Tutorial 1: Optimization techniques

Tutorial 1: Optimization techniques#

Week 1, Day 5: Optimization

By Neuromatch Academy

Content creators: Jose Gallego-Posada, Ioannis Mitliagkas

Content reviewers: Piyush Chauhan, Vladimir Haltakov, Siwei Bai, Kelson Shilling-Scrivo

Content editors: Charles J Edelson, Gagana B, Spiros Chavlis

Production editors: Arush Tagade, R. Krishnakumaran, Gagana B, Spiros Chavlis

Tutorial Objectives#

Objectives:

Necessity and importance of optimization
Introduction to commonly used optimization techniques
Optimization in non-convex loss landscapes
‘Adaptive’ hyperparameter tuning
Ethical concerns

Setup#

Install and import feedback gadget#

# Imports
import copy

import ipywidgets as widgets
import matplotlib.pyplot as plt
import numpy as np

import time
import torch
import torchvision
import torchvision.datasets as datasets
import torch.nn.functional as F
import torch.nn as nn
import torch.optim as optim
from tqdm.auto import tqdm

Figure settings#

Helper functions#

Set random seed#

Executing set_seed(seed=seed) you are setting the seed

Set device (GPU or CPU). Execute `set_device()`#

SEED = 2021
set_seed(seed=SEED)
DEVICE = set_device()

Random seed 2021 has been set.
WARNING: For this notebook to perform best, if possible, in the menu under `Runtime` -> `Change runtime type.`  select `GPU` 

Section 1. Introduction#

Time estimate: ~15 mins

Video 1: Introduction#

Submit your feedback#

Discuss: Unexpected consequences#

Can you think of examples from your own experience/life where poorly chosen incentives or objectives have led to unexpected consequences?

Click for solution

Submit your feedback#

Section 2: Case study: successfully training an MLP for image classification#

Time estimate: ~40 mins

Many of the core ideas (and tricks) in modern optimization for deep learning can be illustrated in the simple setting of training an MLP to solve an image classification task. In this tutorial we will guide you through the key challenges that arise when optimizing high-dimensional, non-convex\(^\dagger\) problems. We will use these challenges to motivate and explain some commonly used solutions.

Disclaimer: Some of the functions you will code in this tutorial are already implemented in Pytorch and many other libraries. For pedagogical reasons, we decided to bring these simple coding tasks into the spotlight and place a relatively higher emphasis in your understanding of the algorithms, rather than the use of a specific library.

In ‘day-to-day’ research projects you will likely rely on the community-vetted, optimized libraries rather than the ‘manual implementations’ you will write today. In Section 8 you will have a chance to ‘put it all together’ and use the full power of Pytorch to tune the parameters of an MLP to classify handwritten digits.

\(^\dagger\): A strictly convex function has the same global and local minimum - a nice property for optimization as it won’t get stuck in a local minimum that isn’t a global one (e.g., \(f(x)=x^2 + 2x + 1\)). A non-convex function is wavy - has some ‘valleys’ (local minima) that aren’t as deep as the overall deepest ‘valley’ (global minimum). Thus, the optimization algorithms can get stuck in the local minimum, and it can be hard to tell when this happens (e.g., \(f(x) = x^4 + x^3 - 2x^2 - 2x\)). See also Section 5 for more details.

Video 2: Case Study - MLP Classification#

Submit your feedback#

Section 2.1: Data#

We will use the MNIST dataset of handwritten digits. We load the data via the Pytorch datasets module, as you learned in W1D1.

Note: Although we can download the MNIST dataset directly from datasets using the optional argument download=True, we are going to download them from NMA directory on OSF to ensure network reliability.

Download MNIST dataset#

Downloading MNIST dataset...

