How to Compute Gradient Between Tensors in PyTorch Using autograd.grad(): Fixing 'Scalar Outputs' RuntimeError (vs TensorFlow tf.gradients)

Table of Contents#

1. Prerequisites
2. Understanding Gradient Computation in PyTorch
3. What is torch.autograd.grad()?
4. The 'Scalar Outputs' RuntimeError: Why It Happens
5. Fixing the Error: Step-by-Step Solutions
   • Solution 1: Ensure Outputs Are Scalars
   • Solution 2: Use grad_outputs for Non-Scalar Outputs
6. Comparing with TensorFlow’s tf.gradients (and GradientTape)
7. Advanced Use Cases
   • Higher-Order Gradients
   • Jacobian and Hessian Matrices
8. Common Pitfalls and Best Practices
9. Conclusion
10. References

Prerequisites#

To follow along, you should:

• Have basic familiarity with PyTorch tensors and computational graphs.
• Understand the concept of gradients and automatic differentiation.
• (Optional) Basic knowledge of TensorFlow for the comparison section.

Understanding Gradient Computation in PyTorch#

PyTorch’s autograd engine dynamically builds a computational graph where nodes are operations and edges are tensors. When requires_grad=True is set on a tensor, autograd tracks all operations involving it, enabling gradient computation later.

Traditionally, gradients are computed using tensor.backward(), which accumulates gradients into the .grad attribute of leaf tensors. However, torch.autograd.grad() offers more control: it directly returns gradients instead of accumulating them, making it ideal for advanced use cases like computing Jacobians or higher-order derivatives.

What is torch.autograd.grad()?#

torch.autograd.grad() computes the gradient of output tensors with respect to input tensors. Its signature is:

torch.autograd.grad(
    outputs,
    inputs,
    grad_outputs=None,
    retain_graph=None,
    create_graph=False,
    only_inputs=True,
    allow_unused=False
)

Key Parameters:#

• outputs: Tensors for which gradients are computed (e.g., loss).
• inputs: Tensors with respect to which gradients are computed (e.g., model weights).
• grad_outputs: "Vector" for the vector-Jacobian product (explained later). Defaults to None, requiring outputs to be scalars.
• create_graph: If True, retains the graph for higher-order differentiation.

Basic Example: Scalar Output#

Let’s compute the gradient of $y = x^2$ with respect to $x$ at $x = 3$:

import torch
 
x = torch.tensor([3.0], requires_grad=True)  # Input tensor with requires_grad=True
y = x ** 2  # Output tensor: y = 3^2 = 9 (scalar)
 
# Compute dy/dx
grad = torch.autograd.grad(outputs=y, inputs=x)
print(grad)  # Output: (tensor([6.]),)  # dy/dx = 2x = 6 at x=3

Here, y is a scalar, so autograd.grad() works seamlessly.

The 'Scalar Outputs' RuntimeError: Why It Happens#

A common error occurs when outputs are non-scalar (e.g., a tensor with shape (2,)). For example:

x = torch.tensor([1.0, 2.0], requires_grad=True)
y = x ** 2  # y = [1.0, 4.0] (non-scalar output)
 
try:
    grad = torch.autograd.grad(outputs=y, inputs=x)
except RuntimeError as e:
    print(e)  # RuntimeError: grad can be implicitly created only for scalar outputs

Why This Error?#

Gradients of a tensor with respect to another tensor are represented by a Jacobian matrix. For example, if $\mathbf{y} = [y_1, y_2]$ and $\mathbf{x} = [x_1, x_2]$, the Jacobian is:

Computing the full Jacobian is expensive. Instead, autograd.grad() computes the vector-Jacobian product (VJP): $v^T \cdot J$, where $v$ is a vector (from grad_outputs). If grad_outputs=None, autograd assumes $v$ is a vector of ones, but this requires $\mathbf{y}$ to be a scalar (so $J$ is a vector, and $v^T \cdot J$ is a scalar). For non-scalar $\mathbf{y}$, grad_outputs must be explicitly provided to define $v$.

Fixing the Error: Step-by-Step Solutions#

Solution 1: Ensure Outputs Are Scalars#

The simplest fix is to convert non-scalar outputs to scalars using operations like sum(), mean(), or indexing.

Example: Sum Non-Scalar Outputs#

x = torch.tensor([1.0, 2.0], requires_grad=True)
y = x ** 2  # y = [1.0, 4.0] (non-scalar)
y_scalar = y.sum()  # Sum to scalar: 5.0
 
grad = torch.autograd.grad(outputs=y_scalar, inputs=x)
print(grad)  # (tensor([2.0, 4.0]),)  # dy_scalar/dx = [2*1, 2*2]

Here, summing $y$ converts it to a scalar, allowing autograd.grad() to compute the gradient.

Example: Select a Single Element#

If you only need the gradient of a specific element in outputs:

y_scalar = y[0]  # Select first element: 1.0
grad = torch.autograd.grad(outputs=y_scalar, inputs=x)
print(grad)  # (tensor([2.0, 0.0]),)  # dy[0]/dx = [2*1, 0] (since y[0] = x[0]^2)

Solution 2: Use grad_outputs to Handle Non-Scalar Outputs#

For non-scalar outputs, provide grad_outputs—a tensor of the same shape as outputs that defines the "vector" $v$ in the VJP $v^T \cdot J$.

