • Only use allocentric
  • Only use egocentric
  • Average the mappings
  • Dynamically change weights

There are, of course, many more ways to weight these systems, but these are some basic ones and they’re what we’re going to look at here.

For testing each of these these different setups we are going to run the mouse on a map for a set amount of time, measure the wins, accidental deaths, and intentional deaths, and then gather statistics (mean value and confidence intervals) across 10 trials.

A side note: The rate of exploration (epsilon value) starts out at .1 and is decreased by half every 100,000 simulation time step, so there are a number of jumps where you see the number of wins go up every 100,000, and that’s an artifact of the rate of exploration dropping (i.e. increasing your exploitation of what you already know about the world).

OK, we’re going to do this across 3 different maps, where the size and complexity increases with each map.

Map 1: the basic cliff map (which should look familiar):

cliffmap
Map 2: a middlesized map with several areas that are identical for the egocentric state:

barrier3
and Map 3: a pretty large map with even more ambiguous areas:

bigmap

Expectations

Now, what we expect from our learning systems is that the egocentric learner with have very few ‘suicide’ deaths, where it’s slow to learn that jumping off a cliff is a terrible idea. But on the flip side of that, since we’re going to set the gamma value of the egocentric learner to 0, (which means that it doesn’t incorporate anything other than the immediate reward into it’s learning, as discussed in the previous post), we expect that it will only rarely find the goal. Also, as the map gets larger, we expect that the goal will be found less and less, as it will have to randomly stumble around avoiding cliffs until it ends up in sight of the cheese.

From the allocentric learner we expect the death rate to be much higher than the egocentric learner, but we also expect the success rate to be significantly higher, as it will be able to identify how to move based on its current (x,y) position, rather than based on its immediate surroundings. As the map gets larger though, we expect the number of deaths to be larger, since there is no transfer of knowledge that jumping off a cliff in this position is as bad as jumping off a cliff in that position.

Alright! Let’s get to it!

Results

Only use allocentric

Not much to explain about this, the mouse is going to be learning Q-values where each state is its (x,y) position on the map. The bigger the map, the more deaths we expect this learner to have. So, here are the results:

Alright, so one of the most salient things in these plots is the stepping up of the number of wins at each of the 100,000 marks. Again, this is due to a transfer from exploration to exploitation of what’s already been learned as the epsilon value is decreased. And, also, as the map gets bigger, the number of wins takes longer to max out, but it’s pretty interesting how consistent the learning is (as seen by the very small confidence intervals on the runs) across trials given the random exploration.
Another point of note is that the number of deaths goes way up with the size of the map, as expected.

Only use egocentric

Again, this one is another straightforward case, the mouse is going to be learning Q-values where each state is its set of immediate neighbors (e.g. cliff to the right, cheese in front, empty cells left and behind). Here we expect that it will have a minimal number of deaths (not much more than a death for each of the situations it can encounter a cliff or cliffs in), and that the number of successes should be pretty low and very inconsistent, since it will be entirely based on randomly stumbling into a spot near the cheese while avoiding cliffs. Additionally, we expect the number of results to drop with increases in the size of the map. The results are below:

The results are as expected, lots of variation in the wins number, low suicide numbers, and the number of wins decreases with map sizes. Something that’s kind of interesting here is that you don’t see any real effects of the epsilon value as it decreases as you do in the egocentric case. I believe that this is because it achieves a balance of exploration that throws you off a cliff and exploration that moves you into the vicinity of the cheese. So it’s dying less when the exploration rate drops, but it also has less chance of reaching the cheese now. That’s my guess, at least. Alright! Next.

Average the allo and ego Q-values

Here we’re going to do the most basic combination of the allo and ego-centric learning possible. Both learners will run simultaneously, and generate a set of Q-values over the set of possible actions at each state, and we’ll average the output of both then choose the max Q-value to choose which action is taken. The hope here is that the ego-centric learner will very quickly learn not to jump off cliffs, and the allocentric learner will more slowly learn how to move in the map to get cheese. So we expect few deaths and more wins. Let’s look at the results:

Damn! So now we have the same (or at least very similar) win rate as the allocentric learner (awesome), and the same death rate as the egocentric learner (also awesome)! We have achieved some sort of super-human…or, super-mouse mouse. A super mouse. This is really cool, it tells us that these two systems really do compliment each other, and that it can be incredibly straightforward to implement a combination of the two.

