The Concept of Neural Network and Technique

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1. Neuron

A neuron is a generic computational unit that takes n inputs and produces a single output. What differentiates the outputs of different neurons is their parameters (also referred to as their weights). One of the most popular choices for neurons is the “sigmoid” or “binary logistic regression” unit. This unit takes an n-dimensional input vector x and produces the scalar activation (output) a. This neuron is also associated with an n-dimensional weight vector, w, and a bias scalar, b. The output of this neuron is then:

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2. A Single Layer of Neurons

If we refer to the different neurons’ weights as {$w_{(1)}, \cdots , w_{(m)}$} and the biases as {$b_1, \cdots , b_m$}, we can say the respective activations are {$a_1, \cdots ,a_m$}:

Let us define the following abstractions to keep the notation simple and useful for more complex networks:

We can now write the output of scaling and biases as:

The activations of the sigmoid function can then be written as:

3. Feed-forward Computation

So far we have seen how an input vector $x \in \mathbb{R}^n$ can be fed to a layer of sigmoid units to create activations $a \in \mathbb{R}^m$. But what is the intuition behind doing so? Let us consider the following named entity recognition (NER) problem in NLP as an example:

"Museums in Paris are amazing"

Here, we want to classify whether or not the center word “Paris” is a named-entity. In such cases, it is very likely that we would not just want to capture the presence of words in the window of word vectors but some other interactions between the words in order to make the classification. For instance, maybe it should matter that “Museums”is the first word only if “in” is the second word. Such non-linear decisions can often not be captured by inputs fed directly to a Softmax function but instead require the scoring of the intermediate layer. We can thus use another matrix $U \in R^{m×1}$ to generate an unnormalized score for a classification task from the activations:

$\checkmark$ Analysis of Dimensions: If we represent each word using a 4-dimensional word vector and we use a 5-word window as input (as in the above example), then the input $x \in \mathbb{R}^{20}.$ If we use 8 sigmoid units in the hidden layer and generate 1 score output from the activations, then $W\in \mathbb{R}^{8 \times 20}$, $b\in \mathbb{R}^{8}$, $U\in \mathbb{R}^{8 \times 1}$, $s\in \mathbb{R}$.

4. Maximum Margin Objective Function

Like most machine learning models, neural networks also need an optimization objective, a measure of error or goodness which we want to minimize or maximize respectively. Here, we will discuss a popular error metric known as the maximum margin objective. The idea behind using this objective is to ensure that the score computed for “true” labeled data points is higher than the score computed for “false” labeled data points.

Using the previous example, if we call the score computed for the “true” labeled window “Museums in Paris are amazing” as s and the score computed for the “false” labeled window ”Not all museums in Paris”* as $s_c$ (subscripted as c to signify that the window is “corrupt”).

Then, our objective function would be to maximize (s − $s_c$) or to minimize ($s_c$ − s). However, we modify our objective to ensure that error is only computed if $s_c$ > s $\rightarrow$ ($s_c$ − s) > 0. The intuition behind doing this is that we only care the the “true” data point have a higher score than the “false” data point and that the rest does not matter. Thus, we want our error to be ($s_c$ − s) if $s_c$ > s else 0. Thus, our optimization objective is now:

However, the above optimization objective is risky in the sense that it does not attempt to create a margin of safety. We would want the “true” labeled data point to score higher than the “false” labeled data point by some positive margin ∆. In other words, we would want error to be calculated if (s−s_c < ∆) and not just when (s−s_c < 0). Thus, we modify the optimization objective:

We can scale this margin such that it is ∆ = 1 and let the other parameters in the optimization problem adapt to this without any change in performance. For more information on this, read about functional and geometric margins - a topic often covered in the study of Support Vector Machines. Finally, we define the following optimization objective which we optimize over all training windows:

In the above formulation

5. Training with Backpropation - Elemental

In this section we discuss how we train the different parameters in the model when the cost J(minimize J = max($s_c$ − s, 0) discussed in Section #4 is positive. No parameter updates are necessary if the cost is 0. Since we typically update parameters using gradient descent (or a variant such as SGD, Stochastic Gradient Descent), we typically need the gradient information for any parameter as required in the update equation:

Backpropagation is technique that allows us to use the chain rule of differentiation to calculate loss gradients for any parameter used in the feed-forward computation on the model. To understand this further, let us understand the toy network shown in Figure 5 for which we will perform backpropagation.

