Learning Rate Scheduling

Run Jupyter Notebook

You can run the code for this section in this jupyter notebook link.

Optimization Algorithm: Mini-batch Stochastic Gradient Descent (SGD)

• We will be using mini-batch gradient descent in all our examples here when scheduling our learning rate
• Combination of batch gradient descent & stochastic gradient descent
  • $\theta = \theta - \eta \cdot \nabla J(\theta, x^{i: i+n}, y^{i:i+n})$
• Characteristics
  • Compute the gradient of the lost function w.r.t. parameters for n sets of training sample (n input and n label), $\nabla J(\theta, x^{i: i+n}, y^{i:i+n})$
  • Use this to update our parameters at every iteration
• Typically in deep learning, some variation of mini-batch gradient is used where the batch size is a hyperparameter to be determined

Learning Intuition Recap

• Learning process
  • Original parameters $\rightarrow$ given input, get output $\rightarrow$ compare with labels $\rightarrow$ get loss with comparison of input/output $\rightarrow$ get gradients of loss w.r.t parameters $\rightarrow$ update parameters so model can churn output closer to labels $\rightarrow$ repeat
• For a detailed mathematical account of how this works and how to implement from scratch in Python and PyTorch, you can read our forward- and back-propagation and gradient descent post.

Learning Rate Pointers

• Update parameters so model can churn output closer to labels, lower loss
  • $\theta = \theta - \eta \cdot \nabla J(\theta, x^{i: i+n}, y^{i:i+n})$
• If we set $\eta$ to be a large value $\rightarrow$ learn too much (rapid learning)
  • Unable to converge to a good local minima (unable to effectively gradually decrease your loss, overshoot the local lowest value)
• If we set $\eta$ to be a small value $\rightarrow$ learn too little (slow learning)
  • May take too long or unable to convert to a good local minima

Need for Learning Rate Schedules

• Benefits
  • Converge faster
  • Higher accuracy

Top Basic Learning Rate Schedules

1. Step-wise Decay
2. Reduce on Loss Plateau Decay

Step-wise Learning Rate Decay

Step-wise Decay: Every Epoch

• At every epoch,
  • $\eta_t = \eta_{t-1}\gamma$
  • $\gamma = 0.1$
• Optimization Algorithm 4: SGD Nesterov
  • Modification of SGD Momentum
    • $v_t = \gamma v_{t-1} + \eta \cdot \nabla J(\theta - \gamma v_{t-1}, x^{i: i+n}, y^{i:i+n})$
    • $\theta = \theta - v_t$
• Practical example
  • Given $\eta_t = 0.1$ and $ \gamma = 0.01$
  • Epoch 0: $\eta_t = 0.1$
  • Epoch 1: $\eta_{t+1} = 0.1 (0.1) = 0.01$
  • Epoch 2: $\eta_{t+2} = 0.1 (0.1)^2 = 0.001$
  • Epoch n: $\eta_{t+n} = 0.1 (0.1)^n$

Code for step-wise learning rate decay at every epoch

import torch
import torch.nn as nn
import torchvision.transforms as transforms
import torchvision.datasets as dsets

# Set seed
torch.manual_seed(0)

# Where to add a new import
from torch.optim.lr_scheduler import StepLR

'''
STEP 1: LOADING DATASET
'''

train_dataset = dsets.MNIST(root='./data', 
                            train=True, 
                            transform=transforms.ToTensor(),
                            download=True)

test_dataset = dsets.MNIST(root='./data', 
                           train=False, 
                           transform=transforms.ToTensor())

'''
STEP 2: MAKING DATASET ITERABLE
'''

batch_size = 100
n_iters = 3000
num_epochs = n_iters / (len(train_dataset) / batch_size)
num_epochs = int(num_epochs)

train_loader = torch.utils.data.DataLoader(dataset=train_dataset, 
                                           batch_size=batch_size, 
                                           shuffle=True)

test_loader = torch.utils.data.DataLoader(dataset=test_dataset, 
                                          batch_size=batch_size, 
                                          shuffle=False)