Dynamic weighting

Here we’ll look at the results from a variation of an algorithm I’ve implemented based on this really interesting paper by Dr. Sakya Dasgupta et al where they got a lot of really neat results implemented on a little robot critter. We can’t quite use the algorithm as presented here for a couple of reasons.

In the paper, the authors design the algorithm for weighting the output from classical conditioning and operant conditioning systems. I’m taking these to be pretty close to analogous to an ego-centric learner (with no lookahead) and an allo-centric learner, respectively. The idea being that classical conditioning is based on immediate associations developed from rewards, and operant conditioning allows the longer-term associations to be formed. Specifically, in operant you learn to associate the conditioned stimulus with the unconditioned stimulus, which leads to a a conditioned response. The classic example being the dog salivating when hearing the bell, because it knows food is coming. Here the allocentric learner is doing a similar thing, using a look-ahead to start associating stimuli (in this case different states) with a reward or punishment. So, this could be a wildly inappropriate application of this algorithm, and the comparison definitely warrants further discussion.

The more immediate reason for varying the algorithm is because the output of their system is a continuous ‘left/right’ decision, and the output from the mouse here is a discrete ‘left/right/up/down’ decision. So the systems in the paper output a number from -1 to 1 that indicates which way to turn, and this is used in the update. So the algorithm needs to be changed up a little.

First, here’s the original algorithm:

weight_allo += eta * reward * [output_allo - filtered_output_allo] * output_ego
weight_ego += eta * reward * [output_ego - filtered_output_ego] * output_allo

where eta is the learning rate, reward is the immediate reward of the state you move into, output_ is the -1 to 1 value from the learners, and filtered_output_ is a low pass filtered version of the output. So the first part is standard, learning rate multiplied by the reward, and then the part inside the brackets is calculating a derivative of the output of the learners. This is again standard, did your learner change? If it did change and a reward was received, then increase the weight of that learner. The last term, which multiplies it all by the output from the other learner is for scaling.

And then normalize the weights relative to each other, and it isn’t explicitly stated in the paper, but also lower bound the weights at 0. The reason you need to do this lower bounding is that if you don’t, then when you use the weight as a gain on the learner output all of the Q-values are inverted! Not easy to recover from that.

The dynamic weighting equations that I’m using are:

weight_allo += eta * reward * dq_allo * output_ego
weight_ego += eta * reward * dq_ego * output_allo

and then lower bound at 0 and normalize.

There is still a lot in these equations that I’m unsure about (for example scaling by the output of the other learner instead of by the other weight, and how the signs of the reward, derivative, and other learner output all interact), but I’m going to leave delving into that for a future post, since this one is getting long already and I don’t have a full understanding of it yet. But! I have enough of an understanding to implement the above and run simulations! Here are the results:

I should note that the learning rate here for the weight equations is 5e-4. When you decrease this weight you get improved performance, but the best I’ve gotten is just reducing it to a point that it’s pretty much constantly using .5 for each. And that’s boring.

Alright, comparing to basic averaging here we notice that we have about the same performance for the small map, and then on the bigger maps significantly worse performance and also a lot more variability between runs. Basically what seems to happen is that the system quickly identifies that the allocentric learner should not have control at the beginning, because it keeps killing the mouse, and then it takes a lot longer to start handing control back over from the egocentric learner to the allocentric learner.

Conclusions

Alright, so that’s all! Bottom line is that allocentric and egocentric learning really work well together, and basic averaging of these gives great results. Whether this is the best means of combining them is unclear, however. This was just a really basic test of different combinations, and there is some really neat work looking at different ways to do it. It’s super fun to mess around with and there’s a ton of room for experimentation. In the code up on my Github, the code for weight learning can be found by searching for weight_learning. If anyone plays around and figures out a kick-ass dynamic weighting algorithm please pass it along!