Here, we use a neural network with a single hidden layer and a single unit output. Let us establish some notation that will make it easier to generalize this model later:

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• $x_i$ is an input to the neural network.
• s is the output of the neural network.
• Each layer (including the input and output layers) has neurons which receive an input and produce an output. The j-th neuron of layer k receives the scalar input $z_j^{(k)}$ and produces the scalar activation output $a_j^{(k)}$
• We will call the backpropagated error calculated at $z_j^{(k)}$ as $\sigma_j^{(k)}$.
• Layer 1 refers to the input layer and not the first hidden layer. For the input layer, $x_j = z_j^{(1)} = a_j^{(1)}$
• $W^{(k)}$ is the transfer matrix that maps the output from the $k$ th layer to the input to the $(k+1)$ th Thus, $W^{(1)} =W$and $W^{(2)} =U$ to put this new generalized notation in perspective of Section #3

$\checkmark$ Let us begin to Calculate!: 

Suppose the cost $J = (1 + s_c − s)$ is positive and we want to perform the update of parameter $W_{14}^{(1)}$ (Figure 6 as below), we must realize that $W_{14}^{(1)}$ only contributes to $z_1^{(2)}$ and thus $a_1^{(2)}$. This fact is crucial to understanding backpropagation-backpropagated gradients are only affected by values they contribute to. $a_1^{(2)}$ is consequently used in the forward computation of score by multiplication with $W_1^{(2)}$We can see from the max-margin loss that:

Therefore we will work with $\dfrac{\partial s}{\partial W_{ij}^{(1)}}$ here for simplicity. Thus,

We see above that the gradient reduces to the product $\delta_i^{(2)} · a_j^{(1)}$ where $\delta_i^{(2)}$ is essentially the error propagating backwards from the i-th neuron in layer 2. $a_j^{(1)}$ is an input fed to i-th neuron in layer 2 when scaled by $W_{ij}$. Let us discuss the “error sharing / distribution” interpretation of backpropagation better using Figure 6 as an example. Say we were to update $W_{14}^{(1)}$:

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$\checkmark$ Sample Sequence

1. We start with the an error signal of 1 propagating backwards from $a_1^{(3)}$.
2. We then multiply this error by the local gradient of the neuron which maps $z_1^{(3)}$ to $a_1^{(3)}$. This happens to be 1 in this case and thus, the error is still 1. This is now known as $\sigma_1^{(3)}$ = 1
3. At this point, the error signal of 1 has reached $z_1^{(3)}$. We now need to distribute the error signal so that the “fair share” of the error reaches to $a_1^{(2)}$. 

4. This amount is the $W_1^{(2)}$= (error signal at $z_1^{(3)}$) $\sigma_1^{(3)}\times W_1^{(2)}$.

   Thus, the error at $a_1^{(2)}$ = $W_1^{(2)}$.

5. As we did in step 2, we need to move the error across the neuron which maps $z_1^{(2)}$ to $a_1^{(2)}$. We do this by multiplying the error signal at $a_1^{(2)}$ by the local gradient of the neuron which happens to be f′($z_1^{(2)}$).
6. Thus, the error signal at $z_1^{(2)}$ is f′($z_1^{(2)}$)$W_1^{(2)}$. This is known as $\sigma_1^{(2)}$.
7. Finally, we need to distribute the “fair share” of the error to $W_{14}^{(1)}$ by simply multiplying it by the input it was responsible for forwarding, which happens to be $a_4^{(1)}$.
8. Thus, the gradient of the loss with respect to W(1) is calculated to be $a_4^{(1)}f’(z_1^{(2)})W_1^{(2)}$

Notice that the result we arrive at using this approach is exactly the same as that we arrived at using explicit differentiation earlier. Thus, we can calculate error gradients with respect to a parameter in the network using either the chain rule of differentiation or using an error sharing and distributed flow approach – both of these ap- proaches happen to do the exact same thing but it might be helpful to think about them one way or another.