'''
STEP 3: CREATE MODEL CLASS
'''
class FeedforwardNeuralNetModel(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(FeedforwardNeuralNetModel, self).__init__()
        # Linear function
        self.fc1 = nn.Linear(input_dim, hidden_dim) 
        # Non-linearity
        self.relu = nn.ReLU()
        # Linear function (readout)
        self.fc2 = nn.Linear(hidden_dim, output_dim)  

    def forward(self, x):
        # Linear function
        out = self.fc1(x)
        # Non-linearity
        out = self.relu(out)
        # Linear function (readout)
        out = self.fc2(out)
        return out
'''
STEP 4: INSTANTIATE MODEL CLASS
'''
input_dim = 28*28
hidden_dim = 100
output_dim = 10

model = FeedforwardNeuralNetModel(input_dim, hidden_dim, output_dim)

'''
STEP 5: INSTANTIATE LOSS CLASS
'''
criterion = nn.CrossEntropyLoss()


'''
STEP 6: INSTANTIATE OPTIMIZER CLASS
'''
learning_rate = 0.1

optimizer = torch.optim.SGD(model.parameters(), lr=learning_rate, momentum=0.9, nesterov=True)

'''
STEP 7: INSTANTIATE STEP LEARNING SCHEDULER CLASS
'''
# step_size: at how many multiples of epoch you decay
# step_size = 1, after every 1 epoch, new_lr = lr*gamma 
# step_size = 2, after every 2 epoch, new_lr = lr*gamma 

# gamma = decaying factor
scheduler = StepLR(optimizer, step_size=1, gamma=0.1)

'''
STEP 7: TRAIN THE MODEL
'''
iter = 0
for epoch in range(num_epochs):
    # Decay Learning Rate
    scheduler.step()
    # Print Learning Rate
    print('Epoch:', epoch,'LR:', scheduler.get_lr())
    for i, (images, labels) in enumerate(train_loader):
        # Load images
        images = images.view(-1, 28*28).requires_grad_()

        # Clear gradients w.r.t. parameters
        optimizer.zero_grad()

        # Forward pass to get output/logits
        outputs = model(images)

        # Calculate Loss: softmax --> cross entropy loss
        loss = criterion(outputs, labels)

        # Getting gradients w.r.t. parameters
        loss.backward()

        # Updating parameters
        optimizer.step()

        iter += 1

        if iter % 500 == 0:
            # Calculate Accuracy         
            correct = 0
            total = 0
            # Iterate through test dataset
            for images, labels in test_loader:
                # Load images to a Torch Variable
                images = images.view(-1, 28*28)

                # Forward pass only to get logits/output
                outputs = model(images)

                # Get predictions from the maximum value
                _, predicted = torch.max(outputs.data, 1)

                # Total number of labels
                total += labels.size(0)

                # Total correct predictions
                correct += (predicted == labels).sum()

            accuracy = 100 * correct / total

            # Print Loss
            print('Iteration: {}. Loss: {}. Accuracy: {}'.format(iter, loss.item(), accuracy))

Epoch: 0 LR: [0.1]
Iteration: 500. Loss: 0.15292978286743164. Accuracy: 96
Epoch: 1 LR: [0.010000000000000002]
Iteration: 1000. Loss: 0.1207798570394516. Accuracy: 97
Epoch: 2 LR: [0.0010000000000000002]
Iteration: 1500. Loss: 0.12287932634353638. Accuracy: 97
Epoch: 3 LR: [0.00010000000000000003]
Iteration: 2000. Loss: 0.05614742264151573. Accuracy: 97
Epoch: 4 LR: [1.0000000000000003e-05]
Iteration: 2500. Loss: 0.06775809079408646. Accuracy: 97
Iteration: 3000. Loss: 0.03737065941095352. Accuracy: 97

Step-wise Decay: Every 2 Epochs

• At every 2 epoch,
  • $\eta_t = \eta_{t-1}\gamma$
  • $\gamma = 0.1$
• Optimization Algorithm 4: SGD Nesterov
  • Modification of SGD Momentum
    • $v_t = \gamma v_{t-1} + \eta \cdot \nabla J(\theta - \gamma v_{t-1}, x^{i: i+n}, y^{i:i+n})$
    • $\theta = \theta - v_t$
• Practical example
  • Given $\eta_t = 0.1$ and $\gamma = 0.01$
  • Epoch 0: $\eta_t = 0.1$
  • Epoch 1: $\eta_{t+1} = 0.1$
  • Epoch 2: $\eta_{t+2} = 0.1 (0.1) = 0.01$

