I have implemented a neural network with 3 layers Input to Hidden Layer with 30 neurons(Relu Activation) to Softmax Output layer. I am using the cross entropy cost function. No outside libraries are being used. This is working on the NMIST dataset so 784 input neurons and 10 output neurons.
I have got about 96% accuracy with hyperbolic tangent as my hidden layer activation.
When I try to switch to relu activation my activations grow very fast which cause my weights grow unbounded as well until it blows up!
Is this a common problem to have when using relu activation?
I have tried L2 Regularization with minimal success. I end up having to set the learning rate lower by a factor of ten compared to the tanh activation and I have tried adjusting the weight decay rate accordingly and still the best accuracy I have gotten is about 90%. The rate of weight decay is still outpaced in the end by the updating of certain weights in the network which lead to an explosion.
It seems everyone is just replacing their activation functions with relu and they experience better results, so I keep looking for bugs and validating my implementation.
Is there more that goes into using relu as an activation function? Maybe I have problems in my implemenation, can someone validate accuracy with the same neural net structure?
as you can see the Relu function is unbounded on positive values, thus creating the weights to grow
in fact, that's why hyperbolic tangent and alike function are being used in those cases, to bound the output value between a certain range (-1 to 1 or 0 to 1 in most cases)
there is another approach to deal with this phenomenon called weights decay
the basic motivation is to get a more generalised model (avoid overfitting) and make sure the weights won't blow up you use a regulation value depending on the weight itself when update them
meaning that bigger weights get bigger penalty
you can farther read about it here
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!
In neural nets, regularization (e.g. L2, dropout) is commonly used to reduce overfitting. For example, the plot below shows typical loss vs epoch, with and without dropout. Solid lines = Train, dashed = Validation, blue = baseline (no dropout), orange = with dropout. Plot courtesy of Tensorflow tutorials.
Weight regularization behaves similarly.
Regularization delays the epoch at which validation loss starts to increase, but regularization apparently does not decrease the minimum value of validation loss (at least in my models and the tutorial from which the above plot is taken).
If we use early stopping to stop training when validation loss is minimum (to avoid overfitting) and if regularization is only delaying the minimum validation loss point (vs. decreasing the minimum validation loss value) then it seems that regularization does not result in a network with greater generalization but rather just slows down training.
How can regularization be used to reduce the minimum validation loss (to improve model generalization) as opposed to just delaying it? If regularization is only delaying minimum validation loss and not reducing it, then why use it?
Over-generalizing from a single tutorial plot is arguably not a good idea; here is a relevant plot from the original dropout paper:
Clearly, if the effect of dropout was to delay convergence it would not be of much use. But of course it does not work always (as your plot clearly suggests), hence it should not be used by default (which is arguably the lesson here)...
In ANN, we know that to make it "learn", we need to adjust the weights of the inputs to a particular neuron.
total_input=summation(w(j,i).a(j))
During adjustment, some weights are to be reduced while others to be increased.
Is the total weight of all j inputs to the i-th neuron should be 1?
There's absolutely no reason for the weights in the linear layer (a.k.a. dense or fully-connected layer) to sum up to anything specific, such as 1.0. They are usually initialized with small random numbers (so initial sum is unlikely to be 1.0) and then get tweaked somehow (not completely independently, but at least differently).
If the neural network doesn't use any regularization, it's often possible to train the network to large weight values, much larger than 1.0 (see also this question).
There are particular cases, when an analogous condition is true, for example softmax layer, which mathematically guarantees that the sum of outputs is 1.0. But the linear layer doesn't guarantee anything like that.
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?