I'm trying to navigate an agent in a n*n gridworld domain by using Q-Learning + a feedforward neural network as a q-function approximator. Basically the agent should find the best/shortest way to reach a certain terminal goal position (+10 reward). Every step the agent takes it gets -1 reward. In the gridworld there are also some positions the agent should avoid (-10 reward, terminal states,too).
So far I implemented a Q-learning algorithm, that saves all Q-values in a Q-table and the agent performs well.
In the next step, I want to replace the Q-table by a neural network, trained online after every step of the agent. I tried a feedforward NN with one hidden layer and four outputs, representing the Q-values for the possible actions in the gridworld (north,south,east, west).
As input I used a nxn zero-matrix, that has a "1" at the current positions of the agent.
To reach my goal I tried to solve the problem from the ground up:
Explore the gridworld with standard Q-Learning and use the Q-map as training data for the Network once Q-Learning is finished
--> worked fine
Use Q-Learning and provide the updates of the Q-map as trainingdata
for NN (batchSize = 1)
--> worked good
Replacy the Q-Map completely by the NN. (This is the point, when it gets interesting!)
-> FIRST MAP: 4 x 4
As described above, I have 16 "discrete" Inputs, 4 Output and it works fine with 8 neurons(relu) in the hidden layer (learning rate: 0.05). I used a greedy policy with an epsilon, that reduces from 1 to 0.1 within 60 episodes.
The test scenario is shown here. Performance is compared beetween standard qlearning with q-map and "neural" qlearning (in this case i used 8 neurons and differnt dropOut rates).
To sum it up: Neural Q-learning works good for small grids, also the performance is okay and reliable.
-> Bigger MAP: 10 x 10
Now I tried to use the neural network for bigger maps.
At first I tried this simple case.
In my case the neural net looks as following: 100 input; 4 Outputs; about 30 neurons(relu) in one hidden layer; again I used a decreasing exploring factor for greedy policy; over 200 episodes the learning rate decreases from 0.1 to 0.015 to increase stability.
At frist I had problems with convergence and interpolation between single positions caused by the discrete input vector.
To solve this I added some neighbour positions to the vector with values depending on thier distance to the current position. This improved the learning a lot and the policy got better. Performance with 24 neurons is seen in the picture above.
Summary: the simple case is solved by the network, but only with a lot of parameter tuning (number of neurons, exploration factor, learning rate) and special input transformation.
Now here are my questions/problems I still haven't solved:
(1) My network is able to solve really simple cases and examples in a 10 x 10 map, but it fails as the problem gets a bit more complex. In cases where failing is very likely, the network has no change to find a correct policy.
I'm open minded for any idea that could improve performace in this cases.
(2) Is there a smarter way to transform the input vector for the network? I'm sure that adding the neighboring positons to the input vector on the one hand improve the interpolation of the q-values over the map, but on the other hand makes it harder to train special/important postions to the network. I already tried standard cartesian two-dimensional input (x/y) on an early stage, but failed.
(3) Is there another network type than feedforward network with backpropagation, that generally produces better results with q-function approximation? Have you seen projects, where a FF-nn performs well with bigger maps?
It's known that Q-Learning + a feedforward neural network as a q-function approximator can fail even in simple problems [Boyan & Moore, 1995].
Rich Sutton has a question in the FAQ of his web site related with this.
A possible explanation is the phenomenok known as interference described in [Barreto & Anderson, 2008]:
Interference happens when the update of one state–action pair changes the Q-values of other pairs, possibly in the wrong direction.
Interference is naturally associated with generalization, and also happens in conventional supervised learning. Nevertheless, in the reinforcement learning paradigm its effects tend to be much more harmful. The reason for this is twofold. First, the combination of interference and bootstrapping can easily become unstable, since the updates are no longer strictly local. The convergence proofs for the algorithms derived from (4) and (5) are based on the fact that these operators are contraction mappings, that is, their successive application results in a sequence converging to a fixed point which is the solution for the Bellman equation [14,36]. When using approximators, however, this asymptotic convergence is lost, [...]
Another source of instability is a consequence of the fact that in on-line reinforcement learning the distribution of the incoming data depends on the current policy. Depending on the dynamics of the system, the agent can remain for some time in a region of the state space which is not representative of the entire domain. In this situation, the learning algorithm may allocate excessive resources of the function approximator to represent that region, possibly “forgetting” the previous stored information.