Code for step-wise learning rate decay at every 2 epoch

import torch
import torch.nn as nn
import torchvision.transforms as transforms
import torchvision.datasets as dsets

# Set seed
torch.manual_seed(0)

# Where to add a new import
from torch.optim.lr_scheduler import StepLR

'''
STEP 1: LOADING DATASET
'''

train_dataset = dsets.MNIST(root='./data', 
                            train=True, 
                            transform=transforms.ToTensor(),
                            download=True)

test_dataset = dsets.MNIST(root='./data', 
                           train=False, 
                           transform=transforms.ToTensor())

'''
STEP 2: MAKING DATASET ITERABLE
'''

batch_size = 100
n_iters = 3000
num_epochs = n_iters / (len(train_dataset) / batch_size)
num_epochs = int(num_epochs)

train_loader = torch.utils.data.DataLoader(dataset=train_dataset, 
                                           batch_size=batch_size, 
                                           shuffle=True)

test_loader = torch.utils.data.DataLoader(dataset=test_dataset, 
                                          batch_size=batch_size, 
                                          shuffle=False)

'''
STEP 3: CREATE MODEL CLASS
'''
class FeedforwardNeuralNetModel(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(FeedforwardNeuralNetModel, self).__init__()
        # Linear function
        self.fc1 = nn.Linear(input_dim, hidden_dim) 
        # Non-linearity
        self.relu = nn.ReLU()
        # Linear function (readout)
        self.fc2 = nn.Linear(hidden_dim, output_dim)  

    def forward(self, x):
        # Linear function
        out = self.fc1(x)
        # Non-linearity
        out = self.relu(out)
        # Linear function (readout)
        out = self.fc2(out)
        return out
'''
STEP 4: INSTANTIATE MODEL CLASS
'''
input_dim = 28*28
hidden_dim = 100
output_dim = 10

model = FeedforwardNeuralNetModel(input_dim, hidden_dim, output_dim)

'''
STEP 5: INSTANTIATE LOSS CLASS
'''
criterion = nn.CrossEntropyLoss()

'''
STEP 6: INSTANTIATE OPTIMIZER CLASS
'''
learning_rate = 0.1

optimizer = torch.optim.SGD(model.parameters(), lr=learning_rate, momentum=0.9, nesterov=True)

'''
STEP 7: INSTANTIATE STEP LEARNING SCHEDULER CLASS
'''
# step_size: at how many multiples of epoch you decay
# step_size = 1, after every 2 epoch, new_lr = lr*gamma 
# step_size = 2, after every 2 epoch, new_lr = lr*gamma 

# gamma = decaying factor
scheduler = StepLR(optimizer, step_size=2, gamma=0.1)

'''
STEP 7: TRAIN THE MODEL
'''
iter = 0
for epoch in range(num_epochs):
    # Decay Learning Rate
    scheduler.step()
    # Print Learning Rate
    print('Epoch:', epoch,'LR:', scheduler.get_lr())
    for i, (images, labels) in enumerate(train_loader):
        # Load images as Variable
        images = images.view(-1, 28*28).requires_grad_()

        # Clear gradients w.r.t. parameters
        optimizer.zero_grad()

        # Forward pass to get output/logits
        outputs = model(images)

        # Calculate Loss: softmax --> cross entropy loss
        loss = criterion(outputs, labels)

        # Getting gradients w.r.t. parameters
        loss.backward()

        # Updating parameters
        optimizer.step()

        iter += 1

        if iter % 500 == 0:
            # Calculate Accuracy         
            correct = 0
            total = 0
            # Iterate through test dataset
            for images, labels in test_loader:
                # Load images to a Torch Variable
                images = images.view(-1, 28*28).requires_grad_()

