I've noticed that a frequent occurrence during training is NANs being introduced.
Often times it seems to be introduced by weights in inner-product/fully-connected or convolution layers blowing up.
Is this occurring because the gradient computation is blowing up? Or is it because of weight initialization (if so, why does weight initialization have this effect)? Or is it likely caused by the nature of the input data?
The overarching question here is simply: What is the most common reason for NANs to occurring during training? And secondly, what are some methods for combatting this (and why do they work)?
I came across this phenomenon several times. Here are my observations:
Gradient blow up
Reason: large gradients throw the learning process off-track.
What you should expect: Looking at the runtime log, you should look at the loss values per-iteration. You'll notice that the loss starts to grow significantly from iteration to iteration, eventually the loss will be too large to be represented by a floating point variable and it will become nan.
What can you do: Decrease the base_lr (in the solver.prototxt) by an order of magnitude (at least). If you have several loss layers, you should inspect the log to see which layer is responsible for the gradient blow up and decrease the loss_weight (in train_val.prototxt) for that specific layer, instead of the general base_lr.
Bad learning rate policy and params
Reason: caffe fails to compute a valid learning rate and gets 'inf' or 'nan' instead, this invalid rate multiplies all updates and thus invalidating all parameters.
What you should expect: Looking at the runtime log, you should see that the learning rate itself becomes 'nan', for example:
... sgd_solver.cpp:106] Iteration 0, lr = -nan
What can you do: fix all parameters affecting the learning rate in your 'solver.prototxt' file.
For instance, if you use lr_policy: "poly" and you forget to define max_iter parameter, you'll end up with lr = nan...
For more information about learning rate in caffe, see this thread.
Faulty Loss function
Reason: Sometimes the computations of the loss in the loss layers causes nans to appear. For example, Feeding InfogainLoss layer with non-normalized values, using custom loss layer with bugs, etc.
What you should expect: Looking at the runtime log you probably won't notice anything unusual: loss is decreasing gradually, and all of a sudden a nan appears.
What can you do: See if you can reproduce the error, add printout to the loss layer and debug the error.
For example: Once I used a loss that normalized the penalty by the frequency of label occurrence in a batch. It just so happened that if one of the training labels did not appear in the batch at all - the loss computed produced nans. In that case, working with large enough batches (with respect to the number of labels in the set) was enough to avoid this error.
Faulty input
Reason: you have an input with nan in it!
What you should expect: once the learning process "hits" this faulty input - output becomes nan. Looking at the runtime log you probably won't notice anything unusual: loss is decreasing gradually, and all of a sudden a nan appears.
What can you do: re-build your input datasets (lmdb/leveldn/hdf5...) make sure you do not have bad image files in your training/validation set. For debug you can build a simple net that read the input layer, has a dummy loss on top of it and runs through all the inputs: if one of them is faulty, this dummy net should also produce nan.
stride larger than kernel size in "Pooling" layer
For some reason, choosing stride > kernel_size for pooling may results with nans. For example:
layer {
name: "faulty_pooling"
type: "Pooling"
bottom: "x"
top: "y"
pooling_param {
pool: AVE
stride: 5
kernel: 3
}
}
results with nans in y.
Instabilities in "BatchNorm"
It was reported that under some settings "BatchNorm" layer may output nans due to numerical instabilities.
This issue was raised in bvlc/caffe and PR #5136 is attempting to fix it.
Recently, I became aware of debug_info flag: setting debug_info: true in 'solver.prototxt' will make caffe print to log more debug information (including gradient magnitudes and activation values) during training: This information can help in spotting gradient blowups and other problems in the training process.
In my case, not setting the bias in the convolution/deconvolution layers was the cause.
Solution: add the following to the convolution layer parameters.
bias_filler {
type: "constant"
value: 0
}
This answer is not about a cause for nans, but rather proposes a way to help debug it.