One way to alleviate the interference problem is to use a local function approximator. The more independent each basis function is from each other, the less severe this problem is (in the limit, one has one basis function for each state, which corresponds to the lookup-table case) [86]. A class of local functions that have been widely used for approximation is the radial basis functions (RBFs) [52].
So, in your kind of problem (n*n gridworld), an RBF neural network should produce better results.
References
Boyan, J. A. & Moore, A. W. (1995) Generalization in reinforcement learning: Safely approximating the value function. NIPS-7. San Mateo, CA: Morgan Kaufmann.
André da Motta Salles Barreto & Charles W. Anderson (2008) Restricted gradient-descent algorithm for value-function approximation in reinforcement learning, Artificial Intelligence 172 (2008) 454–482
Related
I'm working on my bachelor thesis.
My topic is reinforcement learning. The Setup:
Unity3D (C#)
Own neural network framework
Confirmed the network working by testing to training a sine-function.
It can approximate it. Well. there are some values which won't get to their desired value but it's good enough.
When training it with single Values it always converges.
Here is my problem:
I try to teach my network the Q-Value-Function of a simple game,
catch balls:
In this game it just has to catch a ball dropping from random position and with random angle.
+1 if catch
-1 if failed
My network-model has 1 hidden layer with neurons ranging from 45-180 (i tested this numbers with no success)
It uses replay with 32 samples from a 100k memory with a learning-rate of 0.0001
It learns for 50000 frames then tests for 10000 frames. This happens 10 times.
Inputs are PlatformPosX, BallPosX, BallPosY from the last 4 frames
Pseudocode:
Choose action (e-greedy)
Do action,
Store state action, CurrentReward. Done in memory
if in learnphase: Replay
My problem is:
Its actions starts clipping to either 0 or 1 with some variance sometimes.
It never has a ideal policy like if the platform would just follow the ball.
EDIT:
Sorry for cheap info...
My Quality-Function is trained by:
Reward + Gamma(nextEstimated_Reward)
So its discounting.
Why would you possibly expect that to work?
Your training can barely approximate a 1-dimensional function. And now you expect it to solve a 12-dimensional function which involves a differential equation? You should have verified first whether your training does even converge for a multi dimensional function at all, with the chosen training parameters.
Your training, given the little detail you provided, also appears to be unsuitable. There is hardly a chance it ever successfully catches the ball, and even when it does, you are rewarding it mostly for random outputs. Only correlation between in- and output is in the last few frames when the pad can only reach the target in time by a limited set of possible actions.
Then there is the choice of inputs. Don't require your model to differentiate by itself. Relevant inputs would had been x, y, dx, dy. Preferably even x, y relative to pad position, not world. Should have a much better chance to converge. Even if it was only learning to keep x minimal.
Working with absolute world coordinates is pretty much bound to fail, as it would require the training to cover the entire range of possible input combinations. And also the network to be big enough to even store all the combinations. Be aware that the network isn't learning the actual function, it's learning an approximation for every single possible set of inputs. Even if the ideal solution is actually just a linear equation, the non linear properties of the activation function make it impossible to learn it in a generalized form for unbound inputs.
While trying to implement the Episodic Semi-gradient Sarsa with a Neural Network as the approximator I wondered how I choose the optimal action based on the currently learned weights of the network. If the action space is discrete I can just calculate the estimated value of the different actions in the current state and choose the one which gives the maximimum. But this seems to be not the best way of solving the problem. Furthermore, it does not work if the action space can be continous (like the acceleration of a self-driving car for example).
So, basicly I am wondering how to solve the 10th line Choose A' as a function of q(S', , w) in this pseudo-code of Sutton:
How are these problems typically solved? Can one recommend a good example of this algorithm using Keras?
Edit: Do I need to modify the pseudo-code when using a network as the approximator? So, that I simply minimize the MSE of the prediction of the network and the reward R for example?
I wondered how I choose the optimal action based on the currently learned weights of the network
You have three basic choices:
Run the network multiple times, once for each possible value of A' to go with the S' value that you are considering. Take the maximum value as the predicted optimum action (with probability of 1-ε, otherwise choose randomly for ε-greedy policy typically used in SARSA)
Design the network to estimate all action values at once - i.e. to have |A(s)| outputs (perhaps padded to cover "impossible" actions that you need to filter out). This will alter the gradient calculations slightly, there should be zero gradient applied to last layer inactive outputs (i.e. anything not matching the A of (S,A)). Again, just take the maximum valid output as the estimated optimum action. This can be more efficient than running the network multiple times. This is also the approach used by the recent DQN Atari games playing bot, and AlphaGo's policy networks.
