I'm wondering what if I have a layer generating a bottom blob that is further consumed by two subsequent layers, both of which will generate some gradients to fill bottom.diff in the back propagation stage. Will both two gradients be added up to form the final gradient? Or, only one of them can live? In my understanding, Caffe layers need to memset the bottom.diff to all zeros before filling it with some computed gradients, right? Will the memset flush out the already computed gradients by the other layer? Thank you!
Using more than a single loss layer is not out-of-the-ordinary, see GoogLeNet for example: It has three loss layers "pushing" gradients at different depths of the net.
In caffe, each loss layer has a associated loss_weight: how this particular component contribute to the loss function of the net. Thus, if your net has two loss layers, Loss1 and Loss1 the overall loss of your net is
Loss = loss_weight1*Loss1 + loss_weight2*Loss2
The backpropagation uses the chain rule to propagate the gradient of Loss (the overall loss) through all the layers in the net. The chain rule breaks down the derivation of Loss into partial derivatives, i.e., the derivatives of each layer, the overall effect is obtained by propagating the gradients through the partial derivatives. That is, by using top.diff and the layer's backward() function to compute bottom.diff one takes into account not only the layer's derivative, but also the effect of ALL higher layers expressed in top.diff.
TL;DR
You can have multiple loss layers. Caffe (as well as any other decent deep learning framework) handles it seamlessly for you.
Related
I have read that the " He weight Initialization" (He et al., 2015) built on the Lecun weight initialization and suggested a zero-mean Gaussian distribution where the standard deviation is
enter image description here
and this function should be used with ReLU to solve the vanishing/exploding gradient problem. For me, it does make sense because the way ReLu was built makes it no bothered with vanishing/exploding gradient problem. Since, if the input is less than 0 the derivative would be zero otherwise the derivative would be one. So, whatever the variance is, the gradient would be zero or one. Therefore, the He weight Initialization is useless. I know that I am missing something, that's why I am asking if anyone would tell me the usefulness of that weight initialization?
Weight initialization is applied, in general terms, to weights of layers that have learnable / trainable parameters, just like dense layers, convolutional layers, and other layers. ReLU is an activation function, fully deterministic, and has no initialization.
Regarding to the vanishing gradient problem, the backpropagation step is funded by computing the gradients by the chain rule (partial derivatives) for each weight (see here):
(...) each of the neural network's weights receive an update
proportional to the partial derivative of the error function with
respect to the current weight in each iteration of training.
The more deep a network is, the smaller these gradients get, and when a network becomes deep enough, the backprop step is less effective (in the worst case, it stops learning) and this becomes a problem:
This has the effect of multiplying n of these small numbers to compute
gradients of the "front" layers in an n-layer network, meaning that
the gradient (error signal) decreases exponentially with n while the
front layers train very slowly.
Choosing a proper activation function, like ReLU, help avoiding this to happen, as you mentioned in the OP, by making partial derivatives of this activation not too small:
Rectifiers such as ReLU suffer less from the vanishing gradient
problem, because they only saturate in one direction.
Hope this helps!
Iam a little bit confused about how to normalize/standarize image pixel values before training a convolutional autoencoder. The goal is to use the autoencoder for denoising, meaning that my traning images consists of noisy images and the original non-noisy images used as ground truth.
To my knowledge there are to options to pre-process the images:
- normalization
- standarization (z-score)
When normalizing using the MinMax approach (scaling between 0-1) the network works fine, but my question here is:
- When using the min max values of the training set for scaling, should I use the min/max values of the noisy images or of the ground truth images?
The second thing I observed when training my autoencoder:
- Using z-score standarization, the loss decreases for the two first epochs, after that it stops at about 0.030 and stays there (it gets stuck). Why is that? With normalization the loss decreases much more.
Thanks in advance,
cheers,
Mike
[Note: This answer is a compilation of the comments above, for the record]
MinMax is really sensitive to outliers and to some types of noise, so it shouldn't be used it in a denoising application. You can use quantiles 5% and 95% instead, or use z-score (for which ready-made implementations are more common).