                # Forward pass only to get logits/output
                outputs = model(images)

                # Get predictions from the maximum value
                _, predicted = torch.max(outputs.data, 1)

                # Total number of labels
                total += labels.size(0)

                # Total correct predictions
                correct += (predicted == labels).sum()

            accuracy = 100 * correct / total

            # Print Loss
            print('Iteration: {}. Loss: {}. Accuracy: {}'.format(iter, loss.item(), accuracy))

Epoch: 0 LR: [0.1]
Iteration: 500. Loss: 0.15292978286743164. Accuracy: 96
Epoch: 1 LR: [0.1]
Iteration: 1000. Loss: 0.11253029108047485. Accuracy: 96
Epoch: 2 LR: [0.010000000000000002]
Iteration: 1500. Loss: 0.14498558640480042. Accuracy: 97
Epoch: 3 LR: [0.010000000000000002]
Iteration: 2000. Loss: 0.03691177815198898. Accuracy: 97
Epoch: 4 LR: [0.0010000000000000002]
Iteration: 2500. Loss: 0.03511016443371773. Accuracy: 97
Iteration: 3000. Loss: 0.029424520209431648. Accuracy: 97

Step-wise Decay: Every Epoch, Larger Gamma

• At every epoch,
  • $\eta_t = \eta_{t-1}\gamma$
  • $\gamma = 0.96$
• Optimization Algorithm 4: SGD Nesterov
  • Modification of SGD Momentum
    • $v_t = \gamma v_{t-1} + \eta \cdot \nabla J(\theta - \gamma v_{t-1}, x^{i: i+n}, y^{i:i+n})$
    • $\theta = \theta - v_t$
• Practical example
  • Given $\eta_t = 0.1$ and $\gamma = 0.96$
  • Epoch 1: $\eta_t = 0.1$
  • Epoch 2: $\eta_{t+1} = 0.1 (0.96) = 0.096$
  • Epoch 3: $\eta_{t+2} = 0.1 (0.96)^2 = 0.092$
  • Epoch n: $\eta_{t+n} = 0.1 (0.96)^n$

Code for step-wise learning rate decay at every epoch with larger gamma

import torch
import torch.nn as nn
import torchvision.transforms as transforms
import torchvision.datasets as dsets

# Set seed
torch.manual_seed(0)

# Where to add a new import
from torch.optim.lr_scheduler import StepLR

'''
STEP 1: LOADING DATASET
'''

train_dataset = dsets.MNIST(root='./data', 
                            train=True, 
                            transform=transforms.ToTensor(),
                            download=True)

test_dataset = dsets.MNIST(root='./data', 
                           train=False, 
                           transform=transforms.ToTensor())

'''
STEP 2: MAKING DATASET ITERABLE
'''

batch_size = 100
n_iters = 3000
num_epochs = n_iters / (len(train_dataset) / batch_size)
num_epochs = int(num_epochs)

train_loader = torch.utils.data.DataLoader(dataset=train_dataset, 
                                           batch_size=batch_size, 
                                           shuffle=True)

test_loader = torch.utils.data.DataLoader(dataset=test_dataset, 
                                          batch_size=batch_size, 
                                          shuffle=False)

'''
STEP 3: CREATE MODEL CLASS
'''
class FeedforwardNeuralNetModel(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(FeedforwardNeuralNetModel, self).__init__()
        # Linear function
        self.fc1 = nn.Linear(input_dim, hidden_dim) 
        # Non-linearity
        self.relu = nn.ReLU()
        # Linear function (readout)
        self.fc2 = nn.Linear(hidden_dim, output_dim)  

    def forward(self, x):
        # Linear function
        out = self.fc1(x)
        # Non-linearity
        out = self.relu(out)
        # Linear function (readout)
        out = self.fc2(out)
        return out
'''
STEP 4: INSTANTIATE MODEL CLASS
'''
input_dim = 28*28
hidden_dim = 100
output_dim = 10

model = FeedforwardNeuralNetModel(input_dim, hidden_dim, output_dim)

'''
STEP 5: INSTANTIATE LOSS CLASS
'''
criterion = nn.CrossEntropyLoss()


'''
STEP 6: INSTANTIATE OPTIMIZER CLASS
'''
learning_rate = 0.1

optimizer = torch.optim.SGD(model.parameters(), lr=learning_rate, momentum=0.9, nesterov=True)

