Is gradient descent guaranteed to converge?

Intuitively, this means that gradient descent is guaranteed to converge and that it converges with rate O(1/k). value strictly decreases with each iteration of gradient descent until it reaches the optimal value f(x) = f(x∗).

Does gradient descent always converge to the optimum?

Hence, gradient descent would be guaranteed to converge to a local or global optimum.

Why does gradient descent not converge?

If the execution is not done properly while using gradient descent, it may lead to problems like vanishing gradient or exploding gradient problems. These problems occur when the gradient is too small or too large. And because of this problem the algorithms do not converge.

Do gradient descent methods always converge to the same point?

No, they always don't. That's because in some cases it reaches a local minima or a local optima point.

What does gradient descent converge to?

Setting ∇f(w)=0 gives a system of transcendental equations. But this objective function is convex and differentiable. So gradient descent converges to a global optimum.

How Gradient Descent Works. Simple Explanation

Does gradient descent always converge to a local minimum?

Gradient Descent is an iterative process that finds the minima of a function. This is an optimisation algorithm that finds the parameters or coefficients of a function where the function has a minimum value. Although this function does not always guarantee to find a global minimum and can get stuck at a local minimum.

Can gradient descent converge to zero?

We see above that gradient descent can reduce the cost function, and can converge when it reaches a point where the gradient of the cost function is zero.

Which gradient descent converges the fastest?

Mini Batch gradient descent: This is a type of gradient descent which works faster than both batch gradient descent and stochastic gradient descent.

When the gradient descent method is started from a point near the solution it will converge very quickly?

When Newton's method is started from a point near the solution, it will converge very quickly. True. Correct!

Is gradient descent deterministic?

Stochasticity of Deterministic Gradient Descent: Large Learning Rate for Multiscale Objective Function. This article suggests that deterministic Gradient Descent, which does not use any stochastic gradient approximation, can still exhibit stochastic behaviors.

How do you know when gradient descent converges?

In contrast, if we assume that f is strongly convex, we can show that gradient descent converges with rate O(ck) for 0 <c< 1. This means that a bound of f(x(k)) − f(x∗) ≤ ϵ can be achieved using only O(log(1/ϵ)) iterations. This rate is typically called “linear convergence.”

What is the drawback of gradient descent algorithm?

The disadvantage of Batch gradient descent –

1.It is less prone to local minima but in case it tends to local minima. It has no noisy step hence it will not be able to come out of it. 2. Although it is computationally efficient but not fast.

What are some of the problems of gradient descent?

The problem with gradient descent is that the weight update at a moment (t) is governed by the learning rate and gradient at that moment only. It doesn't take into account the past steps taken while traversing the cost space.

What guarantees convergence to the unique global minimum?

Batch Gradient Descent

It has straight trajectory towards the minimum and it is guaranteed to converge in theory to the global minimum if the loss function is convex and to a local minimum if the loss function is not convex. It has unbiased estimate of gradients.

Why does gradient descent always find the global minima?

Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet).

Is it possible that gradient descent fails to find the minimum of a function?

Gradient descent can't tell whether a minimum it has found is local or global. The step size α controls whether the algorithm converges to a minimum quickly or slowly, or whether it diverges. Many real world problems come down to minimizing a function.

In which case the gradient descent algorithm works best?

Gradient descent is best used when the parameters cannot be calculated analytically (e.g. using linear algebra) and must be searched for by an optimization algorithm.

Is gradient descent greedy?

Gradient descent is an optimization technique that can find the minimum of an objective function. It is a greedy technique that finds the optimal solution by taking a step in the direction of the maximum rate of decrease of the function.

Do all gradient descent algorithms lead to the same model provided you let them run long enough?

Do all Gradient Descent algorithms lead to the same model provided you let them run long enough? No. The issue is that stochastic gradient descent and mini-batch gradient descent have randomness built into them. This means that they can find their way to nearby the global optimum, but they generally don't converge.

Is gradient descent fast?

As we need to calculate the gradients for the whole dataset to perform just one update, batch gradient descent can be very slow and is intractable for datasets that don't fit in memory. Batch gradient descent also doesn't allow us to update our model online, i.e. with new examples on-the-fly.

Is stochastic gradient descent faster?

SGD is much faster but the convergence path of SGD is noisier than that of original gradient descent. This is because in each step it is not calculating the actual gradient but an approximation.

Is gradient descent a heuristic?

Gradient-based methods are not considered heuristics or metaheuristics.

Can gradient descent be applied to non convex functions?

Gradient descent is a generic method for continuous optimization, so it can be, and is very commonly, applied to nonconvex functions.

How does gradient descent avoid local minima?

Momentum, simply put, adds a fraction of the past weight update to the current weight update. This helps prevent the model from getting stuck in local minima, as even if the current gradient is 0, the past one most likely was not, so it will as easily get stuck.

Is Newton's method faster than gradient descent?

The three plots show a comparison of Newton's Method and Gradient Descent. Gradient Descent always converges after over 100 iterations from all initial starting points. If it converges (Figure 1), Newton's Method is much faster (convergence after 8 iterations) but it can diverge (Figure 2).