Optimizing Deep Learning with L-BFGS: A Step-by-Step Explanation

LBFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shanno) is an optimization algorithm well-suited for problems with a large number of parameters. It's a quasi-Newton method, meaning it approximates the Hessian (matrix of second-order derivatives) to update the model's parameters efficiently, using only a limited memory of past gradients and function evaluations.

• closure (Callable) argument
  This is a mandatory function that encapsulates the computation of the loss and its backward pass. It's called multiple times by L-BFGS to refine the optimization:
  • Inside the closure function:
    • Clear the gradients (optimizer.zero_grad()) to ensure gradients from previous computations don't accumulate.
    • Forward pass the input through your model.
    • Calculate the loss using a suitable loss function (e.g., criterion(output, target)).
    • Call loss.backward() to compute the gradients of the loss with respect to the model's parameters.
    • Return the loss value.

Key Points

• Limited Memory
  L-BFGS stores information from past iterations to approximate the Hessian, but it does so with limited memory. This makes it efficient for problems with many parameters.
• Multiple Evaluations
  L-BFGS might call the closure function several times within a single step to refine the search direction and update parameters effectively.
• Gradient Clearing
  The closure function is crucial because L-BFGS needs to re-evaluate the loss and gradients iteratively. Clearing gradients before each computation ensures only the gradients for the current iteration are considered.

import torch
import torch.nn as nn
import torch.optim as optim

# Define your model (replace with your actual model)
class MyModel(nn.Module):
    def __init__(self):
        super().__init__()
        # ... your model architecture

    def forward(self, x):
        # ... your forward pass logic

# Create a loss function (e.g., mean squared error)
criterion = nn.MSELoss()

# Create the model and optimizer
model = MyModel()
optimizer = optim.LBFGS(model.parameters())

# Training loop
for epoch in range(num_epochs):
    for inputs, targets in train_data:
        # Clear gradients before each step
        optimizer.zero_grad()

        def closure():
            outputs = model(inputs)
            loss = criterion(outputs, targets)
            loss.backward()
            return loss

        optimizer.step(closure)

        # ... update progress or perform other training tasks

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Logistic Regression with L-BFGS

This example trains a logistic regression model on a synthetic dataset using L-BFGS:

import torch
from torch import nn
from torch.optim import LBFGS

# Generate some dummy data
x = torch.randn(100, 2)
y = (x[:, 0] > 0.5).float()

# Define the logistic regression model
class LogisticRegression(nn.Module):
    def __init__(self):
        super().__init__()
        self.linear = nn.Linear(2, 1)

    def forward(self, x):
        return torch.sigmoid(self.linear(x))

model = LogisticRegression()

# Define the loss function (binary cross-entropy)
loss_fn = nn.BCELoss()

# Create the L-BFGS optimizer
optimizer = LBFGS(model.parameters())

# Training loop
for epoch in range(100):
    # Forward pass
    y_pred = model(x)

    # Calculate loss
    loss = loss_fn(y_pred, y)

    # Clear gradients before each step
    optimizer.zero_grad()

    def closure():
        loss.backward()
        return loss

    # Perform optimization step
    optimizer.step(closure)

    # Print training progress (optional)
    print(f"Epoch: {epoch+1}, Loss: {loss.item():.4f}")

Customizing L-BFGS Parameters

This example shows how to customize L-BFGS parameters like history_size (number of past iterations to store information) and max_iter (maximum number of iterations per step):

import torch
from torch import nn
from torch.optim import LBFGS

# ... (rest of the code as needed)

# Create the L-BFGS optimizer with customized parameters
optimizer = LBFGS(model.parameters(), history_size=20, max_iter=10)

# ... (rest of the training loop)

L-BFGS with Early Stopping

This example demonstrates how to use early stopping with L-BFGS to prevent overfitting:

import torch
from torch import nn
from torch.optim import LBFGS

# ... (rest of the code as needed)

# Track best validation loss so far
best_val_loss = float('inf')

# Training loop
for epoch in range(100):
    # ... (training steps)

    # Evaluate on validation set (optional)
    val_loss = evaluate_model(model, val_data)

    # Early stopping condition
    if val_loss < best_val_loss:
        best_val_loss = val_loss
    else:
        print("Early stopping triggered!")
        break

    # ... (rest of the epoch)

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Gradient Descent-based Optimizers

These optimizers are simpler than L-BFGS and suitable for a wider range of problems. Common choices include:

• torch.optim.RMSprop (Root Mean Squared Prop)
  Similar to Adam, but uses a decaying average of squared gradients. It can be effective when dealing with sparse gradients.
• torch.optim.Adam (Adaptive Moment Estimation)
  A popular choice that adapts the learning rate for each parameter based on past gradients. It's often efficient for diverse problems.
• torch.optim.SGD (Stochastic Gradient Descent)
  A fundamental optimizer widely used as a baseline. It updates parameters based on the current gradient and a learning rate.

Conjugate Gradient (CG) Optimizers

These are also second-order methods like L-BFGS, but they use a different approach to approximate the Hessian. They can be more efficient for problems with specific structures.

• scipy.optimize.minimize (SciPy library)
  While not directly a PyTorch optimizer, SciPy offers various CG variants like L-BFGS-B that might be suitable for your task.
• torch.optim.CG (Conjugate Gradient)
  A basic CG implementation in PyTorch.

Choosing the Right Alternative

The best alternative depends on several factors:

• Ease of Use
  Gradient descent methods are generally simpler to set up. L-BFGS and CG might require more care in parameter tuning.
• Convergence Speed
  L-BFGS and CG methods often converge faster than basic gradient descent, especially for non-convex problems. Adam can be efficient for diverse scenarios.
• Memory Usage
  L-BFGS uses limited memory, which can be an advantage. Some CG variants might have similar memory requirements. Gradient descent methods generally have lower memory needs.
• Problem Type
  L-BFGS is often preferred for large-scale problems with smooth objective functions. Gradient descent methods tend to be versatile, while CG variants might be more efficient for problems with specific structures.