$\checkmark$ Bias Updates: Bias terms (such as $b_1^{(1)}$) are mathematically equivalent to other weights contributing to the neuron input ($z_1{(2)}$) as long as the input being forwarded is 1. As such, the bias gradients for neuron i on layer k is simply $\sigma_1^{(k)}$. For instance, if we were updating $b_1^{(1)}$ instead of $W_{14}^{(1)}$ above, the gradient would simply be $f’(z_1^{(2)})W_1^{(2)}$.

$\checkmark$ Normalized Sequence

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1. We have error $\sigma_i^{(k)}$ propagating backwards from $z_i^{(k)}$, i.e. neuron at layer k. See Figure 7.
2. We propagate this error backwards to $a_j^{(k−1)}$ by multiplying $\sigma_i^{(k)}$ by the path weight $W_{ij}^{(k−1)}$
3. Thus, the error received at $a_j^{(k−1)}$ is $\sigma_i^{(k)}W_{ij}^{(k−1)}$.
4. However, $a_j^{(k−1)}$ may have been forwarded to multiple nodes in the next layer as shown in Figure 8. It should receive responsibility for errors propagating backward from node m in layer k too, using the exact same mechanism.

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1. Thus, error received at $a_j^{(k−1)}$ is $\sigma_i^{(k)}W_{ij}^{(k−1)} + \sigma_m^{(k)}W_{mj}^{(k−1)}$.
2. In fact, we can generalize this to be $\textstyle\sum_{i}\sigma_i^{(k)}W_{ij}^{(k−1)}$.
3. Now that we have the correct error at $a_j^{(k−1)}$, we move it across neuron j at layer k − 1 by multiplying with with the local gradient $f’(z_j^{(k−1)})$.
4. Thus, the error that reaches $z_j^{(k−1)}$, called $\sigma_j^{(k−1)}$ is $f’(z_j^{(k−1)})\textstyle\sum_i \sigma_i^{(k)}W_{ij}^{(k−1)}$

6. Training with Backpropation - Vectorized

$\checkmark$ Generalize the approach above so that we update weight matrices and bias vectors all at once. Note that these are simply extensions of the above model that will help build intuition for the way error propagation can be done at a matrix-vector level. For a given parameter $W_{ij}^{(k)}$, we identified that the error gradient is simply $\sigma_i^{(k+1)} · a_j^{(k)}$. As a reminder, $W^{(k)}$ is the matrix that maps $a^{(k)}$ to $z^{(k+1)}$. We can thus establish that the error gradient for the entire matrix $W^{(k)}$ is:

Thus, we can write an entire matrix gradient using the outer product of the error vector propagating into the matrix and the activations forwarded by the matrix.

The error vector

The $\circ$ operator corresponds to an element wise product between elements of vectors ($\circ: \mathbb{R}^N \times \mathbb{R}^N \rightarrow \mathbb{R}^N)$

$\checkmark$ Computational efficiency: Having explored element-wise updates as well as vector-wise updates, we must realize that the vectorized implementations run substantially faster in scientific computing environments such as MATLAB or Python (using NumPy/SciPy packages). Thus, we should use vectorized implementation in practice. Furthermore, we should also reduce redundant calculations in backpropagation - for instance, notice that $\sigma^{(k)}$ depends directly on $\sigma^{(k+1)}$. Thus, we should ensure that when we update W(k) using $\sigma^{(k+1)}$, we save $\sigma^{(k+1)}$ to later derive $\sigma^{(k)}$ – and we then repeat this for (k − 1) . . . (1). Such a recursive procedure is what makes backpropagation a computationally affordable procedure.