Use a policy-gradient method, which works by using samples to estimate gradient that would improve a policy estimator. You can see chapter 13 of Sutton and Barto's second edition of Reinforcement Learning: An Introduction for more details. Policy-gradient methods become attractive for when there are large numbers of possible actions and can cope with continuous action spaces (by making estimates of the distribution function for optimal policy - e.g. choosing mean and standard deviation of a normal distribution, which you can sample from to take your action). You can also combine policy-gradient with a state-value approach in actor-critic methods, which can be more efficient learners than pure policy-gradient approaches.
Note that if your action space is continuous, you don't have to use a policy-gradient method, you could just quantise the action. Also, in some cases, even when actions are in theory continuous, you may find the optimal policy involves only using extreme values (the classic mountain car example falls into this category, the only useful actions are maximum acceleration and maximum backwards acceleration)
Do I need to modify the pseudo-code when using a network as the approximator? So, that I simply minimize the MSE of the prediction of the network and the reward R for example?
No. There is no separate loss function in the pseudocode, such as the MSE you would see used in supervised learning. The error term (often called the TD error) is given by the part in square brackets, and achieves a similar effect. Literally the term ∇q(S,A,w) (sorry for missing hat, no LaTex on SO) means the gradient of the estimator itself - not the gradient of any loss function.
There are lots of "introduction to neural networks" articles online, but most are an introduction to the math of artificial neural networks and not an introduction to the actual underlying concepts (even though they should be one and the same). How does a simple network of artificial neurons actually work?
This answer is roughly based on the beginning of "Neural Networks and Deep Learning" by M. A. Nielsen which is definitely worth reading - it's online and free.
The fundamental idea behind all neural networks is this: Each neuron in a neural network makes a decision. Once you understand how they do that, everything else will make sense. Let’s walk through a simple situation which will help us arrive at that understanding.
Let’s say you are trying to decide whether or not to wear a hat today. There are a number of factors which will affect your decision, and perhaps the most important ones are:
Is it sunny?
Do I have a hat to wear?
Would a hat suit my outfit?
For simplicity, we’ll assume these are the only three factors that you’re weighing up during this decision. Forgetting about neural networks for a second, let’s just try to build a ‘decision maker’ to help us answer this question.
First, we can see each question has a certain level of importance, and so we’ll need to use this relative importance of each question, along with the corresponding answer to each question, to make our decision.
Secondly, we’ll need to have some component which interprets each (yes or no) answer along with its importance to produce the final answer. This sounds simple enough to put into an equation, right? Let’s do it. We simply decide how important each factor is and multiply that importance (or ‘weight’) by the answer to the question (which can be 0 or 1):
3a + 5b + 2c > 6
The numbers 3, 5 and 2 are the ‘weights’ of question a, b and c, respectively. a, b and c, themselves can be either zero (the answer to the question was ‘no’), or one (the answer to the question was ‘yes’). If the above equation is true, then the decision is to wear a hat, and if it is false, the decision is to not wear a hat. The equation says that we’ll only wear a hat if the sum of our weights multiplied by our factors is greater than some threshold value. Above, I chose a threshold value of 6. If you think about it, this means that if I don’t have a hat to wear (b=0), no matter what the other answers are, I won’t be wearing a hat today. That is,
3a + 2c > 6
is never true, since a and c are only either 0 or 1. This makes sense – our simple decision model tells us not to wear a hat if we don’t have one! So the weights of 3, 5 and 2, and the threshold value of 6 seem like a good choices for our simple “should I wear a hat” decision-maker. It also means that, as long as I have a hat to wear, the sun shining (a=1) OR the hat suiting my outfit (c=1) is enough to make me wear a hat today. That is,
5 + 3 > 6 and 5 + 2 > 6
are both true. Good! You can see that by adjusting the weighting of each factor and the threshold, and by adding more factors, we can adjust our ‘decision maker’ to approximately model any decision-making process. What we have just demonstrated is the functionality of a simple neuron (a decision-maker!). Let’s put the above equation into ‘neuron-form’:
A neuron which processes 3 factors: a, b, c, with corresponding importance weightings of 3, 5, 2, and with a decision threshold of 6.