Example: Non-Scalar Outputs with grad_outputs#

If outputs is shape (2,), grad_outputs must also be shape (2,):

x = torch.tensor([1.0, 2.0], requires_grad=True)
y = x ** 2  # y = [1.0, 4.0] (shape (2,))
 
# Define grad_outputs as a vector of ones (same shape as y)
grad_outputs = torch.ones_like(y)  # v = [1.0, 1.0]
 
# Compute VJP: v^T * J = [1, 1] * [[2x1, 0], [0, 2x2]] = [2x1, 2x2]
grad = torch.autograd.grad(outputs=y, inputs=x, grad_outputs=grad_outputs)
print(grad)  # (tensor([2.0, 4.0]),)  # Same result as summing y!

Custom grad_outputs#

Use grad_outputs to weight gradients. For example, weight the first element of y by 2 and the second by 3:

grad_outputs = torch.tensor([2.0, 3.0])  # v = [2, 3]
grad = torch.autograd.grad(outputs=y, inputs=x, grad_outputs=grad_outputs)
print(grad)  # (tensor([4.0, 12.0]),)  # [2*2*1, 3*2*2] = [4, 12]

Comparing with TensorFlow’s tf.gradients (and GradientTape)#

TensorFlow historically used tf.gradients, but it’s deprecated in TensorFlow 2.x, replaced by tf.GradientTape. Let’s compare PyTorch’s autograd.grad() with TensorFlow’s gradient tools.

Key Difference: Handling Non-Scalar Outputs#

• PyTorch autograd.grad(): Requires scalar outputs or explicit grad_outputs.
• TensorFlow GradientTape: Implicitly sums non-scalar outputs by default (equivalent to grad_outputs=torch.ones_like(outputs) in PyTorch).

Example: Scalar Output (Both Frameworks)#

# PyTorch
x = torch.tensor([3.0], requires_grad=True)
y = x ** 2
grad_pytorch = torch.autograd.grad(y, x)[0]  # tensor(6.)
 
# TensorFlow
import tensorflow as tf
x = tf.Variable(3.0)
with tf.GradientTape() as tape:
    y = x ** 2
grad_tf = tape.gradient(y, x)  # tf.Tensor(6.0, shape=(), dtype=float32)

Example: Non-Scalar Outputs#

# PyTorch: Explicit grad_outputs
x = torch.tensor([1.0, 2.0], requires_grad=True)
y = x ** 2
grad_pytorch = torch.autograd.grad(y, x, grad_outputs=torch.ones_like(y))[0]  # tensor([2.0, 4.0])
 
# TensorFlow: Implicit sum (no need for grad_outputs)
x = tf.Variable([1.0, 2.0])
with tf.GradientTape() as tape:
    y = x ** 2  # y = [1.0, 4.0]
grad_tf = tape.gradient(y, x)  # tf.Tensor([2. 4.], shape=(2,), dtype=float32)

TensorFlow’s GradientTape automatically uses grad_ys=tf.ones_like(y) when outputs are non-scalar, whereas PyTorch requires explicit grad_outputs.

Advanced Use Cases#

Higher-Order Gradients#

Set create_graph=True to compute second-order gradients:

x = torch.tensor([3.0], requires_grad=True)
y = x ** 3  # y = x³
 
# First-order gradient: dy/dx = 3x²
dy_dx = torch.autograd.grad(y, x, create_graph=True)[0]
 
# Second-order gradient: d²y/dx² = 6x
d2y_dx2 = torch.autograd.grad(dy_dx, x)[0]
print(d2y_dx2)  # tensor([18.0])  # 6*3 = 18

Jacobian and Hessian Matrices#

To compute the Jacobian (gradient of a tensor w.r.t. another tensor), use grad_outputs as basis vectors (e.g., [1, 0], [0, 1] for 2D outputs):

x = torch.tensor([1.0, 2.0], requires_grad=True)
y = torch.stack([x[0]**2, x[1]**3])  # y = [x0², x1³] (shape (2,))
 
# Jacobian J = [[2x0, 0], [0, 3x1²]]
jacobian = []
for i in range(y.shape[0]):
    # Use grad_outputs with 1 at position i, 0 elsewhere
    grad = torch.autograd.grad(y, x, grad_outputs=torch.eye(2)[i])[0]
    jacobian.append(grad)
jacobian = torch.stack(jacobian)
print(jacobian)
# Output:
# tensor([[2.0000, 0.0000],
#         [0.0000, 12.0000]])  # 3x1² = 3*(2)^2 = 12

Common Pitfalls and Best Practices#

1. Forgetting requires_grad=True: Input tensors must have requires_grad=True to track gradients.
2. grad_outputs Shape Mismatch: Ensure grad_outputs has the same shape as outputs.
3. In-Place Operations: These break the computational graph (e.g., x += 1). Use x = x + 1 instead.
4. Retaining the Graph: Use retain_graph=True if you need to compute gradients multiple times on the same graph.

Conclusion#

torch.autograd.grad() is a powerful tool for gradient computation in PyTorch, offering fine-grained control over differentiation. The "Scalar Outputs" error arises because autograd requires clarity on how to handle non-scalar outputs—resolved by either converting outputs to scalars (via sum(), indexing) or using grad_outputs to define the vector-Jacobian product.

Compared to TensorFlow’s GradientTape, PyTorch enforces explicit handling of non-scalar outputs, which can prevent silent errors. Mastering autograd.grad() unlocks advanced workflows like Jacobian computation and higher-order differentiation, making it indispensable for research and complex model development.

References#

• PyTorch Autograd Documentation
• TensorFlow GradientTape Documentation
• PyTorch Autograd Mechanics
• TensorFlow Deprecation Guide: tf.gradients