I'm learning MXNet at the moment and I'm working on a problem using neural nets. I'm interested in observing the curvature of my loss function with respect to the network weights but as best I can tell higher order gradients are not supported for neural network functions at the moment. Is there any (possibly hacky) way that I could still do this?
You can follow the discussion here
The gist of it is that not all operators support higher order gradients at the moment.
In Gluon you can try the following:
with mx.autograd.record():
output = net(x)
loss = loss_func(output)
dz = mx.autograd.grad(loss, [z], create_graph=True) # where [z] is the parameter(s) you want
dz[0].backward() # now the actual parameters should have second order gradients
Taken from this forum thread
Related
Being new to Tensorflow, I am trying to understand the difference between underlying functionality of tf.gradients and tf.keras.backend.gradients.
The latter finds the gradient of input feature values w.r.t cost function.
But I couldn't get a clear idea on the former whether it computes the gradient over cost function or output probabilities (For example, consider the case of binary classification using a simple feed forward network. Output probability here is referred to the Sigmoid activation outcome of final layer with single neuron. Cost is given by Binary cross entropy)
I have referred the official documentation for tf.gradients, but it is short and vague (for me), and I did not get a clear picture - The documentation mentions it as just 'y' - is it cost or output probability?
Why I need the gradients?
To implement a basic gradient based feature attribution.
They are basically the same. tf.keras is TensorFlow's high-level API for building and training deep learning models. It's used for fast prototyping, state-of-the-art research, and production. tf.Keras basically uses Tensorflow in its backend. By looking at tf.Keras source code here, we can see that tf.keras.backend.gradients indeed uses tf.gradients:
# Part of Keras.backend.py
from tensorflow.python.ops import gradients as gradients_module
#keras_export('keras.backend.gradients')
def gradients(loss, variables):
"""Returns the gradients of `loss` w.r.t. `variables`.
Arguments:
loss: Scalar tensor to minimize.
variables: List of variables.
Returns:
A gradients tensor.
"""
# ========
# Uses tensorflow's gradient function
# ========
return gradients_module.gradients(
loss, variables, colocate_gradients_with_ops=True)
I'm learning about Action-Critic Reinforcement Learning techniques, in particular A2C algorithm.
I've found a good description of a simple version of the algorithm (i.e. without experience replay, batching or other tricks) with implementation here: https://link.medium.com/yi55uKWwV2. The complete code from that article is available on GitHub.
I think I understand ok-ish what's happening here, but to make sure I actually do, I'm trying to reimplement it from scratch using higher-level tf.keras APIs. Where I'm getting stuck is how do I implement training loop correctly, and how do I formulate actor's loss function.
What is the correct way to pass action and advantage into the loss function?
Actor's loss function involves computing probability of the action taken given to normal distribution. How can I ensure that mu and sigma of the normal distribution during loss function computation actually match the ones were during prediction?
The way it is in the original, the actor's loss function doesn't care about y_pred, it only does about action that was chosen while interacting with the environment. This seems to be wrong, but I'm not sure how.
The code I have so far: https://gist.github.com/nevkontakte/beb59f29e0a8152d99003852887e7de7
Edit: I suppose some of my confusion stems from a poor understanding of magic behind gradient computation in Keras/TensorFlow, so any pointers there would be appreciated.
First, credit where credit is due: information provided by ralf htp and Simon was instrumental in helping me to figure out the right answers eventually.
Before I go into detailed answers to my own questions, here's the original code I was trying to rewrite in tf.keras terms, and here's my result.
What is the correct way to pass action and advantage into a loss function in Keras?
There is a difference between what raw TF optimizer considers a loss function and what Keras does. When using an optimizer directly, it simply expects a tensor (lazy or eager depending on your configuration), which will be evaluated under tf.GradientTape() to compute the gradient and update weights.
Example from https://medium.com/#asteinbach/actor-critic-using-deep-rl-continuous-mountain-car-in-tensorflow-4c1fb2110f7c:
# Below norm_dist is the output tensor of the neural network we are training.
loss_actor = -tfc.log(norm_dist.prob(action_placeholder) + 1e-5) * delta_placeholder
training_op_actor = tfc.train.AdamOptimizer(
lr_actor, name='actor_optimizer').minimize(loss_actor)
# Later, in the training loop...
_, loss_actor_val = sess.run([training_op_actor, loss_actor],
feed_dict={action_placeholder: np.squeeze(action),
state_placeholder: scale_state(state),
delta_placeholder: td_error})
In this example it computes the whole graph, including making an inference, capture the gradient and adjust weights. So to pass whatever values you need into the loss function/gradient computation you just pass necessary values into the computation graph.