7. Neural Networks: Tips and Tricks

7.1 Gredient Check

In the last section, we discussed in detail how to calculate error gradients/updates for parameters in a neural network model via calculus-based (analytic) methods. Here we now introduce a technique of numerically approximating these gradients though too computationally inefficient to be used directly for training the networks, this method will allow us to very precisely estimate the derivative with respect to any parameter; it can thus serve as a useful sanity check on the correctness of our analytic derivatives. Given a model with parameter vector $\theta$ and loss function $J$, the numerical gradient around $\theta$i is simply given by centered difference formula:

where $\epsilon$ is a small number (usually around 1e−5). The term $J($\theta$^{(i+)})$ is simply the error calculated on a forward pass for a given input when we perturb the parameter $\theta$’s i-th element by +$\epsilon$. Similarly, the term $J(\theta^{(i−)})$ is the error calculated on a forward pass for the same input when we perturb the parameter $\theta$’s i-th element by −$\epsilon$. Thus, using two forward passes, we can approximate the gradient with respect to any given parameter element in the model. We note that this definition of the numerical gradient follows very naturally from the definition of the derivative, where, in the scalar case,

Of course, there is a slight difference – the definition above only perturbs $x$ in the positive direction to compute the gradient. While it would have been perfectly acceptable to define the numerical gradient in this way, in practice it is often more precise and stable to use the centered difference formula, where we perturb a parameter in both directions. The intuition is that to get a better approximation of the derivative/slope around a point, we need to examine the function f’s behavior both to the left and right of that point. It can also be shown using Taylor’s theorem that the centered difference formula has an error proportional to $\epsilon^2$, which is quite small, whereas the derivative definition is more error-prone.**

Now, a natural question you might ask is, if this method is so precise, why do we not use it to compute all of our network gradients instead of applying back-propagation? The simple answer, as hinted earlier, is inefficiency – recall that every time we want to compute the gradient with respect to an element, we need to make two forward passes through the network, which will be computationally expensive. Furthermore, many large-scale neural networks can contain millions of parameters, and computing two passes per parameter is clearly not optimal. And, since in optimization techniques such as SGD, we must compute the gradients once per iteration for several thousands of iterations, it is obvious that this method quickly grows intractable. This inefficiency is why we only use gradient check to verify the correctness of our analytic gradients, which are much quicker to compute.

A standard implementation of gradient check is shown below:

# Snippet 2.1
def eval_numerical_gradient(f, x):
    """
    a naive implementation of numerical gradient of f at x
    - f should be a function that takes a single argument
    - x is the point (numpy array) to evaluate the gradient
    at
    """
    fx = f(x) # evaluate function value at original point
    grad = np.zeros(x.shape)
    h = 0.00001
    # iterate over all indexes in x
    it = np.nditer(x, flags=[’multi_index’],
                    op_flags=[’readwrite’])
    while not it.finished:
    # evaluate function at x+h
    ix = it.multi_index
    old_value = x[ix]
    x[ix] = old_value + h # increment by h
    fxh_left = f(x) # evaluate f(x + h)
    x[ix] = old_value - h # decrement by h
    fxh_right = f(x) # evaluate f(x - h)
    x[ix] = old_value # restore to previous value (very important!)
    # compute the partial derivative
    grad[ix] = (fxh_left - fxh_right) / (2*h) # the slope
    it.iternext() # step to next dimension
return grad

7.2 Regularization

As with many machine learning models, neural networks are highly prone to overfitting, where a model is able to obtain near perfect performance on the training dataset, but loses the ability to generalize to unseen data. A common technique used to address overfitting (an issue also known as the “high-variance problem”) is the incorporation of an $L_2$ regularization penalty. The idea is that we will simply append an extra term to our loss function $J$, so that the overall cost is now calculated as:

In the above formulation, $\lvert \lvert W^{(i)}\lvert \lvert _F$ is the Frobenius norm of the matrix $W^{(i)}$ (the i-th weight matrix in the network) and $\lambda$ is the hyper-parameter controlling how much weight the regularization term has relative to the original cost function.