You can have this python layer:
class checkFiniteLayer(caffe.Layer):
def setup(self, bottom, top):
self.prefix = self.param_str
def reshape(self, bottom, top):
pass
def forward(self, bottom, top):
for i in xrange(len(bottom)):
isbad = np.sum(1-np.isfinite(bottom[i].data[...]))
if isbad>0:
raise Exception("checkFiniteLayer: %s forward pass bottom %d has %.2f%% non-finite elements" %
(self.prefix,i,100*float(isbad)/bottom[i].count))
def backward(self, top, propagate_down, bottom):
for i in xrange(len(top)):
if not propagate_down[i]:
continue
isf = np.sum(1-np.isfinite(top[i].diff[...]))
if isf>0:
raise Exception("checkFiniteLayer: %s backward pass top %d has %.2f%% non-finite elements" %
(self.prefix,i,100*float(isf)/top[i].count))
Adding this layer into your train_val.prototxt at certain points you suspect may cause trouble:
layer {
type: "Python"
name: "check_loss"
bottom: "fc2"
top: "fc2" # "in-place" layer
python_param {
module: "/path/to/python/file/check_finite_layer.py" # must be in $PYTHONPATH
layer: "checkFiniteLayer"
param_str: "prefix-check_loss" # string for printouts
}
}
learning_rate is high and should be decreased
The accuracy in the RNN code was nan, with select the low value for learning rate it fixes
One more solution for anyone stuck like I just was-
I was receiving nan or inf losses on a network I setup with float16 dtype across the layers and input data. After all else failed, it occurred to me to switch back to float32, and the nan losses were solved!
So bottom line, if you switched dtype to float16, change it back to float32.
I was trying to build a sparse autoencoder and had several layers in it to induce sparsity. While running my net, I encountered the NaN's. On removing some of the layers (in my case, I actually had to remove 1), I found that the NaN's disappeared. So, I guess too much sparsity may lead to NaN's as well (some 0/0 computations may have been invoked!?)
Related
Huggigface BERT implementation has a hack to remove the pooler from optimizer.
https://github.com/huggingface/transformers/blob/b832d5bb8a6dfc5965015b828e577677eace601e/examples/run_squad.py#L927
# hack to remove pooler, which is not used
# thus it produce None grad that break apex
param_optimizer = [n for n in param_optimizer if 'pooler' not in n[0]]
We are trying to run pretrining on huggingface bert models. The code always diverges later during the training if this pooler hack is not applied. I also see the pooler layer being used during classification.
pooled_output = outputs[1]
pooled_output = self.dropout(pooled_output)
logits = self.classifier(pooled_output)
The pooler layer is a FFN with tanh activation
class BertPooler(nn.Module):
def __init__(self, config):
super().__init__()
self.dense = nn.Linear(config.hidden_size, config.hidden_size)
self.activation = nn.Tanh()
def forward(self, hidden_states):
# We "pool" the model by simply taking the hidden state corresponding
# to the first token.
first_token_tensor = hidden_states[:, 0]
pooled_output = self.dense(first_token_tensor)
pooled_output = self.activation(pooled_output)
return pooled_output
My question is why this pooler hack solves numeric instability?
Problem seen with pooler
There are quite a few resources out there that probably tackle this issue better than me, see for example here, or here.
Specifically, the problem is that you are dealing with vanishing (or exploding) gradients, specifically when using loss functions that flatten in either direction for very small/large inputs, which is the case for both sigmoid and tanh (the only difference here is the range in which their output lies, which is [0, 1] and [-1, 1], respectively.
Additionally, if you have a low-precision decimal, as is the case with APEX, then the gradient vanishing behavior is much more likely to appear already for relatively moderate outputs, as the precision limits the numbers which it is able to differentiate from zero. One way to deal with this is to have functions that have strictly non-zero and easily computable derivatives, such as Leaky ReLU, or simply avoid the activation function altogether (which I'm assuming is what huggingface is doing here).