Downloading MNIST completed.

def load_mnist_data(change_tensors=False, download=False):
  """
  Load training and test examples for the MNIST handwritten digits dataset
  with every image: 28*28 x 1 channel (greyscale image)

  Args:
    change_tensors: Bool
      Argument to check if tensors need to be normalised
    download: Bool
      Argument to check if dataset needs to be downloaded/already exists

  Returns:
    train_set:
      train_data: Tensor
        training input tensor of size (train_size x 784)
      train_target: Tensor
        training 0-9 integer label tensor of size (train_size)
    test_set:
      test_data: Tensor
        test input tensor of size (test_size x 784)
      test_target: Tensor
        training 0-9 integer label tensor of size (test_size)
  """
  # Load train and test sets
  train_set = datasets.MNIST(root='.', train=True, download=download,
                             transform=torchvision.transforms.ToTensor())
  test_set = datasets.MNIST(root='.', train=False, download=download,
                            transform=torchvision.transforms.ToTensor())

  # Original data is in range [0, 255]. We normalize the data wrt its mean and std_dev.
  # Note that we only used *training set* information to compute mean and std
  mean = train_set.data.float().mean()
  std = train_set.data.float().std()

  if change_tensors:
    # Apply normalization directly to the tensors containing the dataset
    train_set.data = (train_set.data.float() - mean) / std
    test_set.data = (test_set.data.float() - mean) / std
  else:
    tform = torchvision.transforms.Compose([torchvision.transforms.ToTensor(),
                                            torchvision.transforms.Normalize(mean=[mean / 255.], std=[std / 255.])
                                            ])
    train_set = datasets.MNIST(root='.', train=True, download=download,
                               transform=tform)
    test_set = datasets.MNIST(root='.', train=False, download=download,
                              transform=tform)

  return train_set, test_set


train_set, test_set = load_mnist_data(change_tensors=True)

As we are just getting started, we will concentrate on a small subset of only 500 examples out of the 60.000 data points contained in the whole training set.

# Sample a random subset of 500 indices
subset_index = np.random.choice(len(train_set.data), 500)

# We will use these symbols to represent the training data and labels, to stay
# as close to the mathematical expressions as possible.
X, y = train_set.data[subset_index, :], train_set.targets[subset_index]

Run the following cell to visualize the content of three examples in our training set. Note how the preprocessing we applied to the data changes the range of pixel values after normalization.

Run me!#

../../../_images/6a17da77de16610a758d6b30e0402616a1204ddba72f258848556290002ef5ea.png

Section 2.2: Model#

As you will see next week, there are specific model architectures that are better suited to image-like data, such as Convolutional Neural Networks (CNNs). For simplicity, in this tutorial we will focus exclusively on Multi-Layer Perceptron (MLP) models as they allow us to highlight many important optimization challenges shared with more advanced neural network designs.

class MLP(nn.Module):
  """
  This class implements MLPs in Pytorch of an arbitrary number of hidden
  layers of potentially different sizes. Since we concentrate on classification
  tasks in this tutorial, we have a log_softmax layer at prediction time.
  """

  def __init__(self, in_dim=784, out_dim=10, hidden_dims=[], use_bias=True):
    """
    Constructs a MultiLayerPerceptron

    Args:
      in_dim: Integer
        dimensionality of input data (784)
      out_dim: Integer
        number of classes (10)
      hidden_dims: List
        containing the dimensions of the hidden layers,
        empty list corresponds to a linear model (in_dim, out_dim)

    Returns:
      Nothing
    """

    super(MLP, self).__init__()

    self.in_dim = in_dim
    self.out_dim = out_dim

    # If we have no hidden layer, just initialize a linear model (e.g. in logistic regression)
    if len(hidden_dims) == 0:
      layers = [nn.Linear(in_dim, out_dim, bias=use_bias)]
    else:
      # 'Actual' MLP with dimensions in_dim - num_hidden_layers*[hidden_dim] - out_dim
      layers = [nn.Linear(in_dim, hidden_dims[0], bias=use_bias), nn.ReLU()]

      # Loop until before the last layer
      for i, hidden_dim in enumerate(hidden_dims[:-1]):
        layers += [nn.Linear(hidden_dim, hidden_dims[i + 1], bias=use_bias),
                   nn.ReLU()]

      # Add final layer to the number of classes
      layers += [nn.Linear(hidden_dims[-1], out_dim, bias=use_bias)]

    self.main = nn.Sequential(*layers)

  def forward(self, x):
    """
    Defines the network structure and flow from input to output

    Args:
      x: Tensor
        Image to be processed by the network

    Returns:
      output: Tensor
        same dimension and shape as the input with probabilistic values in the range [0, 1]

    """
    # Flatten each images into a 'vector'
    transformed_x = x.view(-1, self.in_dim)
    hidden_output = self.main(transformed_x)
    output = F.log_softmax(hidden_output, dim=1)
    return output