Keras is a bit more formal in what loss function should look like:
loss: String (name of objective function), objective function or tf.keras.losses.Loss instance. See tf.keras.losses. An objective function is any callable with the signature scalar_loss = fn(y_true, y_pred). If the model has multiple outputs, you can use a different loss on each output by passing a dictionary or a list of losses. The loss value that will be minimized by the model will then be the sum of all individual losses.
Keras will do the inference (forward pass) for you and pass the output into the loss function. The loss function is supposed to do some extra computation on the predicted value and y_true label, and return the result. This whole process will be tracked for the purpose of gradient computation.
Although it is very convenient for traditional training, this is a bit restrictive when we want to pass some extra data in, like TD error. It is possible to work around that and shove all the extra data into y_true, and pull it apart inside the loss function (I found this trick somewhere on the web, but unfortunately lost the link to source).
Here's how I rewrote the above in the end:
def loss(y_true, y_pred):
action_true = y_true[:, :n_outputs]
advantage = y_true[:, n_outputs:]
return -tfc.log(y_pred.prob(action_true) + 1e-5) * advantage
# Below, in the training loop...
# A trick to pass TD error *and* actual action to the loss function: join them into a tensor and split apart
# Inside the loss function.
annotated_action = tf.concat([action, td_error], axis=1)
actor_model.train_on_batch([scale_state(state)], [annotated_action])
Actor's loss function involves computing probability of the action taken given to normal distribution. How can I ensure that mu and sigma of the normal distribution during loss function computation actually match the ones were during prediction?
When I asked this question, I didn't understand well enough how TF compute graph works. So the answer is simple: every time sess.run() is invoked, it must compute the whole graph from scratch. Parameters of the distribution would be the same (or similar) as long as graph inputs (e.g. observed state) and NN weights are the same (or similar).
The way it is in the original, the actor's loss function doesn't care about y_pred, it only does about action that was chosen while interacting with the environment. This seems to be wrong, but I'm not sure how.
What's wrong is the assumption "the actor's loss function doesn't care about y_pred" :) Actor's loss function involves norm_dist (which is action probability distribution), which is effectively an analog of y_pred in this context.
As far as i understand A2C it is the machine learning implementation of activator-inhibitor systems that are also called two-component reaction diffusion systems (https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_system). Activator-inhibitor models are important in any field of science as they describe pattern formations like i.e. the Turing mechanism (simply search the net for activator-inhibitor model and you find a vast amount of information, a very common application are predator-prey models). Also cf the graphic
source of graphic : https://www.researchgate.net/figure/Activator-Inhibitor-System_fig1_23671770/
with the explanatory graphic of the A2C algorithm in https://towardsdatascience.com/reinforcement-learning-w-keras-openai-actor-critic-models-f084612cfd69
Activator-inhibitor models are closely linked to the theory of nonlinear dynamical systems (or 'chaos theory') this also becomes obvious in the comparison of the bifurcation tree-like structure in https://medium.com/#asteinbach/rl-introduction-simple-actor-critic-for-continuous-actions-4e22afb712 and the bifurcation tree of a nonlinear dynamical systems like i.e. the logistic map (https://en.wikipedia.org/wiki/Logistic_map, the logistic map is one of the simplest predator-prey models or activator-inhibitor models). Another similarity is the sensitivity to initial condition in A2C models that is described as
This introduces in inherent high variability in log probabilities (log of the policy distribution) and cumulative reward values, because each trajectories during training can deviate from each other at great degrees.
in https://towardsdatascience.com/understanding-actor-critic-methods-931b97b6df3f and the curse of dimensionality appears also in chaos theory, i.e. in attractor reconstruction
From the viewpoint of systems theory the A2C algorithm tries to adapt the initial value (start state) in a way that it ends up at a given endpoint when increasing the growth rate of a dynamical systems i.e. the logistic map (r-value is increased and the initial value (start state) is constantly re-adapted to choose the correct bifurations (actions) in the bifurcation tree )
So A2C tries to numerically solve a chaos theory problem, namely finding the initial value for a given outcome of a nonlinear dynamical system in its chaotic region. Analytically this problem is in most cases not solveable.