'''
STEP 7: INSTANTIATE STEP LEARNING SCHEDULER CLASS
'''
# step_size: at how many multiples of epoch you decay
# step_size = 1, after every 2 epoch, new_lr = lr*gamma 
# step_size = 2, after every 2 epoch, new_lr = lr*gamma 

# gamma = decaying factor
scheduler = StepLR(optimizer, step_size=2, gamma=0.96)

'''
STEP 7: TRAIN THE MODEL
'''
iter = 0
for epoch in range(num_epochs):
    # Decay Learning Rate
    scheduler.step()
    # Print Learning Rate
    print('Epoch:', epoch,'LR:', scheduler.get_lr())
    for i, (images, labels) in enumerate(train_loader):
        # Load images as Variable
        images = images.view(-1, 28*28).requires_grad_()

        # Clear gradients w.r.t. parameters
        optimizer.zero_grad()

        # Forward pass to get output/logits
        outputs = model(images)

        # Calculate Loss: softmax --> cross entropy loss
        loss = criterion(outputs, labels)

        # Getting gradients w.r.t. parameters
        loss.backward()

        # Updating parameters
        optimizer.step()

        iter += 1

        if iter % 500 == 0:
            # Calculate Accuracy         
            correct = 0
            total = 0
            # Iterate through test dataset
            for images, labels in test_loader:
                # Load images to a Torch Variable
                images = images.view(-1, 28*28)

                # Forward pass only to get logits/output
                outputs = model(images)

                # Get predictions from the maximum value
                _, predicted = torch.max(outputs.data, 1)

                # Total number of labels
                total += labels.size(0)

                # Total correct predictions
                correct += (predicted == labels).sum()

            accuracy = 100 * correct / total

            # Print Loss
            print('Iteration: {}. Loss: {}. Accuracy: {}'.format(iter, loss.item(), accuracy))

Epoch: 0 LR: [0.1]
Iteration: 500. Loss: 0.15292978286743164. Accuracy: 96
Epoch: 1 LR: [0.1]
Iteration: 1000. Loss: 0.11253029108047485. Accuracy: 96
Epoch: 2 LR: [0.096]
Iteration: 1500. Loss: 0.11864850670099258. Accuracy: 97
Epoch: 3 LR: [0.096]
Iteration: 2000. Loss: 0.030942382290959358. Accuracy: 97
Epoch: 4 LR: [0.09216]
Iteration: 2500. Loss: 0.04521659016609192. Accuracy: 97
Iteration: 3000. Loss: 0.027839098125696182. Accuracy: 97

Pointers on Step-wise Decay

• You would want to decay your LR gradually when you're training more epochs
  • Converge too fast, to a crappy loss/accuracy, if you decay rapidly
• To decay slower
  • Larger $\gamma$
  • Larger interval of decay

Reduce on Loss Plateau Decay

Reduce on Loss Plateau Decay, Patience=0, Factor=0.1

• Reduce learning rate whenever loss plateaus
  • Patience: number of epochs with no improvement after which learning rate will be reduced
    • Patience = 0
  • Factor: multiplier to decrease learning rate, $lr = lr*factor = \gamma$
    • Factor = 0.1
• Optimization Algorithm: SGD Nesterov
  • Modification of SGD Momentum
    • $v_t = \gamma v_{t-1} + \eta \cdot \nabla J(\theta - \gamma v_{t-1}, x^{i: i+n}, y^{i:i+n})$
    • $\theta = \theta - v_t$

Code for reduce on loss plateau learning rate decay of factor 0.1 and 0 patience

import torch
import torch.nn as nn
import torchvision.transforms as transforms
import torchvision.datasets as dsets

# Set seed
torch.manual_seed(0)

# Where to add a new import
from torch.optim.lr_scheduler import ReduceLROnPlateau

'''
STEP 1: LOADING DATASET
'''

train_dataset = dsets.MNIST(root='./data', 
                            train=True, 
                            transform=transforms.ToTensor(),
                            download=True)

test_dataset = dsets.MNIST(root='./data', 
                           train=False, 
                           transform=transforms.ToTensor())

