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Tutorial 2: Normalization

Week 1, Day 5: Microcircuits

By Neuromatch Academy

Content creators: Alish Dipani, Xaq Pitkow

Content reviewers: Yizhou Chen, RyeongKyung Yoon, Ruiyi Zhang, Lily Chamakura, Hlib Solodzhuk, Patrick Mineault, Alex Murphy

Production editors: Konstantine Tsafatinos, Ella Batty, Spiros Chavlis, Samuele Bolotta, Hlib Solodzhuk, Alex Murphy


Tutorial Objectives

Estimated timing of tutorial: 50 minutes

Recap: the definition of normalization is the process of bringing a set of values, each with potentially differing ranges and offsets, back into a common range such that there is not much of an extreme discrepancy between features. In modeling house prices using linear regression, features on the order of num_bathrooms (e.g. 1,2,3) are overshadowed by features such as square_ft (e.g. from 1,000 to 3,000). After applying normalization, these features are comparable and can be used effectively in such a linear regression.

In this tutorial, you will learn about the microcircuit element of normalization, which is a prominent computation in brains and machines. You will see different types of normalization, how to implement them, and observe some of its benefits for generalization.

Tutorial Learning Objectives

  • Understand how nonlinearities may be universal function approximators, but not all functions are simple to learn

  • Implement a family of normalization mechanisms

  • Demonstrate how normalization helps in learning and information transmission


Setup

Install and import feedback gadget

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Imports

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Figure settings

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Helper functions

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Set random seed

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Set device (GPU or CPU)

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GPU is not enabled in this notebook. 
If you want to enable it, in the menu under `Runtime` -> 
`Hardware accelerator.` and select `GPU` from the dropdown menu

Section 1: Can ReLUs implement normalization?

In this section we will explore how feasible it is to estimate a normalization-like function.

Video 1: Introduction

Youtube
Bilibili

The general form of normalization is:

x^=xf(∣∣x∣∣)\hat{x} = \frac{x}{f(||x||)}

There are indeed many options for the specific form of the denominator here; still, what we want to highlight is the essential nature of the normalization, that it is achieved by dividing the values of the vector by some function ff of its magnitude.

Evidence suggests that normalization provides a useful inductive bias in artificial and natural systems. However, do we need a dedicated computation (a microcircuit) that implements normalization? Can we not just let the network take care of it during the learning of the weights?

Let’s explore if a fully-connected network with ReLU nonlinearities can estimate a normalization-like function. Specifically, we will see if a fully-connected one-layer network can estimate the function y=1x+ϵy=\frac{1}{x+\epsilon}.

In the cell below, we visualize train and test data.

Generate y=1x+ϵy=\frac{1}{x+\epsilon} train and test dataloaders

ϵ=0.01\epsilon = 0.01

Source
<Figure size 800x600 with 1 Axes>

Coding Exercise 1: ReLUNet

Let’s define a simple one-layer neural network using the following equation, with our single scalar input xx:

y^=∑iwiReLU(θi−x)\hat{y} = \sum_{i}w_{i} \text{ReLU}(\theta_{i} - x)

Here θi\theta_{i} is the threshold, and wiw_{i} is the slope of neuron ii. θi\theta_{i} & wiw_{i} are learned parameters. Our network has a total of 100 neurons (the summation and aggregation over index ii reflects each of these 100 neurons). Your task is to complete the forward pass of the model.

---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
Cell In[8], line 3
      1 ###################################################################
      2 ## Fill out the following then remove
----> 3 raise NotImplementedError("Student exercise: complete forward pass.")
      4 ###################################################################
      6 class ReLUNet(nn.Module):

NotImplementedError: Student exercise: complete forward pass.

Now, let’s train the model and evaluate it.

Training & Evaluating model

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<Figure size 1000x500 with 2 Axes>

While the model learns, we see that it does not fit the test data well. Let’s find out where the model struggles during training.

Here, we plot the log mean-squared errors for values of xx between 0 and 1 and their progression with epochs. These are log errors (clipped at e5e^{5} represented with blue color).

Plot Training Loss Dynamics

Source
<Figure size 1000x500 with 2 Axes>

We can see that the model has higher errors for lower values of xx, and as the training progresses, the errors for lower xx values start to decrease. Note that the losses are huge for very small values of xx (>e5> e^5).