💡 The Frobenius Norm of a matrix U is defined as follows**

Since we are trying to minimize $J_R$, what regularization is essentially doing is penalizing weights for being too large while optimizing over the original cost function. Due to the quadratic nature of the Frobenius norm (which computes the sum of the squared elements of a matrix), L2 regularization effectively reduces the flexibility of the model and thereby reduces the overfitting phenomenon. Imposing such a con- straint can also be interpreted as the prior Bayesian belief that the optimal weights are close to zero – how close depends on the value of $\lambda$. Choosing the right value of $\lambda$ is critical, and must be chosen via hyperparameter-tuning. Too high a value of $\lambda$ causes most of the weights to be set too close to 0, and the model does not learn anything meaningful from the training data, often obtaining poor ac- curacy on training, validation, and testing sets. Too low a value, and we fall into the domain of overfitting once again. It must be noted that the bias terms are not regularized and do not contribute to the cost term above.

$\checkmark$ Try thinking about why this is the case!

There are indeed other types of regularization that are sometimes used, such as L1 regularization, which sums over the absolute values (rather than squares) of parameter elements – however, this is less commonly applied in practice since it leads to sparsity of parameter weights. In the next section, we discuss dropout, which effectively acts as another form of regularization by randomly dropping (i.e. setting to zero) neurons in the forward pass.

$\checkmark$ L1 norm

\(\left\lvert\left\lvert x\right\lvert\right\lvert = \left\lvert x \right\lvert\) is a norm on the one-dimensional vector spaces formed by the real or complex numbers

$\checkmark$ L2 norm

$\left\lvert\left\lvert E(a) \right\lvert\right\lvert^2_2 $ means “$L^2$norm”, so it defines $\left\lvert\left\lvert x \right\lvert\right\lvert_2 := \sqrt{x^H x}$

• A full loss function includes regularization over all parameters 𝜃, e.g., L2 regularization

  $J(\theta) = \dfrac{1}{N}\displaystyle\sum_{i=1}^N - log(\dfrac{e^{f_{y_i}}}{\sum_{c=1}^C e^{f_c}})+\lambda\displaystyle\sum_k \theta_k^2$

• Classic view: Regularization works to prevent overfitting when we have a lot of features (or later a very powerful/deep model, etc.)
• Now: Regularization produces models that generalize well when we have a “big” model. We do not care that our models overfit on the training data ,even though they are hugely overfit

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7.3 Dropout

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Dropout is a powerful technique for regularization, first introduced by Srivastava et al. in Dropout: A Simple Way to Prevent Neural Networks from Overfitting. The idea is simple yet effective – during train- ing, we will randomly “drop” with some probability (1 − p) a subset of neurons during each forward/backward pass (or equivalently,

We will keep alive each neuron with a probability p). Then, during testing, we will use the full network to compute our predictions. The result is that the network typically learns more meaningful informa- tion from the data, is less likely to overfit, and usually obtains higher performance overall on the task at hand. One intuitive reason why this technique should be so effective is that what dropout is doing is essentially doing is training exponentially many smaller networks at once and averaging over their predictions.

In practice, the way we introduce dropout is that we take the output h of each layer of neurons, and keep each neuron with probability p, and else set it to 0. Then, during back-propagation, we only pass gradients through neurons that were kept alive duringthe forward pass. Finally, during testing, we compute the forward pass using all of the neurons in the network. However, a key sub- tlety is that in order for dropout to work effectively, the expected output of a neuron during testing should be approximately the same as it was during training – else the magnitude of the outputs could be radically different, and the behavior of the network is no longer well-defined. Thus, we must typically divide the outputs of each neuron during testing by a certain value – it is left as an exercise to the reader to determine what this value should be in order for the expected outputs during training and testing to be equivalent.

$\checkmark$ Srivastava, Hinton, Krizhevsky, Sutskever, & Salakhutdinov 2012/JMLR 2014

Preventing Feature Co-adaptation = Good Regularization Method!