The neuron has 3 input connections (the factors) and 1 output connection (the decision). Each input connection has a weighting which encodes the importance of that connection. If the weighting of that connection is low (relative to the other weights), then it won’t have much effect on the decision. If it’s high, the decision will heavily depend on it.
This is great, we’ve got a fully working neuron that weights inputs and makes decisions. So here’s the next though: What if the output (our decision) was fed into the input of another neuron? That neuron would be using our decision about our hat to make a more abstract decision. And what if the inputs a, b and c are themselves the outputs of other neurons which compute lower-level decisions? We can see that neural networks can be interpreted as networks which compute decisions about decisions, leading from simple input data to more and more complex ‘meta-decisions’. This, to me, is an incredible concept. All the complexity of even the human brain can be modelled using these principles. From the level of photons interacting with our cone-cells right up to our pondering of the meaning of life, it’s just simple little decision-making neurons.
Below is a diagram of a simple neural network which essentially has 3 layers of abstraction:
A simple neural network with 2 inputs and 2 outputs.
As an example, the above inputs could be 2 infrared distance sensors, and the outputs might control control the on/off switch for 2 motors which drive the wheels of a robot.
In our simple hat example, we could pick the weights and the threshold quite easily, but how do we pick the weights and thresholds in this example so that, say, the robot can follow things that move? And how do we know how many neurons we need to solve this problem? Could we solve it with just 1 neuron, maybe 2? Or do we need 20? And how do we organise them? In layers? Modules? These questions are the questions in the field of neural networks. Techniques such as ‘backpropagation’ and (more recently) ‘neuroevolution’ are used effectively to answer some of these troubling questions, but these are outside the scope of this introduction – Wikipedia and Google Scholar and free online textbooks like “Neural Networks and Deep Learning” by M. A. Nielsen are great places to start learning about these concepts.
Hopefully you now have some intuition for how neural networks work, but if you’re interested in actually implementing a neural network there are a few optimisations and extensions to our concept of a neuron which.will make our neural nets more efficient and effective.
Firstly, notice that if we set the threshold value of the neuron to zero, we can always adjust the weightings of the inputs to account for this – only, we’ll also need to allow negative values for our weights. This is great since it removes one variable from our neuron. So we’ll allow negative weights and from now on we won’t need to worry about setting a threshold – it’ll always be zero.
Next, we’ll notice that the weights of the input connections are all relative to one-another, so we can actually normalise these to a value between -1 and 1. Cool. That simplifies things a little.
We can make a further, more substantial improvement to our decision-maker by realising that the inputs themselves (a, b and c in the above example) need not just be 0 or 1. For example, what if today is really sunny? Or maybe there’s scattered clouds, do it’s intermittently sunny? We can see that by allowing values between 0 and 1, our neuron gets more information and can therefore make a better decision – and the good news is, we don’t need to change anything in our neuron model!
So far, we’ve allowed the neuron to accept inputs between 0 and 1, and we’ve normalised the weights between -1 and 1 for convenience.
The next question is: why do we need such certainty in our final decision (i.e. the output of the neuron)? Why can’t it, like the inputs, also be a value between 0 and 1? If we did allow this, the decision of whether or not to wear a hat would become a level of certainty that wearing a hat is the right choice. But if this is a good idea, why did I introduce a threshold at all? Why not just directly pass on the sum of the weighted inputs to the output connection? Well, because, for reasons beyond the scope of this simple introduction to neural networks, it turns out that a neural network works better if the neurons are allowed to make something like an ‘educated guess’, rather than just presenting a raw probability. A threshold gives the neurons a slight bias toward certainty and allows them to be more ‘assertive’, and doing so makes neural networks more efficient. So in that sense, a threshold is good. But the problem with a threshold is that it doesn’t let us know when the neuron is uncertain about its decision – that is, if the sum of the weighted inputs is very close to the threshold, the neuron makes a definite yes/no answer where a definite yes/no answer is not ideal.
So how can we overcome this problem? Well it turns out that if we replace our “greater than zero” condition with a continuous function (called an ‘activation function’), then we can choose non-binary and non-linear reactions to the neuron’s weighted inputs. Let’s first look at our original “greater than zero” condition as a function:
‘Step’ function representing the original neuron’s ‘activation function’.
In the above activation function, the x-axis represents the sum of the weighted inputs and the y-axis represents the neuron’s output. Notice that even if the inputs sum to 0.01, the output is a very certain 1. This is not ideal, as we’ve explained earlier. So we need another activation function that only has a bias towards certainty. Here’s where we welcome the ‘sigmoid’ function:
The ‘sigmoid’ function; a more effective activation function for our artificial neural networks.