For more realistic training, normalization should be performed on the noisy images.
Because the last layer uses sigmoid activation (info from your comments), the network's outputs will be forced between 0 and 1. Hence it is not suited for an autoencoder on z-score-transformed images (because target intensities can take arbitrary positive or negative values). The identity activation (called linear in Keras) is the right choice in this case.
Note however that this remark on activation only concerns the output layer, any activation function can be used in the hidden layers. Rationale: negative values in the output can be obtained through negative weights multiplying the ReLU output of hidden layers.
I'm trying to understand "Back Propagation" as it is used in Neural Nets that are optimized using Gradient Descent. Reading through the literature it seems to do a few things.
Use random weights to start with and get error values
Perform Gradient Descent on the loss function using these weights to arrive at new weights.
Update the weights with these new weights until the loss function is minimized.
The steps above seem to be the EXACT process to solve for Linear Models (Regression for e.g.)? Andrew Ng's excellent course on Coursera for Machine Learning does exactly that for Linear Regression.
So, I'm trying to understand if BackPropagation does anything more than gradient descent on the loss function.. and if not, why is it only referenced in the case of Neural Nets and why not for GLMs (Generalized Linear Models). They all seem to be doing the same thing- what might I be missing?
The main division happens to be hiding in plain sight: linearity. In fact, extend to question to continuity of the first derivative, and you'll encapsulate most of the difference.
First of all, take note of one basic principle of neural nets (NN): a NN with linear weights and linear dependencies is a GLM. Also, having multiple hidden layers is equivalent to a single hidden layer: it's still linear combinations from input to output.
A "modern' NN has non-linear layers: ReLUs (change negative values to 0), pooling (max, min, or mean of several values), dropouts (randomly remove some values), and other methods destroy our ability to smoothly apply Gradient Descent (GD) to the model. Instead, we take many of the principles and work backward, applying limited corrections layer by layer, all the way back to the weights at layer 1.
Lather, rinse, repeat until convergence.
Does that clear up the problem for you?
You got it!
A typical ReLU is
f(x) = x if x > 0,
0 otherwise
A typical pooling layer reduces the input length and width by a factor of 2; in each 2x2 square, only the maximum value is passed through. Dropout simply kills off random values to make the model retrain those weights from "primary sources". Each of these is a headache for GD, so we have to do it layer by layer.
So, I'm trying to understand if BackPropagation does anything more than gradient descent on the loss function.. and if not, why is it only referenced in the case of Neural Nets
I think (at least originally) back propagation of errors meant less than what you describe: the term "backpropagation of errors" only refered to the method of calculating derivatives of the loss function, instead of e.g. automatic differentiation, symbolic differentiation, or numerical differentiation. No matter what the gradient was then used for (e.g. Gradient Descent, or maybe Levenberg/Marquardt).
They all seem to be doing the same thing- what might I be missing?
They're using different models. If your neural network used linear neurons, it would be equivalent to linear regression.
I think I read somewhere that convolutional neural networks do not suffer from the vanishing gradient problem as much as standard sigmoid neural networks with increasing number of layers. But I have not been able to find a 'why'.
Does it truly not suffer from the problem or am I wrong and it depends on the activation function?
[I have been using Rectified Linear Units, so I have never tested the Sigmoid Units for Convolutional Neural Networks]
Convolutional neural networks (like standard sigmoid neural networks) do suffer from the vanishing gradient problem. The most recommended approaches to overcome the vanishing gradient problem are:
Layerwise pre-training
Choice of the activation function
You may see that the state-of-the-art deep neural network for computer vision problem (like the ImageNet winners) have used convolutional layers as the first few layers of the their network, but it is not the key for solving the vanishing gradient. The key is usually training the network greedily layer by layer. Using convolutional layers have several other important benefits of course. Especially in vision problems when the input size is large (the pixels of an image), using convolutional layers for the first layers are recommended because they have fewer parameters than fully-connected layers and you don't end up with billions of parameters for the first layer (which will make your network prone to overfitting).