• Training time: at each instance of evaluation (in online SGD-training), randomly set 50% of the inputs to each neuron to 0
• Test time: halve the model weights (now twice as many)
• (Except usually only drop first layer inputs a little (~15%) or not at all)
• This prevents feature co-adaptation: A feature cannot only be useful in the presence of particular other features
• In a single layer: A kind of middle-ground between Naïve Bayes (where all feature weights are set independently) and logistic regression models (where weights are set in the context of all others)
• Can be thought of as a form of model bagging (i.e., like an ensemble model)
• Nowadays usually thought of as strong, feature-dependent regularizer[Wager, Wang, & Liang 2013]

$\checkmark$ Dropout Method

Reference: https://towardsdatascience.com/12-main-dropout-methods-mathematical-and-visual-explanation-58cdc2112293

7.4 Neuron Unit

$\checkmark$ Sigmoid

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the activation function $\sigma$ is given by:

The gradient of $\sigma$(z) is:

$\checkmark$ Tanh

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The tanh function is an alternative to the sigmoid function that is often found to converge faster in practice. The primary difference between tanh and sigmoid is that tanh output ranges from −1 to 1 while the sigmoid ranges from 0 to 1.

$\checkmark$ Hard tanh

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The hard tanh function is sometimes preferred over the tanh function since it is computationally cheaper. It does however saturate for magnitudes of z greater than 1. The activation of the hard tanh is:

The derivative can also be expressed in a piecewise functional form:

$\checkmark$ Soft sign

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The soft sign function is another nonlinearity which can be considered an alternative to tanh since it too does not saturate as easily as hard clipped functions:

The derivative is the expressed as:

where sgn is the signum function which returns $\pm$1 depending on the sign of z.

$\checkmark$ ReLU

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The ReLU (Rectified Linear Unit) function is a popular choice of activation since it does not saturate even for larger values of z and has found much success in computer vision applications:

The derivative is then the piecewise function:

$\checkmark$ Leaky ReLU

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Traditional ReLU units by design do not propagate any error for non-positive z – the leaky ReLU modifies this such that a small error is allowed to propagate backwards even when z is negative:

This way, the derivative is representable as:

$\checkmark$ Exponential Linear Unit (ELU)

Reference. Stanford CS224n, 2021

• Reference. Clevert, Djork-Arné, Thomas Unterthiner, and Sepp Hochreiter. “Fast and accurate deep network learning by exponential linear units (elus).” arXiv preprint arXiv:1511.07289 (2015)

7.5 Data Processing

$\checkmark$ Mean Subtraction

Given a set of input data X, it is customary to zero-center the data by subtracting the mean feature vector of X from X. An important point is that in practice, the mean is calculated only across the training set, and this mean is subtracted from the training, validation, and testing sets.

$\checkmark$ Normalization

Another frequently used technique (though perhaps less so than mean subtraction) is to scale every input feature dimension to have similar ranges of magnitudes. This is useful since input features are often measured in different “units”, but we often want to initially consider all features as equally important. The way we accomplish this is by simply dividing the features by their respective standard deviation calculated across the training set.

$\checkmark$ Whitening

Not as commonly used as mean-subtraction + normalization, whitening essentially converts the data to have an identity covariance matrix – that is, features become uncorrelated and have a variance of 1. This is done by first mean-subtracting the data, as usual, to get X′. We can then take the Singular Value Decomposition (SVD) of X′ to get matrices U, S, V. We then compute UX′ to project X′ into the basis defined by the columns of U. We finally divide each dimension of the result by the corresponding singular value in S to scale our data appropriately (if a singular value is zero, we can divide by a small number instead). This makes inputs have small correlation and same variance. Make $X’ = U^TX$, then X’ components have uncorrelation, because SVD include the meaning of linear transformation and uncorrelated vectors($AV = US,\ A=USV^T$, $A$ is linear transformation, $V$ and $U$ includes uncorrelated vectors, and $S$ includes singular value).

Reference

• What is PCA(Principle Components Analysis)?
• What is SVD?
• How to get SVD?
• ZCA Whitening

7.6 Parameter Initialization

A key step towards achieving superlative performance with a neu- ral network is initializing the parameters in a reasonable way. A good starting strategy is to initialize the weights to small random numbers normally distributed around 0

# Xavier Initialization
# Gaussian with zero mean and 1e-2 standard deviation
W = np.random.randn(fan_in, fan_out) / np.sqrt(fan_in)

and in practice, this often words acceptably well. However, in Understanding the difficulty of training deep feedforward neural networks (2010), Xavier et al study the effect of different weight and bias initialization schemes on training dynamics. The empirical findings suggest that for sigmoid and tanh activation units, faster convergence and lower error rates are achieved when the weights of a matrix $W \in \mathbb{R}^{n(l+1) \times n(l)}$ are initialized randomly with a uniform distribution as follows:

Where $n^{(l)}$ is the number of input units to W (fan-in) and $n^{(l+1)}$is the number of output units from W (fan-out).