'''
STEP 2: MAKING DATASET ITERABLE
'''

batch_size = 100
n_iters = 6000
num_epochs = n_iters / (len(train_dataset) / batch_size)
num_epochs = int(num_epochs)

train_loader = torch.utils.data.DataLoader(dataset=train_dataset, 
                                           batch_size=batch_size, 
                                           shuffle=True)

test_loader = torch.utils.data.DataLoader(dataset=test_dataset, 
                                          batch_size=batch_size, 
                                          shuffle=False)

'''
STEP 3: CREATE MODEL CLASS
'''
class FeedforwardNeuralNetModel(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(FeedforwardNeuralNetModel, self).__init__()
        # Linear function
        self.fc1 = nn.Linear(input_dim, hidden_dim) 
        # Non-linearity
        self.relu = nn.ReLU()
        # Linear function (readout)
        self.fc2 = nn.Linear(hidden_dim, output_dim)  

    def forward(self, x):
        # Linear function
        out = self.fc1(x)
        # Non-linearity
        out = self.relu(out)
        # Linear function (readout)
        out = self.fc2(out)
        return out
'''
STEP 4: INSTANTIATE MODEL CLASS
'''
input_dim = 28*28
hidden_dim = 100
output_dim = 10

model = FeedforwardNeuralNetModel(input_dim, hidden_dim, output_dim)

'''
STEP 5: INSTANTIATE LOSS CLASS
'''
criterion = nn.CrossEntropyLoss()


'''
STEP 6: INSTANTIATE OPTIMIZER CLASS
'''
learning_rate = 0.1

optimizer = torch.optim.SGD(model.parameters(), lr=learning_rate, momentum=0.9, nesterov=True)

'''
STEP 7: INSTANTIATE STEP LEARNING SCHEDULER CLASS
'''
# lr = lr * factor 
# mode='max': look for the maximum validation accuracy to track
# patience: number of epochs - 1 where loss plateaus before decreasing LR
        # patience = 0, after 1 bad epoch, reduce LR
# factor = decaying factor
scheduler = ReduceLROnPlateau(optimizer, mode='max', factor=0.1, patience=0, verbose=True)

'''
STEP 7: TRAIN THE MODEL
'''
iter = 0
for epoch in range(num_epochs):
    for i, (images, labels) in enumerate(train_loader):
        # Load images as Variable
        images = images.view(-1, 28*28).requires_grad_()

        # Clear gradients w.r.t. parameters
        optimizer.zero_grad()

        # Forward pass to get output/logits
        outputs = model(images)

        # Calculate Loss: softmax --> cross entropy loss
        loss = criterion(outputs, labels)

        # Getting gradients w.r.t. parameters
        loss.backward()

        # Updating parameters
        optimizer.step()

        iter += 1

        if iter % 500 == 0:
            # Calculate Accuracy         
            correct = 0
            total = 0
            # Iterate through test dataset
            for images, labels in test_loader:
                # Load images to a Torch Variable
                images = images.view(-1, 28*28)

                # Forward pass only to get logits/output
                outputs = model(images)

                # Get predictions from the maximum value
                _, predicted = torch.max(outputs.data, 1)

                # Total number of labels
                total += labels.size(0)

                # Total correct predictions
                # Without .item(), it is a uint8 tensor which will not work when you pass this number to the scheduler
                correct += (predicted == labels).sum().item()

            accuracy = 100 * correct / total

            # Print Loss
            # print('Iteration: {}. Loss: {}. Accuracy: {}'.format(iter, loss.data[0], accuracy))

    # Decay Learning Rate, pass validation accuracy for tracking at every epoch
    print('Epoch {} completed'.format(epoch))
    print('Loss: {}. Accuracy: {}'.format(loss.item(), accuracy))
    print('-'*20)
    scheduler.step(accuracy)