Does it mean the model employs more resources to learn the function between 0 and 1? To check this out, let’s visualize the ReLU thresholds. Concretely, let’s look at the theta values representing the 100 neurons in our one-layer neural network. This will allow us to see where the model is placing most of the emphasis in order to learn the target function.

If you’re unsure about the terminology being used (threshold and ReLU value) then it’s useful to remember that a neural network is modelled by weights and biases (slopes and intercepts). Each layer of a neural network has an associated slope/weight which is combined with the input value and is then added to a bias (intercept) which then goes through the nonlinearity (ReLU here). The red lines in the figure below represent the bias/intercept values, while the teal lines represent the slopes/weights.

Vizualize ReLUs

Source
<Figure size 800x600 with 1 Axes>

We can see that the thresholds (red lines) are bunched up between 0 and 1, which means that the model dedicates the most resources to learning the function on this interval. Let’s quantify the learning by plotting the threshold distributions and dynamics with epochs.

Here we plot the cumulative distributions of θi\theta_{i} & wiw_{i}. We also plot the values of the parameters as they change across epochs.

Plot Weight Dynamics

Source
<Figure size 1250x1000 with 4 Axes>

From the cumulative distribution plot of the thresholds (θi\theta_{i}), we can see that around 8080% of them are below x=1x=1. Hence, the model majorly struggles to learn the function between 0 and 1, where the slope changes a lot.

Since the slope changes infinite times between x=0x=0 and 1, and ReLUs implement a linear function with a single slope, we would ideally need an infinite number of ReLUs units to fit the y=1x+ϵy=\frac{1}{x+\epsilon} function. Hence, even though theoretically we can estimate the function, it is not empirically feasible to do so. This suggests that a microcircuit able to implement this computation would be a very useful thing to have.

Coding Exercise 1 Discussion

  1. Do you think that having more slope changes in the activation function would help?

Take a minute to think on your own, then discuss in a group.

Coding Exercise 2: Test other non-linear activation functions

Let’s see if other non-linear activation functions perform better. Specifically, we test:

  1. ReLU(x)=(x)+=max⁡(0,x)\text{ReLU}(x) = (x)^{+} = \max(0,x)

  2. ReLU6(x)=min⁡(max⁡(0,x),6)\text{ReLU6}(x) = \min(\max(0,x),6)

  3. SoftPlus(x,β=1)=1βlog⁡(1+eβx)\text{SoftPlus}(x, \beta=1) = \frac{1}{\beta} \log(1+e^{βx})

  4. Sigmoid(x)=σ(x)=11+e−x\text{Sigmoid}(x) = \sigma(x)= \frac{1}{1+e^{-x}}

  5. Tanh(x)=ex−e−xex+e−x\text{Tanh}(x) = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}

We still consider the case of the one-layer neural network, except we change the activation function to be one of the above:

y^=∑iwiActivation(θi−x)\hat{y} = \sum_{i}w_{i} \text{Activation}(\theta_{i} - x)

Here θi\theta_{i} is the threshold, and wiw_{i} is the slope of neuron ii. We train and evaluate each model three times and plot the mean performance across runs in order to make sure our results are not highly dependent on any specific initialization setting of the network. Your task is to complete the dictionary of the proposed non-linear functions (by defining them using torch.nn library).

---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
Cell In[13], line 3
      1 ###################################################################
      2 ## Fill out the following then remove
----> 3 raise NotImplementedError("Student exercise: complete non-linearities.")
      4 ###################################################################
      5 non_linearities = {
      6     'ReLU': nn.ReLU(),
      7     'ReLU6': ...,
   (...)
     10     'Tanh': ...
     11 }

NotImplementedError: Student exercise: complete non-linearities.

Now, let’s train different networks and evaluate them. Notice that the cell below will run for 1 minute approximately.

Train & Evaluate

Source
<Figure size 1000x500 with 2 Axes>

We can see that all of the proposed non-linear activation functions do not perform very well. Hence, it is beneficial to have dedicated computation that implements normalization.

Video 2: Summary

Youtube
Bilibili

Section 2: Benefits of normalization

Estimated timing to here from start of tutorial: 20 minutes.