In this parameter initialization scheme, bias units are initialized to 0. This approach attempts to maintain activation variances as well as backpropagated gradient variances across layers. Without such initialization, the gradient variances (which are a proxy for information) generally decrease with backpropagation across layers.

• You normally must initialize weights to small random values (i.e., not zero matrices!) to avoid symmetries that prevent learning/specialization.
• Initialize hidden layer biases to 0 and output (or reconstruction) biases to optimal value if weights were 0 (e.g., mean target or inverse sigmoid of mean target)
• Initialize all other weights ~ Uniform(–r, r), with r chosen so numbers get neither too big or too small [later the need for this is removed with use of layer normalization]
• Xavier initialization has variance inversely proportional to fan-in nin (previous layer size) and fan-out nout (next layer size):

• CS231N: neural-networks, init
• Practical example

7.7 Learning Stategies

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The rate/magnitude of model parameter updates during training can be controlled using the learning rate. In the following naive Gradient Descent formulation, \alpha is the learning rate:

You might think that for fast convergence rates, we should set $\alpha$ to larger values – however faster convergence is not guaranteed with larger convergence rates. In fact, with very large learning rates, we might experience that the loss function actually diverges because the parameters update causes the model to overshoot the convex minima as shown in Figure 15. In non-convex models (most of those we work with), the outcome of a large learning rate is unpredictable, but the chances of diverging loss functions are very high.

The simple solution to avoiding a diverging loss is to use a very small learning rate so that we carefully scan the parameter space – of course, if we use too small a learning rate, we might not converge in a reasonable amount of time, or might get caught in local minima. Thus, as with any other hyperparameter, the learning rate must be tuned effectively.

Since training is the most expensive phase in a deep learning system, some research has attempted to improve this naive approach to setting learning learning rates. For instance, Ronan Collobert scales the learning rate of a weight $W_{ij}$ (where $W \in R^{n(l+1) \times n(l)}$) by the inverse square root of the fan-in of the neuron ($n^{(l)}$).

There are several other techniques that have proven to be effective as well – one such method is annealing, where, after several iterations, the learning rate is reduced in some way – this method ensures that we start off with a high learning rate and approach a minimum quickly; as we get closer to the minimum, we start lowering our learning rate so that we can find the optimum under a more fine-grained scope. A common way to perform annealing is to reduce the learning rate $\alpha$ by a factor $x$ after every $n$ iterations of learning.

Exponential decay is also common, where, the learning rate $\alpha$ at iteration $t$ is given by $α(t) = α_0e^{−kt}$, where $\alpha_0$ is the initial learning rate, and $k$ is a hyper-parameter. Another approach is to allow the learning rate to decrease over time such that:

In the above scheme, $\alpha_0$ is a tunable parameter and represents the starting learning rate. $\tau$ is also a tunable parameter and represents the time at which the learning rate should start reducing. In practice, this method has been found to work quite well. In the next section, we discuss another method for adaptive gradient descent that does not require hand-set learning rates.

$\checkmark$ Strategy

• You can just use a constant learning rate. Start around lr = 0.001?