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Reinforcement Learning part 3: Egocentric learning

When an agent is learning how to move, it’s learning a set of values to associate with possible actions for each situation it encounters. We say that it’s learning the action values. When the agent learns to associate the values of taking different actions in different states, it is learning the state-action values, also referred to as the Q-values. How the agent identifies ‘a situation’ can be handled in a couple of different ways.

If the agent is performing egocentric learning, then it distinguishes situations based on it’s immediately surrounding environment. If the agent is perform allocentric learning, then it has an internal map of the environment, and each different position on the map is considered a different situation. Whether an agent is using allocentric or egocentric learning ends up influencing its behavior quite a bit, and both have their pros and cons. We’ve some of both ego and allo-centric in the previous RL posts; in this post we’re going to examine egocentric learning specifically a little closer. And the code for this post can be found up on my GitHub.

Difference between ego and allo-centric

Let’s look at when a new set of action values is learned in these two cases. Going back to the ever wonderful mouse and cheese scenario, we’ll create a map for our mouse to learn to navigate.

map
In egocentric learning, each situation is determined by what is near the mouse, and so new action values for each of the possible movements ({W,NW,N,NE,E,SE,S,SW}) have to be learned whenever the immediate environment is not something the agent has seen before. For example, the mouse keeps a different set of action values for each of the following environments:

mouse states
On the other hand, in the case of allocentric learning, each situation is determined by the (x,y) coordinates of the mouse’s current position. That means that the mouse keeps a different set of action values for every possible (x,y) position:

mouse allo
so even if the environment in each position is the same:

mouse states2
with different (x,y) coordinates the mouse maintains different action value maps for all of them.

With egocentric learning, the agent tends to develop immediately beneficial behavior more quickly, to avoid running off cliffs and to move towards cheese when it’s in sight. The agent can even learn to move towards cheese that’s outside of the range of its sensors in simple maps, essentially by learning heuristics like ‘always moving right leads to reward.’ But this falls apart if there are several areas in the map where the immediate environment is the same, as in the case above with states 2 and 3.

With allocentric learning, the agent develops action value mappings that are much better suited to the overall environment and long-term goals (e.g. if the cheese is always in the center of the map, if you are in the bottom-right move up and left, and if you are in the top-left move down and right), but you can’t generalize across local situations (for example when there is a cliff one space to the right), the consequences of movement at each (x,y) position must be learned independently.

Clearly a combination of these two learning types is desirable, but that is for a future post! There is still a lot to examine in ego-centric learning, and so we’re going to dig deeper.

Thinking about the future

In allocentric learning, you don’t often perform an action, successfully move, and then still wind up in the same state. This makes transmitting information about rewards received in the future backwards to states that are still far away fairly stable, because a specific state is some constant distance away from getting reward. In egocentric learning, however, this isn’t the case. So when you start trying to propagate reward information backwards you can get some unstable behavior; let’s look at a very basic scenario:

cliffs
There’s the mouse in blue, on the left, and the reward, in green, on the right, and red represents cliffs / certain death. So now, whenever the mouse isn’t beside one of the walls he will be in one of three states, cliffs above, nothing around him, or cliffs below:

3states

OK. Now let’s look at the update formula for Q-learning again:

Q(s, a) += alpha * (reward(s') + gamma * max(Q(s')) - Q(s, a))

where s is the previous system state, a is the action just taken, s' is the current state, reward(s') is the reward for being in the current state, and max(Q(s')) is the maximum reward obtainable by performing an action in the current state.

Informally, it says that the change to make to the Q-value for the action just taken, Q(s, a), is equal to your learning rate, alpha, multiplied by the reward for the state that you moved into,reward(s'), plus the lookahead reward,max(Q(s')), which relays the expected reward now that you’re in state s', minus the previous Q-value,Q(s, a).

