Will gradient descent work for non-convex functions? Hint: see if you can come up with some loss functions where gradient descent will fail.
The correct answer and explanation is:
Gradient descent is an optimization algorithm that iteratively moves towards the minimum of a function by following the direction of the negative gradient. While it is effective for many problems, its performance on non-convex functions can be problematic.
A non-convex function is one that has multiple local minima and potentially saddle points, which means there are several places where the gradient is zero but these points are not necessarily the global minimum. Gradient descent can get stuck in one of these local minima or saddle points, making it difficult to find the global minimum.
To illustrate this, let’s consider the example of a simple non-convex loss function, such as a multi-modal function. A typical function with this characteristic might look like a series of peaks and valleys. Gradient descent may start at a point on one of the valleys and find a local minimum, but this might not be the lowest valley or the global minimum. Even if it is a local minimum, it may not represent the optimal solution.
A classic example where gradient descent can fail is the Rosenbrock function often used for testing optimization algorithms. The function has a narrow, curved valley leading towards the global minimum. If the gradient descent starts too far away from the valley, it may fail to find the optimal path and get stuck at a local minimum.
Another example is the sine wave loss function, which oscillates between multiple peaks and valleys. Gradient descent will tend to fall into a local minimum, ignoring the global minimum that might be far away in the direction of the oscillation.
Gradient descent can still work in non-convex settings, but it’s highly sensitive to the initial starting point and can easily converge to a local minimum or saddle point. Techniques like random restarts, momentum, or using more advanced optimizers (e.g., Adam) are used to mitigate this problem and improve convergence in these cases.