However, it has been shown (like this paper) for several tasks that using Rectified linear units alleviates the problem of vanishing gradients (as oppose to conventional sigmoid functions).
Recent advances had alleviate the effects of vanishing gradients in deep neural networks. Among contributing advances include:
Usage of GPU for training deep neural networks
Usage of better activation functions. (At this point rectified linear units (ReLU) seems to work the best.)
With these advances, deep neural networks can be trained even without layerwise pretraining.
Source:
http://devblogs.nvidia.com/parallelforall/deep-learning-nutshell-history-training/
we do not use Sigmoid and Tanh as Activation functions which causes vanishing Gradient Problems. Mostly nowadays we use RELU based activation functions in training a Deep Neural Network Model to avoid such complications and improve the accuracy.
It’s because the gradient or slope of RELU activation if it’s over 0, is 1. Sigmoid derivative has a maximum slope of .25, which means that during the backward pass, you are multiplying gradients with values less than 1, and if you have more and more layers, you are multiplying it with values less than 1, making gradients smaller and smaller. RELU activation solves this by having a gradient slope of 1, so during backpropagation, there isn’t gradients passed back that are progressively getting smaller and smaller. but instead they are staying the same, which is how RELU solves the vanishing gradient problem.
One thing to note about RELU however is that if you have a value less than 0, that neuron is dead, and the gradient passed back is 0, meaning that during backpropagation, you will have 0 gradient being passed back if you had a value less than 0.
An alternative is Leaky RELU, which gives some gradient for values less than 0.
The first answer is from 2015 and a bit of age.
Today, CNNs typically also use batchnorm - while there is some debate why this helps: the inventors mention covariate shift: https://arxiv.org/abs/1502.03167
There are other theories like smoothing the loss landscape: https://arxiv.org/abs/1805.11604
Either way, it is a method that helps to deal significantly with vanishing/exploding gradient problem that is also relevant for CNNs. In CNNs you also apply the chain rule to get gradients. That is the update of the first layer is proportional to the product of N numbers, where N is the number of inputs. It is very likely that this number is either relatively big or small compared to the update of the last layer. This might be seen by looking at the variance of a product of random variables that quickly grows the more variables are being multiplied: https://stats.stackexchange.com/questions/52646/variance-of-product-of-multiple-random-variables
For recurrent networks that have long sequences of inputs, ie. of length L, the situation is often worse than for CNN, since there the product consists of L numbers. Often the sequence length L in a RNN is much larger than the number of layers N in a CNN.
I am attempting to train a 2 hidden layer tanh neural neural network on the MNIST data set using the ADADELTA algorithm.
Here are the parameters of my setup:
Tanh activation function
2 Hidden layers with 784 units (same as the number of input units)
I am using softmax with cross entropy loss on the output layer
I randomly initialized weights with a fanin of ~15, and gaussian distributed weights with standard deviation of 1/sqrt(15)
I am using a minibatch size of 10 with 50% dropout.
I am using the default parameters of ADADELTA (rho=0.95, epsilon=1e-6)
I have checked my derivatives vs automatic differentiation
If I run ADADELTA, at first it makes gains in the error, and it I can see that the first layer is learning to identify the shapes of digits. It does a decent job of classifying the digits. However, when I run ADADELTA for a long time (30,000 iterations), it's clear that something is going wrong. While the objective function stops improving after a few hundred iterations (and the internal ADADELTA variables stop changing), the first layer weights still have the same sparse noise they were initialized with (despite real features being learned on top of that noise).
To illustrate what I mean, here is the example output from the visualization of the network.
Notice the pixel noise in the weights of the first layer, despite them having structure. This is the same noise that they were initialized with.
None of the training examples have discontinuous values like this noise, but for some reason the ADADELTA algorithm never reduces these outlier weights to be in line with their neighbors.
What is going on?