Epoch 0 completed
Loss: 0.17087846994400024. Accuracy: 96.26
--------------------
Epoch 1 completed
Loss: 0.11688263714313507. Accuracy: 96.96
--------------------
Epoch 2 completed
Loss: 0.035437121987342834. Accuracy: 96.78
--------------------
Epoch     2: reducing learning rate of group 0 to 1.0000e-02.
Epoch 3 completed
Loss: 0.0324370414018631. Accuracy: 97.7
--------------------
Epoch 4 completed
Loss: 0.022194599732756615. Accuracy: 98.02
--------------------
Epoch 5 completed
Loss: 0.007145566865801811. Accuracy: 98.03
--------------------
Epoch 6 completed
Loss: 0.01673538237810135. Accuracy: 98.05
--------------------
Epoch 7 completed
Loss: 0.025424446910619736. Accuracy: 98.01
--------------------
Epoch     7: reducing learning rate of group 0 to 1.0000e-03.
Epoch 8 completed
Loss: 0.014696130529046059. Accuracy: 98.05
--------------------
Epoch     8: reducing learning rate of group 0 to 1.0000e-04.
Epoch 9 completed
Loss: 0.00573748117312789. Accuracy: 98.04
--------------------
Epoch     9: reducing learning rate of group 0 to 1.0000e-05.

Reduce on Loss Plateau Decay, Patience=0, Factor=0.5

• Reduce learning rate whenever loss plateaus
  • Patience: number of epochs with no improvement after which learning rate will be reduced
    • Patience = 0
  • Factor: multiplier to decrease learning rate, $lr = lr*factor = \gamma$
    • Factor = 0.5
• Optimization Algorithm 4: SGD Nesterov
  • Modification of SGD Momentum
    • $v_t = \gamma v_{t-1} + \eta \cdot \nabla J(\theta - \gamma v_{t-1}, x^{i: i+n}, y^{i:i+n})$
    • $\theta = \theta - v_t$

Code for reduce on loss plateau learning rate decay with factor 0.5 and 0 patience

import torch
import torch.nn as nn
import torchvision.transforms as transforms
import torchvision.datasets as dsets

# Set seed
torch.manual_seed(0)

# Where to add a new import
from torch.optim.lr_scheduler import ReduceLROnPlateau

'''
STEP 1: LOADING DATASET
'''

train_dataset = dsets.MNIST(root='./data', 
                            train=True, 
                            transform=transforms.ToTensor(),
                            download=True)

test_dataset = dsets.MNIST(root='./data', 
                           train=False, 
                           transform=transforms.ToTensor())

'''
STEP 2: MAKING DATASET ITERABLE
'''

batch_size = 100
n_iters = 6000
num_epochs = n_iters / (len(train_dataset) / batch_size)
num_epochs = int(num_epochs)

train_loader = torch.utils.data.DataLoader(dataset=train_dataset, 
                                           batch_size=batch_size, 
                                           shuffle=True)

test_loader = torch.utils.data.DataLoader(dataset=test_dataset, 
                                          batch_size=batch_size, 
                                          shuffle=False)

'''
STEP 3: CREATE MODEL CLASS
'''
class FeedforwardNeuralNetModel(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(FeedforwardNeuralNetModel, self).__init__()
        # Linear function
        self.fc1 = nn.Linear(input_dim, hidden_dim) 
        # Non-linearity
        self.relu = nn.ReLU()
        # Linear function (readout)
        self.fc2 = nn.Linear(hidden_dim, output_dim)  

    def forward(self, x):
        # Linear function
        out = self.fc1(x)
        # Non-linearity
        out = self.relu(out)
        # Linear function (readout)
        out = self.fc2(out)
        return out
'''
STEP 4: INSTANTIATE MODEL CLASS
'''
input_dim = 28*28
hidden_dim = 100
output_dim = 10

model = FeedforwardNeuralNetModel(input_dim, hidden_dim, output_dim)

'''
STEP 5: INSTANTIATE LOSS CLASS
'''
criterion = nn.CrossEntropyLoss()


'''
STEP 6: INSTANTIATE OPTIMIZER CLASS
'''
learning_rate = 0.1

optimizer = torch.optim.SGD(model.parameters(), lr=learning_rate, momentum=0.9, nesterov=True)

'''
STEP 7: INSTANTIATE STEP LEARNING SCHEDULER CLASS
'''
# lr = lr * factor 
# mode='max': look for the maximum validation accuracy to track
# patience: number of epochs - 1 where loss plateaus before decreasing LR
        # patience = 0, after 1 bad epoch, reduce LR
# factor = decaying factor
scheduler = ReduceLROnPlateau(optimizer, mode='max', factor=0.5, patience=0, verbose=True)