• It must be order of magnitude right – try powers of 10
  • Too big: model may diverge or not converge
  • Too small: your model may not have trained by the assignment deadline
• Better results can generally be obtained by allowing learning rates to decrease as you train
  • By hand: halve the learning rate every k epochs
  • An epoch = a pass through the data (shuffled or sampled–not in same order each time)
  • By a formula: $lr = lr_o e^{-kt}$, for epoch t
  • There are fancier methods like cyclic learning rates (q.v.)
• Fancier optimizers still use a learning rate but it may be an initial rate that the optimizer shrinks – so you may want to start with a higher learning rate → You can meet Adam in Assignment 3

7.8 Momentum Updates

Momentum methods, a variant of gradient descent inspired by the study of dynamics and motion in physics, attempt to use the “velocity” of updates as a more effective update scheme. Pseudocode for momentum updates is shown below:

# Reference. Stanford CS224n, 2021
# Snippet 2.2
# Computes a standard momentum update # on parameters x
v = mu*v - alpha*grad_x
x += v

7.9 Adaptive Optimization

$\checkmark$ AdaGrad

AdaGrad is an implementation of standard stochastic gradient descent (SGD) with one key difference: the learning rate can vary for each parameter. The learning rate for each parameter depends on the history of gradient updates of that parameter in a way such that parameters with a scarce history of updates are updated faster using a larger learning rate. In other words, parameters that have not been updated much in the past are likelier to have higher learning rates now. Formally:

In this technique, we see that if the RMS of the history of gradients is extremely low, the learning rate is very high. A simple implementation of this technique is:

# Reference. Stanford CS224n, 2021
# Snippet 2.3
# Assume the gradient dx and parameter vector x 
cache += dx**2
x += - learning_rate * dx / np.sqrt(cache + 1e-8)

$\checkmark$ RMSProp and Adam

Other common adaptive methods are RMSProp and Adam, whose update rules are shown below (courtesy of Andrej Karpathy):

# Reference. Stanford CS224n, 2021
#Snippet 2.4
# Update rule for RMS prop
cache = decay_rate * cache + (1 - decay_rate) * dx**2 
x += - learning_rate * dx / (np.sqrt(cache) + eps)

#Snippet 2.5
# Update rule for Adam
m = beta1*m + (1-beta1)*dx
v = beta2*v + (1-beta2)*(dx**2)
x += - learning_rate * m / (np.sqrt(v) + eps)

RMSProp is a variant of AdaGrad that utilizes a moving average of squared gradients – in particular, unlike AdaGrad, its updates do not become monotonically smaller. The Adam update rule is in turn a variant of RMSProp, but with the addition of momentum-like updates. We refer the reader to the respective sources of these methods for more detailed analyses of their behavior. It is simply similar to velocity and accelerates such as considering derivate and derivate’s derivate.

• Other method: AdamW, AdamWR, SGDW

$\checkmark$ Strategy

• Usually, plain SGD will work just fine! However, getting good results will often require hand-tuning the learning rate
• For more complex nets and situations, or just to avoid worry, you often do better with one of a family of more sophisticated “adaptive” optimizers that scale the parameter adjustment by an accumulated gradient.

• Example(These models give differential per-parameter learning rates)

    Adagrad, RMSprop, Adam(A fairly good, safe place to begin in many cases), SparseAdam
  

8. Vertorization

E.g., looping over word vectors versus concatenating them all into one large matrix and then multiplying the softmax weights with that matrix:

from numpy import random
N = 500 # number of windows to classify
d = 300 # dimensionality of each window
C = 5 # number of classes
W = random.rand(C, d)
wordvectors_list = [random.rand(d,1) for i in range(N)]
wordvectors_one_matrix = random.rand(d,N)

%timeit [W.dot(wordvecotrs_list[i]) for i in range(N)]
%timeit W.dot(wordvectros_one_matrix)

• 1000 loops, best of 3: 639 μs per loop
• 10000 loops, best of 3: 53.8 μs per loop
• Always try to use vectors and matrices rather than for loops
• The speed gain goes from 1 to 2 orders of magnitude with GPUs

9. Non-linearities, old and new

• Logistic(“sigmoid”), tanh, hard tans, ReLU(Rectified Linear Unit)
• tanh is just a rescaled and shifted sigmoid (2 × as steep, [−1,1]):

• Both logistic and tanh are still used in various places (e.g., to get a probability), but are no longer the defaults for making deep networks
• For building a deep network, the first thing you should try is ReLU — it trains quickly and performs well due to good gradient backflow

Reference. Stanford CS224n, 2021