In this section, we will propose a simple normalization function, which you are going to test in different environments and observe how it is connected to the concept of generalization, the overarching theme of this entire course.

Video 3: Introduction to Normalization

Youtube
Bilibili

Subsection 2.1: Explore normalization

This subsection is devoted to the definition of simple normalization function and the exploration of hyperparameters’ impact on its result.

Coding Exercise 2.1: Implement normalization

Let’s implement the normalization we saw in the video above. Specifically, we will use the following function:

xnorm=gxσ+Σi=1N∣xi∣pp\mathbf{x}_{norm} = \frac{g \mathbf{x}}{\sigma + \sqrt[p]{\Sigma_{i = 1}^{N} |x_{i}|^{p}}}

Where

  • xx is an NN-dimensional vector (x∈RNx \in \mathbb{R}^N),

  • gg is a scaling factor,

  • σ\sigma is a smoothing (translation) factor,

  • pp defines pp-norm of the input vector (e.g. if p=2p=2 this is an L2L2 norm)

Note that we are normalizing across dimensions (similar to Layer Normalization).

You can test your implementation by running the code cell below!

Test normalize() function

Source
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
Cell In[16], line 19
     16   assert np.array_equal(np_norm(test_x, 2.4, 3.2, 1.5), normalize(test_x, 2.4, 3.2, 1.5)), incorrect_message
     17   print('Normalize function works correctly!')
---> 19 check_normalize(normalize)

Cell In[16], line 10, in check_normalize(func)
      8 test_x = np.random.rand(3, 3)
      9 # Test 1
---> 10 assert np.array_equal(np_norm(test_x, 1, 1, 1), normalize(test_x, 1, 1, 1)), incorrect_message
     11 # Test 2
     12 assert np.array_equal(np_norm(test_x, 2, 0.3, 1.2), normalize(test_x, 2, 0.3, 1.2)), incorrect_message

Cell In[15], line 15, in normalize(x, sigma, p, g)
      2 """
      3 Inputs:
      4 - x(np.ndarray): Input array (n_samples * n_dim)
   (...)
     10 - xnorm (np.ndarray): normalized values.
     11 """
     12 #################################################
     13 ## TODO: Implement the normalization example equation ##
     14 # Fill remove the following line of code once you have completed the exercise:
---> 15 raise NotImplementedError("Student exercise: complete normalization function.")
     16 #################################################
     17 # Raise the absolute value of x to the power p
     18 xp = ...

NotImplementedError: Student exercise: complete normalization function.

Interactive Demo 2.1

Let’s explore the effect of smoothing factor (σ\sigma), p-norm (pp) and scaling factor (gg) in our normalization function:

xnorm=gxσ+Σi=1N∣xi∣pp\mathbf{x}_{norm} = \frac{g \mathbf{x}}{\sigma + \sqrt[p]{\Sigma_{i = 1}^{N} |x_{i}|^{p}}}

We will see the effect of normalization being induced on the points sampled from a 2-dimensional normal distribution.

Take a minute to play around with the values and then discuss them in the group.

Effect of smoothing factor (σ\sigma), p-norm (pp) and scaling factor (gg)

Source
<Figure size 500x500 with 1 Axes>
Loading...

Video 4: Effect of smoothing factor, p-norm and scaling factor

Youtube
Bilibili

Subsection 2.2: Estimating latent properties

In this subsection, we will use the normalization function to retrieve the target variable being corrupted with scaling.

Video 5: Normalization example

Youtube
Bilibili

Interactive Demo 2.2.1

For this demo, we have a target image (x\mathbf{x}), which we would like to infer, and a visible image (v\mathbf{v}), which is a scaled version of the target one: x=sv\mathbf{x} = s \mathbf{v}. We will generate 300 different examples (we will visualize only 25 of them) of 5-dimensional vectors x\mathbf{x} (each of the components of the vectors is generated from an exponential distribution with λ=1\lambda = 1). Then, the scaling factor ss is generated from an exponential distribution with λ=1\lambda = 1 as well.

Play around with different hyperparameter values to get the best R-squared value.