Let’s break this down a bit more, because this can be complicated to think about still. If we set gamma = 0, then our update is

Q(s, a) += alpha * (reward(s') - Q(s, a)).

Here it’s more apparent what the dynamics are doing, they’re driving the Q-value to the reward value of the state you end up in. So you take an action a in state s, you find the reward for where you ended up, and you learn the reward for performing action a in state s. Now if gamma is not equal to 0, then you can think of the Q-value dynamics as trying to drive the Q-values to

reward(s') + gamma * max(Q(s'))

The larger that gamma is, the farther into the future you look. This can lead to unstable / non-convergent systems when you repeatedly move into the same state again and again. For example, if you’re in State 2 from above, and you move right often enough you’ll end up in a state that leads to getting a reward. This boosts the state before the reward, and now the next time you happen to move to a state where the reward is next to you, the Q-values of State 2 are now updated. And so you get your reward and you reset. Now you move into State 2 again and hey! Moving right leads to a reward. So you move right, but now you’re back in State 2 again! And so the reward for moving right decreases a bit. But it’s still the best decision, so you move right. And you’re in State 2 again, and so the reward for moving right goes down again, etc etc. Eventually moving right brings you to the target though and the Q-value for moving right gets a boost. So it’s going to forever be fluctuating.

In this example, it doesn’t seem like a big deal, because the mouse still learns to move towards the cheese and the Q-values have entered a kind of rhythmic limit cycle. There are a few problems that arise, however, when counting the future to heavily in egocentric learning.

Problem: Death-wish

Basically, the mouse decides that certain death is a better option than wandering around in uncertainty forever and starts deciding to commit suicide. Let’s say that to encourage exploration, the reward for being in an unoccupied square is -1, and to encourage not dying the reward for jumping off a cliff is -10. The problem arises in that jumping off the cliff is a constant punishment, -10. It’s always -10 and it will never be any worse than -10. There is no further future past death to influence the reward. When the mouse is just exploring, however, it will explore, and get -1, and then explore more, and get -1, etc etc. This causes the expected reward of actions that lead you into the same state keep dropping. If the weight of gamma is too large, then without all that much exploring the reward for exploring has suddenly dropped to less than -10, and the mouse decides that continuing to explore is worse than death, and jumps.

By setting gamma = 5 (which is pretty high) we then get this behavior very quickly, here are some simulation results, where the first column is the timestep, the second column W denotes how many times the mouse found the cheese, the third column L denotes how many times the mouse chose to jump off a cliff (as dictated by its Q-values), and the last column A denotes how many times random exploration led the mouse to its accidental death:

10000, W: 0, L: 3138, A: 51
20000, W: 0, L: 3207, A: 40
30000, W: 0, L: 3204, A: 40
40000, W: 0, L: 3205, A: 45
50000, W: 0, L: 3208, A: 43
60000, W: 0, L: 3198, A: 49
70000, W: 0, L: 3200, A: 49
80000, W: 0, L: 3207, A: 46
90000, W: 0, L: 3204, A: 46
100000, W: 0, L: 3214, A: 38
110000, W: 0, L: 3211, A: 37
120000, W: 0, L: 3203, A: 44
130000, W: 0, L: 3199, A: 45
140000, W: 0, L: 3225, A: 28
150000, W: 0, L: 3211, A: 39
160000, W: 0, L: 3196, A: 45
170000, W: 0, L: 3208, A: 46
180000, W: 0, L: 3203, A: 47
190000, W: 0, L: 3209, A: 38
200000, W: 0, L: 3216, A: 38

So pretty immediately the mouse gets into a dark place, and starts choosing to jump, without ever having found the cheese even once.

Problem: Confusion

Another problem is that by heavily weighting the immediate future the mouse can get easily confused again about what was good or how it actually found the cheese and got the big reward in the first place. Once State 2 has rewarded moving right, and the mouse moves left, there is now a new max(Q(s')) value for State 2 (the reward for moving right), and so the reward for moving left is now brought up. At the same time, the reward for moving right is dropping every time it moves right and doesn’t get rewarded (as described above). Same with moving up and down, so very quickly the Q-values can be flooded with high values and the optimal direction of movement is lost.

