Use Torch.matmul Achieve Conv Backward

8 min read Sep 30, 2024

How to Achieve Convolution Backward Using `torch.matmul` in PyTorch

In the realm of deep learning, convolutional neural networks (CNNs) are ubiquitous, employed for image recognition, natural language processing, and a plethora of other tasks. At the heart of CNNs lies the convolution operation, which extracts features by applying learned filters to input data. The backpropagation process, essential for training these networks, necessitates the computation of gradients with respect to the convolutional weights and biases.

While PyTorch provides convenient functions like torch.nn.Conv2d for defining and applying convolution layers, understanding how to achieve the backward pass manually can be illuminating. This allows for greater control and opens up opportunities for implementing custom convolution operations.

One approach to performing convolution backward manually is leveraging the powerful torch.matmul function. This article will explore how to utilize torch.matmul to calculate the gradients for convolutional weights and biases.

The Convolution Operation: A Recap

Before delving into the backward pass, let's briefly revisit the convolution operation. Given an input tensor X and a filter W, the convolution output Y is obtained by sliding the filter over the input, performing element-wise multiplication and summation at each location.

In mathematical terms, the convolution output Y at a given location can be expressed as:

Y[i, j, k] = sum(X[i + m, j + n, l] * W[m, n, l, k])

where:

i, j, k are the indices of the output tensor Y
m, n, l are the indices of the filter W
l, k are the input and output channels, respectively

Convolution Backward: Gradients for Weights and Biases

The goal of the backward pass is to compute gradients with respect to the convolutional weights W and biases b. These gradients are then used to update the weights during the training process using an optimization algorithm like stochastic gradient descent (SGD).

Gradients for Weights:

The gradient of the loss function with respect to a specific weight W[m, n, l, k] can be computed by summing the products of the corresponding output gradient dY[i, j, k] and the input X[i + m, j + n, l].

Gradients for Biases:

The gradient of the loss function with respect to a bias b[k] is simply the sum of the output gradients dY[i, j, k] across all output locations.

Implementing Convolution Backward Using `torch.matmul`

Let's outline a procedure to calculate the gradients for convolutional weights and biases using torch.matmul:

Reshape Input and Filter: Reshape the input tensor X and the filter W into matrices. This is crucial to leverage torch.matmul effectively.
Im2Col Transformation: Transform the reshaped input tensor X into a matrix using the "im2col" operation. This operation essentially extracts patches from the input tensor and arranges them as columns in a matrix.
Compute Convolution Output: Perform matrix multiplication (torch.matmul) between the im2col matrix and the reshaped filter matrix to calculate the convolution output Y.
Calculate Output Gradients: Obtain the gradients of the loss function with respect to the output dY.
Compute Gradient for Weights: Reshape the output gradient dY into a matrix, and then perform torch.matmul with the transposed im2col matrix to calculate the gradients for the filter weights.
Compute Gradient for Biases: Sum the output gradient dY along the output dimensions to obtain the gradients for the biases.

Example Code

To illustrate the process, let's provide a Python code snippet using PyTorch:

import torch

# Define input tensor, filter, and output gradient
X = torch.randn(1, 3, 5, 5)  # Input tensor (batch, channels, height, width)
W = torch.randn(2, 3, 3, 3)  # Filter (output_channels, input_channels, height, width)
dY = torch.randn(1, 2, 3, 3)  # Output gradient

# Reshape input and filter
X_reshaped = X.view(1, -1, 9)  # (batch, input_channels * height * width, patch_size)
W_reshaped = W.view(2, -1)  # (output_channels, input_channels * height * width)

# Im2Col transformation
im2col_X = torch.nn.functional.unfold(X, kernel_size=(3, 3), stride=(1, 1))
im2col_X = im2col_X.transpose(1, 2)  # (batch, patch_size, input_channels * height * width)

# Compute convolution output
Y = torch.matmul(im2col_X, W_reshaped.T)
Y = Y.view(1, 2, 3, 3)  # Reshape output

# Reshape output gradient
dY_reshaped = dY.view(1, -1)

# Compute gradient for weights
dW = torch.matmul(dY_reshaped.T, im2col_X).view(2, 3, 3, 3)

# Compute gradient for biases
db = torch.sum(dY, dim=(1, 2, 3))

print("Gradients for Weights:", dW)
print("Gradients for Biases:", db)

Advantages of Using `torch.matmul`

Employing torch.matmul for convolution backward offers several advantages:

Efficiency: Matrix multiplication is highly optimized in libraries like PyTorch, leading to efficient computation of gradients.
Flexibility: The approach allows for customization of convolutional kernels, strides, and other parameters without relying on pre-defined functions.
Understanding: Manual implementation provides a deeper understanding of the underlying mechanics of convolution and backpropagation.

Conclusion

Achieving convolution backward using torch.matmul in PyTorch enables manual gradient computation, offering greater control and flexibility over the backpropagation process. This approach leverages the optimized matrix multiplication capabilities of PyTorch and promotes a deeper understanding of convolutional operations. By combining torch.matmul with techniques like im2col, we can efficiently calculate gradients for convolutional weights and biases, contributing to the training and optimization of convolutional neural networks.

Use Torch.matmul Achieve Conv Backward

How to Achieve Convolution Backward Using torch.matmul in PyTorch