---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
Cell In[18], line 14
      9 visible_image = np.repeat(illuminant_intensity, number_pixels, axis=1) * reflectance
     11 #################################################
     12 ## TODO: Implement the normalization example equation ##
     13 # Fill out and remove the following line of code one you have completed the exercise:
---> 14 raise NotImplementedError("Student exercise: choose your parameters values.")
     15 #################################################
     16 
     17 # Normalized visible image
     18 norm_visible_image = normalize(
     19     visible_image,
     20     sigma = ...,
     21     p = ...,
     22     g = ...
     23 )

NotImplementedError: Student exercise: choose your parameters values.

Click for solution

Example output:

Solution hint

Plot correlation of estimated reflectance with true reflectance

Source
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[19], line 17
     15 # Plot xh vs. x
     16 x_ = reflectance.reshape((-1, 1)).squeeze(-1)
---> 17 y_ = norm_visible_image.reshape((-1, 1)).squeeze(-1)
     18 sns.regplot(x=x_, y=y_, ax=ax[1])
     19 ax[1].set_xlabel('Reflectance')

NameError: name 'norm_visible_image' is not defined
<Figure size 1500x500 with 2 Axes>

Interactive Demo 2.2.2

Now, explore the impact of change in the hyperparemeter values on the R-squared value by using interactive sliders below.

Effect of smoothing factor (σ\sigma), p-norm (pp) and scaling factor (gg)

Source
<Figure size 800x600 with 1 Axes>
Loading...

Video 6: Summary

Youtube
Bilibili

Subsection 2.3: Layer Normalization

In this subsection, we are going to the explore normalization operation on a real application from the CIFAR10 dataset. This data subset has undergone scaling that we will aim to correct for.

Video 7: Image recognition example

Youtube
Bilibili

First, we fetch the dataset and derive only the part corresponding to the following three classes: plane, car, and cat.

Get CIFAR3 scaled and unscaled dataloader

Source
Downloading https://cave.cs.toronto.edu/kriz/cifar-10-python.tar.gz to ./data/cifar-10-python.tar.gz
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Extracting ./data/cifar-10-python.tar.gz to ./data
Files already downloaded and verified

Let us take a look at the original (latent) images and their scaled versions.

Visualize Images

Source
Files already downloaded and verified
Clipping input data to the valid range for imshow with RGB data ([0..1] for floats or [0..255] for integers).
Labels = car  , plane, car  , car  , cat  , cat  , car  , car  
<Figure size 800x600 with 1 Axes>
Scaling factors = 0.14, 0.171, 0.224, 0.148, 0.221, 0.179, 0.099, 0.155
<Figure size 800x600 with 1 Axes>

Here, we define the CNN model with an optional parameter for adding the LayerNorm layer.

Define CIFARNet model

Source

It will take around 3 minutes to complete training on different types of models.

Training & Evaluating the models

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<Figure size 1000x500 with 2 Axes>
<Figure size 1000x500 with 4 Axes>

Let us also check whether normalization improves generalization with different scaling parameters.

Test Generalization

Source
<Figure size 500x500 with 1 Axes>

By adding a normalization layer: (i) the training process converges quicker, (ii) we achieve better test accuracy and (iii) we achieve better out-of-distribution generalization accuracy in the image recognition tasks.

Video 8: Section summary

Youtube
Bilibili

Summary

Estimated timing of tutorial: 50 minutes

In this tutorial, we observed that normalization as an inductive bias is useful. We have implemented the normalization function and explored the examples. Finally, we discovered the benefits of using normalization.

Video 9: Tutorial summary

Youtube
Bilibili

The Big Picture

In this tutorial, we observed that normalization as an inductive bias is useful. We first showed that models struggle to learn the normalization function and it makes sense computationally to implement this operation as a dedicated microcircuit, our topic of the day! We further showed the effect that normalization has on efficient model training and how it enables out-of-distribution accuracy by stabilizing the training process and recovering images that had been corrupted by scaling.

We ended by looking at how normalization processes occur in brains and we prompt you to think whether any of the computational advantages in neural networks might perform similar functions in the brain. Are there things we can learn and apply to studying brains? Perhaps further work in normalization in neuroscience might end up introducing variations of brain-inspired normalization into neural networks, which could help the model learn even better!