# Under What Conditions Does Gradient Descent Converge?

We can use fixed learning rate during training without worrying about learning rate decay. 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.

In addition to, Do gradient descent methods always converge to similar points?

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

Correspondingly, What is the problem with 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.

Furthermore, Does 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.

Can gradient descent fail to converge?

Gradient Descent need not always converge at global minimum. It all depends on following conditions; If the line segment between any two points on the graph of the function lies above or on the graph then it is convex function.

## Related Question for Under What Conditions Does Gradient Descent Converge?

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∗).

Is SGD guaranteed to converge?

In such a context, our analysis shows that SGD, although has long been considered as a randomized algorithm, converges in an intrinsically deterministic manner to a global minimum. Traditional analysis of SGD in nonconvex optimization guarantees the convergence to a stationary point Bottou et al.

What is the drawback of gradient descent algorithm?

Due to frequent updates, the steps taken towards the minima are very noisy. This can often lean the gradient descent into other directions. Also, due to noisy steps, it may take longer to achieve convergence to the minima of the loss function.

decrease with an appropriate rate, and subject to relatively mild assumptions, stochastic gradient descent converges almost surely to a global minimum when the objective function is convex or pseudoconvex, and otherwise converges almost surely to a local minimum.

Why gradient descent isn't enough a comprehensive introduction to optimization algorithms in neural networks?

So after a finite number of updates, the algorithm refuses to learn and converges slowly even if we run it for a large number of epochs. The gradient reaches to a bad minimum (close to desired minima) but not at exact minima. So adagrad results in decaying and decreasing learning rate for bias parameters.

showed even without adding noise, gradient descent will not converge to any strict saddle point if the initial point is chosen randomly. However their result relies on the Stable Manifold Theorem from dynamical systems theory, which inherently does not provide any upperbound on the number of steps.

Can gradient descent get stuck in local minima?

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.

How gradient descent can converge to local minimum even learning rate is fixed?

Batch Gradient Descent uses a whole batch of training data at every training step. Thus it is very slow for larger datasets. The learning rate is fixed. In theory, if the cost function has a convex function, it is guaranteed to reach the global minimum, else the local minimum in case the loss function is not convex.

Does gradient descent reach global minimum?

Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. This structure allows us to show the Gram matrix is stable throughout the training process and this stability implies the global optimality of the gradient descent algorithm.

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!

Can gradient descent get stuck in a local minimum of the logistic regression error function?

Can Gradient Descent get stuck in a local minimum when training a Logistic Regression model? Gradient descent produces a convex shaped graph which only has one global optimum. Therefore, it cannot get stuck in a local minimum.

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.

How many iterations does gradient descent take?

t ≥ 2L[f(w0) − f∗] ϵ , so gradient descent requires t = O(1/ϵ) iterations to achieve ∇f(wk)2 ≤ ϵ.

Does SGD take longer to converge?

SGD is much faster but the convergence path of SGD is noisier than that of original gradient descent. SGD takes a lot of update steps but it will take a lesser number of epochs i.e. the number of times we iterate through all examples will be lesser in this case and thus it is a much faster process.

What is the best gradient descent algorithm?

TensorFlow

• Momentum.
• RMSprop.

Some advantages of batch gradient descent are its computational efficient, it produces a stable error gradient and a stable convergence. Some disadvantages are the stable error gradient can sometimes result in a state of convergence that isn't the best the model can achieve.

The batched updates provide a computationally more efficient process than stochastic gradient descent. Error information must be accumulated across mini-batches of training examples like batch gradient descent.

Why is stochastic gradient descent better?

According to a senior data scientist, one of the distinct advantages of using Stochastic Gradient Descent is that it does the calculations faster than gradient descent and batch gradient descent. Also, on massive datasets, stochastic gradient descent can converges faster because it performs updates more frequently.

On each iteration, we update the parameters in the opposite direction of the gradient of the objective function J(w) w.r.t the parameters where the gradient gives the direction of the steepest ascent. The size of the step we take on each iteration to reach the local minimum is determined by the learning rate α.

SGD often converges much faster compared to GD but the error function is not as well minimized as in the case of GD. Often in most cases, the close approximation that you get in SGD for the parameter values are enough because they reach the optimal values and keep oscillating there.

What does SGD stand for?

Is gradient descent sufficient for neural network?

An important factor that is the basis of any Neural Network is the Optimizer, which is used to train the model. The most prominent optimizer on which almost every Machine Learning algorithm is built is the Gradient Descent.

What is the problem with having too large of a learning rate in gradient descent?

When the learning rate is too large, gradient descent can inadvertently increase rather than decrease the training error. […] When the learning rate is too small, training is not only slower, but may become permanently stuck with a high training error.