'''
STEP 7: TRAIN THE MODEL
'''
iter = 0
for epoch in range(num_epochs):
    for i, (images, labels) in enumerate(train_loader):
        # Load images as Variable
        images = images.view(-1, 28*28).requires_grad_()

        # Clear gradients w.r.t. parameters
        optimizer.zero_grad()

        # Forward pass to get output/logits
        outputs = model(images)

        # Calculate Loss: softmax --> cross entropy loss
        loss = criterion(outputs, labels)

        # Getting gradients w.r.t. parameters
        loss.backward()

        # Updating parameters
        optimizer.step()

        iter += 1

        if iter % 500 == 0:
            # Calculate Accuracy         
            correct = 0
            total = 0
            # Iterate through test dataset
            for images, labels in test_loader:
                # Load images to a Torch Variable
                images = images.view(-1, 28*28)

                # Forward pass only to get logits/output
                outputs = model(images)

                # Get predictions from the maximum value
                _, predicted = torch.max(outputs.data, 1)

                # Total number of labels
                total += labels.size(0)

                # Total correct predictions
                 # Without .item(), it is a uint8 tensor which will not work when you pass this number to the scheduler
                correct += (predicted == labels).sum().item()

            accuracy = 100 * correct / total

            # Print Loss
            # print('Iteration: {}. Loss: {}. Accuracy: {}'.format(iter, loss.data[0], accuracy))

    # Decay Learning Rate, pass validation accuracy for tracking at every epoch
    print('Epoch {} completed'.format(epoch))
    print('Loss: {}. Accuracy: {}'.format(loss.item(), accuracy))
    print('-'*20)
    scheduler.step(accuracy)

Epoch 0 completed
Loss: 0.17087846994400024. Accuracy: 96.26
--------------------
Epoch 1 completed
Loss: 0.11688263714313507. Accuracy: 96.96
--------------------
Epoch 2 completed
Loss: 0.035437121987342834. Accuracy: 96.78
--------------------
Epoch     2: reducing learning rate of group 0 to 5.0000e-02.
Epoch 3 completed
Loss: 0.04893001914024353. Accuracy: 97.62
--------------------
Epoch 4 completed
Loss: 0.020584167912602425. Accuracy: 97.86
--------------------
Epoch 5 completed
Loss: 0.006022400688380003. Accuracy: 97.95
--------------------
Epoch 6 completed
Loss: 0.028374142944812775. Accuracy: 97.87
--------------------
Epoch     6: reducing learning rate of group 0 to 2.5000e-02.
Epoch 7 completed
Loss: 0.013204765506088734. Accuracy: 98.0
--------------------
Epoch 8 completed
Loss: 0.010137186385691166. Accuracy: 97.95
--------------------
Epoch     8: reducing learning rate of group 0 to 1.2500e-02.
Epoch 9 completed
Loss: 0.0035198689438402653. Accuracy: 98.01
--------------------

Pointers on Reduce on Loss Pleateau Decay

• In these examples, we used patience=1 because we are running few epochs
  • You should look at a larger patience such as 5 if for example you ran 500 epochs.
• You should experiment with 2 properties
  • Patience
  • Decay factor

Summary

We've learnt...

Success

• Learning Rate Intuition
  • Update parameters so model can churn output closer to labels
  • Gradual parameter updates
• Learning Rate Pointers
  • If we set $\eta$ to be a large value $\rightarrow$ learn too much (rapid learning)
  • If we set $\eta$ to be a small value $\rightarrow$ learn too little (slow learning)
• Learning Rate Schedules
  • Step-wise Decay
  • Reduce on Loss Plateau Decay
• Step-wise Decay
  • Every 1 epoch
  • Every 2 epoch
  • Every 1 epoch, larger gamma
• Step-wise Decay Pointers
  • Decay LR gradually
    • Larger $\gamma$
    • Larger interval of decay (increase epoch)
• Reduce on Loss Plateau Decay
  • Patience=0, Factor=1
  • Patience=0, Factor=0.5
• Pointers on Reduce on Loss Plateau Decay
  • Larger patience with more epochs
  • 2 hyperparameters to experiment
    • Patience
    • Decay factor

Citation

If you have found these useful in your research, presentations, school work, projects or workshops, feel free to cite using this DOI.

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