The action is the bifurcation points in the bifurcation tree, the states are the future bifurctions.
Both, actions and states, are modeled by two coupled neural networks and this coupling of two neural nets is the great innovation of A2C algorithms.
In https://towardsdatascience.com/reinforcement-learning-w-keras-openai-actor-critic-models-f084612cfd69 is well documented keras code for implementing A2C, so you have a possible implementation there.
The loss function here is defined as the temporal difference (TD) function that is the exact difference between state at the actual bifurcation point and the state at the estimated future one, however this mathematically exactly defined is prone to stochastic error (or noise), so the stochastic error is included in the definition of exact, because in the end machine learning is based on stochastic systems or error calculus, meaning systems that are composed of a deterministic and a stochastic component. To zero this error stochastic gradient descend is used. In keras this is simply implmeneted by choosing optimizer=sge.
This interaction of actual and future step is implemented as memory on https://towardsdatascience.com/reinforcement-learning-w-keras-openai-actor-critic-models-f084612cfd69 in the function remember and this function also links the actor and the critic network (or activator and inhibitor network). This general structure of trial (action), call predict (TD function ), remember and train (i.e. stochastic gradient descent) is fundamental to all reinforcement learning algorithms, and is linked to the structure actual state, action, reward, new state :
The prediction code is also very much the same as it was in previous reinforcement learning algorithms. That is, we just have to iterate through the trial and call predict, remember, and train on the agent:
In the implementation on your first question is solved by applying remember on the critic and the train the critic with these values (this is in the main function), where training always evaluates the loss function, so action and reward are passed to the loss function by remember in this implementation :
actor_critic.remember(cur_state, action, reward, new_state, done)
actor_critic.train()
Because of your second question : i am not sure but i think this is achieved by the optimization algorithm (i.e. stochastic gradient descent)
Third question : In the predator-prey model the actors or activator is the prey and the behavior of the prey is only determined by the size or capacity of the habitat (the amount of grass) and the size of the predator (inhibitor) population, so modelling it in this way is consistent with nature or an activator-inhibitor system again. In the main function in https://towardsdatascience.com/reinforcement-learning-w-keras-openai-actor-critic-models-f084612cfd69 also only the critic or inhibitor / predator is trained.
I am building a vanilla neural network from scratch using NumPy and trialling the model performance for different activation functions. I am especially keen to see how the 'Maxout' activation function would effect my model performance.
After doing some search, I was not able to find an implementation in NumPy except for their definition (https://ibb.co/kXCpjKc). The formula for forward propagation is clear where I would take the max(Z) (where Z = w.T * x + b). But, their derivative that I will be using in backpropogation is not clear to me.
What does j = argmax(z) mean in this context? How do I implement it in NumPy?
Any help would be much appreciated! Thank you!
Changing any of the non maximum values slightly does not affect the output, so their gradient is zero. The gradient is passed from the next layer to only the neuron that achieved the max (gradient = 1 in the link you provided). See this stackoverflow answer: https://datascience.stackexchange.com/a/11703.
In a neural network setting you would need the gradient with respect to every of the x_i, so you would need the full derivative. In the link you provided you can see there is only a partial derivative defined. The partial derivative is a vector (of almost all zeros and 1 where the neuron is maximum), so the full gradient will become a matrix.
You can implement this in numpy using np.argmax.
Recently, I try to do some experiments and I have a neural network D(x) where x is the input image with batch size 64. I want to compute the gradient of D(x) with respect to x. Should I do the computation as the following?
grad = tf.gradients(D(x), [x])
Thank you everybody!
Yes, you will need to use tf.gradients. For more details see https://www.tensorflow.org/api_docs/python/tf/gradients.
During the training of a neural network, the gradient is generally computed of a loss function with respect to the input. This is because, the loss function can be well defined along with its gradient.
However, if you talk about the gradient of your output D(x), this I assume is some set of vector(s). You will need to define how the gradient will be computed with respect to its input (i.e the layer which generates the output).
The exact details of that implementation depends upon the framework which you are using.