Note that the problem of exploding gradients is usually not as tragic, as we can apply gradient clipping (limiting it to a fixed maximum size), but nonetheless the principle is the same. For zeroed gradients, on the other hand, there is no such easy fix, since it causes your neurons to "die" (no active learning is happening with zero backflow), which is why I'm assuming that you see the diverging behavior.
I am trying to fit a Maxwell distribution with the values included in this file.
Since I need to force a loc of 0, I am using the expression:
params = st.maxwell.fit(values, floc=0)
However, this fails with a "divide by zero encountered in log" error:
I estimate that a pdf with loc=0 and a scale around 11000 would be quite good.
How can I bypass this problem and make the fitting work?
The code that maximizes the log-likelihood is triggering an underflow, because its default initial guess for the scale parameter is 1, which is pretty far from the actual optimum. You can pass in a better guess to fit with the scale argument. A good guess is provided by the method of moments, which in this case means using the sample mean to estimate the scale parameter. The wikipedia page on the Maxwell-Boltzmann distribution gives the formula for the mean μ in terms of the scale a. Solving for a gives:
In [19]: a = np.mean(values)/(2*np.sqrt(2/np.pi))
In [20]: a
Out[20]: 10587.275841415043
Now pass that value to fit as the scale argument, which overrides the default initial value used by the optimization code:
In [21]: p = maxwell.fit(values, floc=0, scale=a)
In [22]: p
Out[22]: (0, 10455.558876443883)
That's the maximum likelihood estimate for the location (which you set to 0) and the scale a.
for my input in the feed forward neural network that I have implemented in Keras, I just wanted to check that my understanding is correct.
[[ 25.26000023 26.37000084 24.67000008 23.30999947]
[ 26.37000084 24.67000008 23.30999947 21.36000061]
[ 24.67000008 23.30999947 21.36000061 19.77000046]...]
So in the data above it is a time window of 4 inputs in an array. My input layer is
model.add(Dense(4, input_dim=4, activation='sigmoid'))
model.fit(trainX, trainY, nb_epoch=10000,verbose=2,batch_size=4)
and batch_size is 4, in theory when I call the fit function will the function go over all these inputs in each nb_epoch? and does the batch_size need to be 4 in order for this time window to work?
Thanks John
and batch_size is 4, in theory when I call the fit function will the function go over all these inputs in each nb_epoch?
Yes, each epoch is iteration over all training samples
and does the batch_size need to be 4 in order for this time window to work?
No, these are completely unrelated things. Batch is simply a subset of your training data which is used to compute approximation of the true gradient of the cost function. Bigger the batch - closer you get to the true gradient (and original Gradient Descent), but training gets slower. Closer to 1 you get - it becomes more and more stochastic, noisy approxmation (and closer to Stochastic Gradient Descent). The fact that you matched batch_size and data dimensionality is just an odd-coincidence, and has no meaning.
Let me put this in more generall setting, what you do in gradient descent with additive loss function (which neural nets usually use) is going against the gradient which is
grad_theta 1/N SUM_i=1^N loss(x_i, pred(x_i), y_i|theta) =
= 1/N SUM_i=1^N grad_theta loss(x_i, pred(x_i), y_i|theta)
where loss is some loss function over your pred (prediction) as compared to y_i.
And in batch based scenatio (the rough idea) is that you do not need to go over all examples, but instead some strict subset, like batch = {(x_1, y_1), (x_5, y_5), (x_89, y_89) ... } and use approximation of the gradient of form
1/|batch| SUM_(x_i, y_i) in batch: grad_theta loss(x_i, pred(x_i), y_i|theta)
As you can see this is not related in any sense to the space where x_i live, thus there is no connection with dimensionality of your data.
Let me explain this with an example:
When you have 32 training examples and you call model.fit with a batch_size of 4, the neural network will be presented with 4 examples at a time, but one epoch will still be defined as one complete pass over all 32 examples. So in this case the network will go through 4 examples at a time, and will ,theoretically at least, call the forward pass (and the backward pass) 32 / 4 = 8 times.