Notice how it looks like a halfway point between a step function (which we established as too certain) and a linear x=y line that we’d expect from a neuron which just outputs the raw probability that some some decision is correct. The equation for this sigmoid function is:
where x is the sum of the weighted inputs.
And that’s it! Our new-and-improved neuron does the following:
Takes multiple inputs between 0 and 1.
Weights each one by a value between -1 and 1.
Sums them all together.
Puts that sum into the sigmoid function.
Outputs the result!
It's deceptively simple, but by combining these simple decision-makers together and finding ideal connection weights, we can make arbitrarily complex decisions and calculations which stretch far beyond what our biological brains allow.
I'm trying to implement the Deep q-learning algorithm for a pong game.
I've already implemented Q-learning using a table as Q-function. It works very well and learns how to beat the naive AI within 10 minutes. But I can't make it work
using neural networks as a Q-function approximator.
I want to know if I am on the right track, so here is a summary of what I am doing:
I'm storing the current state, action taken and reward as current Experience in the replay memory
I'm using a multi layer perceptron as Q-function with 1 hidden layer with 512 hidden units. for the input -> hidden layer I am using a sigmoid activation function. For hidden -> output layer I'm using a linear activation function
A state is represented by the position of both players and the ball, as well as the velocity of the ball. Positions are remapped, to a much smaller state space.
I am using an epsilon-greedy approach for exploring the state space where epsilon gradually goes down to 0.
When learning, a random batch of 32 subsequent experiences is selected. Then I
compute the target q-values for all the current state and action Q(s, a).
forall Experience e in batch
if e == endOfEpisode
target = e.getReward
else
target = e.getReward + discountFactor*qMaxPostState
end
Now I have a set of 32 target Q values, I am training the neural network with those values using batch gradient descent. I am just doing 1 training step. How many should I do?
I am programming in Java and using Encog for the multilayer perceptron implementation. The problem is that training is very slow and performance is very weak. I think I am missing something, but can't figure out what. I would expect at least a somewhat decent result as the table approach has no problems.
I'm using a multi layer perceptron as Q-function with 1 hidden layer with 512 hidden units.
Might be too big. Depends on your input / output dimensionality and the problem. Did you try fewer?
Sanity checks
Can the network possibly learn the necessary function?
Collect ground truth input/output. Fit the network in a supervised way. Does it give the desired output?
A common error is to have the last activation function something wrong. Most of the time, you will want a linear activation function (as you have). Then you want the network to be as small as possible, because RL is pretty unstable: You can have 99 runs where it doesn't work and 1 where it works.
Do I explore enough?
Check how much you explore. Maybe you need more exploration, especially in the beginning?
See also
My DQN agent
keras-rl
Try using ReLu (or better Leaky ReLu)-Units in the hidden layer and a Linear-Activision for the output.
Try changing the optimizer, sometimes SGD with propper learning-rate-decay helps.
Sometimes ADAM works fine.
Reduce the number of hidden units. It might be just too much.
Adjust the learning rate. The more units you have, the more impact does the learning rate have as the output is the weighted sum of all neurons before.
Try using the local position of the ball meaning: ballY - paddleY. This can help drastically as it reduces the data to: above or below the paddle distinguished by the sign. Remember: if you use the local position, you won't need the players paddle-position and the enemies paddle position must be local too.
Instead of the velocity, you can give it the previous state as an additional input.
The network can calculate the difference between those 2 steps.
I have a neural network with 2 entry variables, 1 hidden layer with 2 neurons and the output layer with one output neuron. When I start with some randomly (from 0 to 1) generated weights, the network learns the XOR function very fast and good, but in other cases, the network NEVER learns the XOR function! Do you know why this happens and how can I overcome this problem? Could some chaotic behaviour be involved? Thanks!
This is quite normal situation, because error function for multilayer NN is not convex, and optimization converges to local minimum.
You can just keep initial weights that resulted in successful optimization, or run optimizer multiple times starting from different weights, and keep the best solution. Optimization algorithm and learning rate also plays certain role, for example backpropagation with momentum and/or stochastic gradient descent sometimes work better. Also, if you add more neurons, beyond the minimum needed to learn XOR, this also helps.
There exist methodologies designed to find global minimum, such as simulated annealing, but, in practice they are not commonly used for NN optimization, except for some specific cases