What’s the difference between derivative, gradient, and Jacobian?
If you’re in machine learning, you’ve probably heard these three terms.
At a high level, all of them relate to measuring the rate of change of a function.
The difference lies in the number of input (x) and output (f(x)) variables involved.
The derivative
The derivative is the simplest case.
It applies to functions with one input and one output, and it tells us how the output changes as the input changes a little bit.
In mathematical terms, it’s the slope of the tangent line to the function at a given point, representing the rate of change.
For instance, if the curve above is modeling a patient’s pain level over time:
- A positive derivative (green) means the pain is increasing;
- A negative derivative (red) indicates the pain is decreasing;
- A zero derivative (black) means the pain level is constant.
The gradient
Now that we’ve explored the derivative, the gradient becomes easier to understand.
What if, instead of just one variable, our function f(x) depends on multiple inputs x₁, x₂…xₙ?
For instance, what if we wanted to include more variables, like demographics and comorbidities, to see how they affect a patient’s pain level?
This is where the gradient comes in.
The gradient is a generalization of the derivative for functions with multiple input variables and a single output.
Since we have multiple inputs, the gradient forms a vector. Each element in this vector is a partial derivative, which measures how the output changes as one specific input changes while keeping all other inputs constant.
In this way, the gradient tells us not only the direction of the function’s steepest increase, but also the magnitude of this rate of change.
In machine learning, the gradient is crucial for optimization techniques like gradient descent. This method uses the gradient to update the model’s parameters in the direction that reduces the loss function most quickly:
The Jacobian
So, you might be wondering: we’ve explored how to measure the rate of change for functions with both single and multiple input variables, but so far, all examples had a single output.
What if we have multiple inputs and multiple outputs?
In such cases, we work with vector-valued functions, where both the inputs and outputs are vectors. In this universe, we use something called the Jacobian.
The Jacobian is a matrix that organizes the gradients for each output with respect to each input. Each row corresponds to a different output, and each column corresponds to a different input.
And how is this useful in machine learning?
During the training process of a neural network, backpropagation requires computing the gradient of the loss with respect to each parameter in the network.
When we have multiple outputs, such as in multi-class classification, we need to calculate how the inputs (e.g., the model’s weights) affect each of these outputs.
The Jacobian helps us with this: it organizes all the gradients between the inputs and outputs into a matrix, making it easier to understand how changes in parameters impact the overall output of the network.
Not that hard, right? See you next time.