In the extreme case when your batch_size is 1, that is plain old stochastic gradient descent. When your batch_size is greater than 1 then it's called batch gradient descent.
In theano, given a batch cost cost with shape (batch_size,), it is easy to compute the gradient of the mean cost, as in T.grad(T.mean(cost,axis=0),p) with p being a parameter used in the computation of cost. This is done efficiently by backpropagating the gradient through the computational graph. What I would now like to do is to compute the mean of the squared gradients over the batch. This can be done using the following piece of code:
import theano.tensor as T
g_square = T.mean(theano.scan(lambda i:T.grad(cost[i],p)**2,sequences=T.arange(cost.shape[0]))[0],axis=0)
Where for convenience p is assumed to be a single theano tensor and not a list of tensors.
The computation could be performed efficiently by simply backpropagating the gradient until the last step, and squaring the components of the last operation (which should be a sum over the batch index). I might be wrong on this one, but the computation should be as easy, and nearly as fast as a simple backpropagation. However, theano seems unable to optimize the computation, and it keeps using a loop, making computations extremely slow.
Would anyone know of a solution to make the computation efficient, either by forcing optimizations, expressing the computation in a different way, or even going through the backpropagation process ?
Thanks in advance.
Your function g_square happens to have complexity O(batch_size**2) instead of O(batch_size) as expected. This lets it appear incredibly slow for larger batch sizes.
The reason is because in every iteration the forward and backward pass is computed over the whole batch, even though just cost[i] for one data point is needed.
I assume the input to the cost computation graph, x, is a tensor with the first dimension of size batch_size. Theano has no means to automatically slice this tensor along this dimension. Therefore computation is always done over the whole batch.
Unfortunately I see no better solution than slicing your input and doing the loop outside Theano:
# x: input data batch
batch_size = x.shape[0]
g_square_fun = theano.function( [p], T.grad(cost[0],p)**2)
g_square_value = 0
for i in batch_size:
g_square_value += g_square_fun( x[i:i+1])
Perhaps when future versions of Theano come with better build in capabilities to compute Jacobians there will be more elegant solutions.
After digging deeper in the Theano docs I found a solution that would work in the compute graph. Key idea is that you clone the graph of your network inside the scan function, thereby explicitly slicing the input tensor. I tried the following code and empirically it shows O(batch_size) as expected:
# x: input data batch
# assuming cost = network(x,p)
from theano.gof.graph import clone_get_equiv
def g_square(cost,p):
g = T.zeros_like(p)
def scan_fn( i, g, cost, p):
# clone the graph computing cost, but slice it's input
cloned = clone_get_equiv([],[cost],
copy_inputs_and_orphans=False,
memo={x: x[i:i+1]})
cost_slice = cloned[cost].reshape([])
return g+T.grad(cost_slice,p)**2
result,updates = theano.reduce( scan_fn,
outputs_info=g,
sequences=[T.arange(cost.size)],
non_sequences=[cost.flatten(),p])
return result
I used scipy.optimize.fmin_bfgs to minimize the hinge loss (SVM). However, there are errors :
Divide-by-zero encountered: rhok assumed large.
Somebody said that “It had to do with the training data set”， anybody knows how to deal with the problem?
From the source code of scipy, rhok is,
rhok = 1.0 / (numpy.dot(yk, sk))
where both yk and sk depend on intput array x0.
A possible causes of this error may be a bad choice of initial condition x0 which tends to singularities in your function f. I would suggest plotting your function and maybe ensuring initial conditions are always away from possible divergent values. If this is part of a larger training routine, you could possibly use try and on catching an ZeroDivisionError try shifting the initial condition shifted by some amount. You may also find a different minimisation method is more robust from scipy minimize.
If you add the full_output option to scipy.optimize.fmin_bfgs it should give you more information about you particular case.