I've been trying to build a model using 'Deep Q-Learning' where I have a large number of actions (2908). After some limited success with using standard DQN:
(https://www.cs.toronto.edu/~vmnih/docs/dqn.pdf), I decided to do some more research because I figured the action space was too large to do effective exploration.
I then discovered this paper: https://arxiv.org/pdf/1512.07679.pdf where they use an actor-critic model and policy gradients, which then led me to: https://arxiv.org/pdf/1602.01783.pdf where they use policy gradients to get much better results then DQN overall.
I've found a few sites where they have implemented policy gradients in Keras, https://yanpanlau.github.io/2016/10/11/Torcs-Keras.html and https://oshearesearch.com/index.php/2016/06/14/kerlym-a-deep-reinforcement-learning-toolbox-in-keras/ however I'm confused how they are implemented. In the former (and when I read the papers) it seems like instead of providing an input and output pair for the actor network, you provide the gradients for the all the weights and then use the network to update it, whereas, in the latter they just calculate an input-output pair.
Have I just confused myself? Am I just supposed to be training the network by providing an input-output pair and use the standard 'fit', or do I have to do something special? If it's the latter, how do I do it with the Theano backend? (the examples above use TensorFlow).
TL;DR
Learn how to implement custom loss functions and gradients using Keras.backend. You will need it for more advanced algorithms and it's actually much easier once you get the hang of it
One CartPole example of using keras.backend could be https://gist.github.com/kkweon/c8d1caabaf7b43317bc8825c226045d2 (though its backend used Tensorflow but it should be very similar if not the same)
Problem
When playing,
the agent needs a policy that is basically a function that maps a state into a policy that is a probability for each action. So, the agent will choose an action according to its policy.
i.e, policy = f(state)
When training,
Policy Gradient does not have a loss function. Instead, it tries to maximize the expected return of rewards. And, we need to compute the gradients of log(action_prob) * advantage
advantage is a function of rewards.
advantage = f(rewards)
action_prob is a function of states and action_taken. For example, we need to know which action we took so that we can update parameters to increase/decrease a probability for the action we took.
action_prob = sum(policy * action_onehot) = f(states, action_taken)
I'm assuming something like this
policy = [0.1, 0.9]
action_onehot = action_taken = [0, 1]
then action_prob = sum(policy * action_onehot) = 0.9
Summary
We need two functions
update function: f(state, action_taken, reward)
choose action function: f(state)
You already know it's not easy to implement like typical classification problems where you can just model.compile(...) -> model.fit(X, y)
However,
In order to fully utilize Keras, you should be comfortable with defining custom loss functions and gradients. This is basically the same approach the author of the former one took.
You should read more documentations of Keras functional API and keras.backend
Plus, there are many many kinds of policy gradients.
The former one is called DDPG which is actually quite different from regular policy gradients
The latter one I see is a traditional REINFORCE policy gradient (pg.py) which is based on Kapathy's policy gradient example. But it's very simple for example it only assumes only one action. That's why it could have been implemented somehow using model.fit(...) instead.
References
Schulman, "Policy Gradient Methods", http://rll.berkeley.edu/deeprlcourse/docs/lec2.pdf
The seemingly conflicting implementations you encountered are both valid implementations. They are two equivalent ways two implement the policy gradients.
In the vanilla implementation, you calculate the gradients of the policy network w.r.t. rewards and directly update the weights in the direction of the gradients. This would require you to do the steps described by Mo K.
The second option is arguably a more convenient implementation for autodiff frameworks like keras/tensorflow. The idea is to implement an input-output (state-action) function like supervised learning, but with a loss function who's gradient is identical to the policy gradient. For a softmax policy, this simply means predicting the 'true action' and multiplying the (cross-entropy) loss with the observed returns/advantage. Aleksis Pirinen has some useful notes about this [1].
The modified loss function for option 2 in Keras looks like this:
import keras.backend as K
def policy_gradient_loss(Returns):
def modified_crossentropy(action,action_probs):
cost = K.categorical_crossentropy(action,action_probs,from_logits=False,axis=1 * Returns)
return K.mean(cost)
return modified_crossentropy
where 'action' is the true action of the episode (y), action_probs is the predicted probability (y*). This is based on another stackoverflow question [2].
References
https://aleksispi.github.io/assets/pg_autodiff.pdf
Make a custom loss function in keras