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Yes, you can fix (or freeze) some of the weights during the training of a neural network. In fact, this is done in the most common form of transfer learning (which is described here). I don't know exactly how this affects learning in general. In transfer learning, this is definitely beneficial, as we are freezing the weights that are associated with the ...

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Yes, it is not unusual to omit the bias by adding a neuron which always outputs a constant 1, which will then be multiplied by an appropriate weight to give the same formula as you would get using an explicit bias. One notable text using this convention is Understanding Machine Learning: From Theory to Algorithms by Shai Shalev-Shwartz and Shai Ben-David. ...

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According to wikipedia of backpropagation: In fitting a neural network, backpropagation computes the gradient of the loss function during supervised learning with respect to the weights of the network for a single input–output example, and does so efficiently, unlike a naive direct computation of the gradient with respect to each weight individually. ...

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By convention, the $\mathrm{ReLU}$ activation is treated as if it is differentiable at zero (e.g. in [1]). Therefore it makes sense for TensorFlow to adopt this convention for tf.nn.relu. As you've found, of course, it's not true in general that we treat the gradient of the absolute value function as zero in the same situation; it makes sense for it to be an ...

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Second-order optimization algorithms like Hessian optimization have more information on the curvature of the loss function, so converge much, much faster than first-order optimization algorithms like gradient descent. I remember reading somewhere that if you have $n$ weights in the neural network, one iteration of a second-order optimization algorithm will ...

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When you use the softmax activation function is usually as a last layer of your network and to get an output that is a vector. Now your confusion is about shapes, so let's review a bit of calculus. If you have a function $$f:\mathbb{R}\rightarrow\mathbb{R}$$ the derivative is a function on its own and you have $$f':\mathbb{R}\rightarrow\mathbb{R}.$$ If you ...

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Imagine we have the curve $f(x) = x^2$, and we want to find the minimum of this function. The derivate of $f$ with respect to $x$ is $2x$. Now, gradient descent works by updating our current estimate of the minimum, say $c_t$, by the following iterative process $$c_{t+1} = c_t - \alpha \times \nabla_xf(x=c_t),$$ where $\alpha$ is some constant to control how ...

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If anything, you want the learning rate to decrease as the number of iterations increases. When you're looking for a good spot and you're clueless, take large steps. When you've found a pretty good spot, take small steps, so you don't end up far away. In other fields of machine learning, there are studies of how the learning rate should scale. For example, ...

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Yes, if the activation function of the network is not zero centered, $y = f(x^{T}w)$ is always positive or always negative. Thus, the output of a layer is always being moved to either the positive values or the negative values. As a result, the weight vector needs more update to be trained properly and the number of epochs needed for the network to get ...

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All modern frameworks for deep learning (PyTorch, Jax, Tensorflow) support automatic differentation. These operations can be easily implemented. Here I write, how it would look like in PyTorch: class Net(nn.Module): def __init__(self): super().__init__() self.a = nn.Parameter(torch.randn(1)) self.b = nn.Parameter(torch.randn(...

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Without the specific context, I cannot give a definitive answer, but it's very likely that a "differentiable architecture" refers to a neural network that represents/computes a differentiable function (so you need to use differentiable activation functions, such as the sigmoid), i.e. you can take the partial derivatives of the loss function with ...

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The basic idea of gradient descent is: Calculate the gradient of some score with respect to parameters that you can control Take a step in the direction of that gradient that improves the score (subract a multiple of gradient - for gradient descent - if you want to minimise some cost function) The backpropagation using chain rule in neural network layers ...

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Alright. Consider an ordinary neural network, so, in the last layer, we have, $z^{[L]} = W^{[L]} a^{[L-1]} + b^{[L]}$, where $a^{[L]} = \sigma(z^{[L]})$, where $\sigma$ is the softmax activation: $$\sigma(\mathbf z)_{i} = \frac{e^{z_i}}{\sum_k e^{z_k}}$$ I think, one of the most effective ways of not to get confused about all these matrices with different ...

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That is exactly a neural network works like. Suppose you have a 1000 examples. How you train a network is: First, you divide these 1000 into maybe 100 batches (10 each). After that's done, you feed a batch to the network get its output and compare it with the ground truth, whatever is the error gets backpropagated. Then, for the next batch and then another. ...

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Here is a paper that explains why ReLU rules. What we want is to disentangle data of different classes. In order to do that, we need a discontinuous mapping for the data. ReLU easily allows for that. It is even better than LeakyReLU, sigmoid and tanh in that regard. Also, the reason any of the activations work is because of the floating point error, there is ...

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Okay - the answer is here https://explained.ai/matrix-calculus/#sec6.2 and it is pretty involved. Basically, there is a difference when you derive the equation for one neuron and when you have to do practically for a set of neurons. The answer is matrix calculus. Here goes from what I could make out. Feel free to correct if I am wrong Gradient Vector/Matrix/...

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Deep Learning by Goodfellow et. al is a good book for anything related neural networks, and it's freely available online. Backpropagation is covered in Chapter 6.5.

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I will first address your main question "Why did the development of neural networks stop between 50s and 80s?" In 40-50s there was a lot of progress (McCulloch and Pitts); the perceptron was invented (Rosenblatt). That gave rise to an AI hype giving many promises (exactly like today)! However, Minsky and Papert have proved in 1969 that a single-...

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Creating custom gradient for tf.abs may solve the problem: @tf.custom_gradient def abs_with_grad(x): y = tf.abs(x); def grad(div): # Derivation intermediate value g = 1; # Use 1 to make the chain rule just skip abs return div*g; return y,grad;

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I assume you are considering a network where the activation function of the last layer is a sigmoid, so the output of your network is $$\tilde{y}=\sigma(W^{L}\cdot f(X, W^1, \dots, W^{L-1})),$$ where $X$ is the input vector, and $f$ is obtained by feeding the input to the network up to the layer $L-1$. Let's also call $Z:= W^{L}\cdot f(X, W^1, \dots, W^{L-1})... 1 As you say, the outputs are modeled as a vector, each output in one vector component. In regression problems: The most common loss function, like in the scalar case, is the square error. Skipping constants, it is defined as: $$E=\sum_i ||\mathbf{y_i}-\mathbf{\hat{y_i}}||^2 = \sum_i (\mathbf{y_i}-\mathbf{\hat{y_i}})(\mathbf{y_i}-\mathbf{\hat{y_i}})$$ where:$...

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Assuming you're using softmax on the last layer for classification, it sounds like a simple application of cross entropy loss from here on out: https://datascience.stackexchange.com/questions/20296/cross-entropy-loss-explanation Edit:

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I have not implement the backprop of a bi-directional RNN from scratch so I can't be sure my answer is correct but I hope it helps. You can see how bi-directional RNN works from this video from Andrew NG. I got the image below from that video: For more clarity: So if you know how to backprop through a simple RNN, you should be able to do so for bi-...

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