We can see this case by setting gamma = .8. Now, the mouse will find the cheese and learns to move right, but there are also these blips of confusion where it forgets how to find the cheese,

10000, W: 0, L: 4, A: 78
20000, W: 0, L: 0, A: 78
30000, W: 87, L: 0, A: 67
40000, W: 810, L: 0, A: 16
50000, W: 841, L: 0, A: 1
60000, W: 843, L: 0, A: 2
70000, W: 847, L: 0, A: 3
80000, W: 816, L: 0, A: 10
90000, W: 667, L: 0, A: 17
100000, W: 0, L: 0, A: 68
110000, W: 0, L: 0, A: 68
120000, W: 34, L: 0, A: 68
130000, W: 748, L: 0, A: 43
140000, W: 853, L: 0, A: 0
150000, W: 846, L: 0, A: 2
160000, W: 847, L: 0, A: 3
170000, W: 856, L: 0, A: 2
180000, W: 853, L: 0, A: 2
190000, W: 120, L: 0, A: 62
200000, W: 0, L: 0, A: 70

So it’s getting there, but then it gets confused and forgets how to find it, then it remembers, then it forgets…etc.

Solution: Have a healthy fear of death

If instead now we set gamma = .05, the mouse has a much stronger repulsion to imminent death than to ennui for a much longer time, and will go around exploring things (note that this is the same as changing the reward for dying to be a much higher penalty). As a result the mouse will find the cheese faster. As a bonus, we also don’t have the fits of forgetfulness seem in the previous example:

10000, W: 313, L: 4, A: 50
20000, W: 845, L: 0, A: 6
30000, W: 848, L: 0, A: 1
40000, W: 850, L: 0, A: 1
50000, W: 850, L: 0, A: 1
60000, W: 846, L: 0, A: 2
70000, W: 844, L: 0, A: 6
80000, W: 843, L: 0, A: 2
90000, W: 846, L: 0, A: 2
100000, W: 847, L: 0, A: 5
110000, W: 844, L: 0, A: 4
120000, W: 849, L: 0, A: 3
130000, W: 846, L: 0, A: 5
140000, W: 842, L: 0, A: 4
150000, W: 845, L: 0, A: 4
160000, W: 847, L: 0, A: 5
170000, W: 848, L: 0, A: 10
180000, W: 844, L: 0, A: 7
190000, W: 840, L: 0, A: 8
200000, W: 848, L: 0, A: 1

And so in this case now we’ve developed a pretty good, stable, egocentric RL mouse, that can learn to avoid jumping off cliffs and find the cheese in this very simple map.

We can still, of course, drop the gamma value down to the extreme, 0, such that our egocentric learner would have no knowledge of any future further than the reward it gets for taking the next step. In this case it would know that going off a cliff is bad, but it won’t know that moving to the right will eventually lead to cheese / a reward.

However, that can be OK! Warding off any immediate danger is and capitalizing on any immediate reward is the perfect role for this kind of short-sighted system. Trying to make it learn how to run through a complicated maze without a lot of very specific landmarks is just not something that’s going to happen.

Conclusions

So, recapping, the egocentric system learns Q-values for states that are based on the environment relative to the agent. This has some important consequences, in that it allows it to learn very quickly not to move forward if there’s a cliff in front of you, and that if there’s a reward in the immediate vicinity to go get it. Setting the weight on the expected reward in the Q-values update can lead to some unstable behavior, and it’s important to have a good concept of how the behavior of the agent changes with different gamma values. These undesirable behaviors can be avoided by setting gamma = 0, but the agent will then only learn the consequences of its immediate action, and be unable to look into the future at all to see how to bring about a reward or avoid getting trapped.

The egocentric learner has strengths where the allocentric learner fails, and the allocentric learner has strengths where the egocentric learner fails. The natural thought is to combine these two learners, and that will be the subject of the next RL post!

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