Linear models constitute a very special kind of MLPs: they are equivalent to an MLP with zero hidden layers. This is simply an affine transformation, in other words a ‘linear’ map \(W x\) with an ‘offset’ \(b\); followed by a softmax function.

\[f(x) = \text{softmax}(W x + b)\]

Here \(x \in \mathbb{R}^{784}\), \(W \in \mathbb{R}^{10 \times 784}\) and \(b \in \mathbb{R}^{10}\). Notice that the dimensions of the weight matrix are \(10 \times 784\) as the input tensors are flattened images, i.e., \(28 \times 28 = 784\)-dimensional tensors and the output layer consists of \(10\) nodes. Also, note that the implementation of softmax encapsulates b in W i.e., It maps the rows of the input instead of the columns. That is, the i’th row of the output is the mapping of the i’th row of the input under W, plus the bias term. Refer Affine maps here: https://pytorch.org/tutorials/beginner/nlp/deep_learning_tutorial.html#affine-maps

# Empty hidden_dims means we take a model with zero hidden layers.
model = MLP(in_dim=784, out_dim=10, hidden_dims=[])

# We print the model structure with 784 inputs and 10 outputs
print(model)

MLP(
  (main): Sequential(
    (0): Linear(in_features=784, out_features=10, bias=True)
  )
)

Section 2.3: Loss#

While we care about the accuracy of the model, the ‘discrete’ nature of the 0-1 loss makes it challenging to optimize. In order to learn good parameters for this model, we will use the cross entropy loss (negative log-likelihood), which you saw in the last lecture, as a surrogate objective to be minimized.

This particular choice of model and optimization objective leads to a convex optimization problem with respect to the parameters \(W\) and \(b\).

loss_fn = F.nll_loss

Section 2.4: Interpretability#

In the last lecture, you saw that inspecting the weights of a model can provide insights on what ‘concepts’ the model has learned. Here we show the weights of a partially trained model. The weights corresponding to each class ‘learn’ to fire when an input of the class is detected.

# Show class filters of a trained model
W = partial_trained_model.main[0].weight.data.numpy()

fig, axs = plt.subplots(1, 10, figsize=(15, 4))
for class_id in range(10):
  axs[class_id].imshow(W[class_id, :].reshape(28, 28), cmap='gray_r')
  axs[class_id].axis('off')
  axs[class_id].set_title('Class ' + str(class_id) )

plt.show()

../../../_images/53945e5d9ecd7944a885d94fbea0b76e11b08e1dd985fcc0c0776ac95bc88bb1.png

Section 3: High dimensional search#

Time estimate: ~25 mins

We now have a model with its corresponding trainable parameters as well as an objective to optimize. Where do we goto next? How do we find a ‘good’ configuration of parameters?

One idea is to choose a random direction and move only if the objective is reduced. However, this is inefficient in high dimensions and you will see how gradient descent (with a suitable step-size) can guarantee consistent improvement in terms of the objective function.

Video 3: Optimization of an Objective Function#

Submit your feedback#

Coding Exercise 3: Implement gradient descent#

In this exercise you will use PyTorch automatic differentiation capabilities to compute the gradient of the loss with respect to the parameters of the model. You will then use these gradients to implement the update performed by the gradient descent method.

def zero_grad(params):
  <

Tutorial 1: Optimization techniques

Contents

Tutorial 1: Optimization techniques#

Tutorial Objectives#

Setup#

Install and import feedback gadget#

Figure settings#

Helper functions#

Set random seed#

Set device (GPU or CPU). Execute set_device()#

Section 1. Introduction#

Video 1: Introduction#

Submit your feedback#

Discuss: Unexpected consequences#

Submit your feedback#

Section 2: Case study: successfully training an MLP for image classification#

Video 2: Case Study - MLP Classification#

Submit your feedback#

Section 2.1: Data#

Download MNIST dataset#

Run me!#

Section 2.2: Model#

Section 2.3: Loss#

Section 2.4: Interpretability#

Section 3: High dimensional search#

Video 3: Optimization of an Objective Function#

Submit your feedback#

Coding Exercise 3: Implement gradient descent#

Set device (GPU or CPU). Execute `set_device()`#