# On uniform boundedness of the GD algorithm

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• Vulture1991
In summary, we are considering a function ##f## with Lipschitz continuous gradient and at least one minimum, and a sequence ##\{x^k\}_k## generated by Gradient Descent with initial point ##x^0## and step-size ##0<\alpha<2/L##. We know that the sequence will converge to a critical point. We then introduce a new function ##\tilde{f}(x)=f(x)+x'Ax## with some ##A\succeq\mathbf{0}## and a sequence ##\{\tilde{x}^k\}_k## generated by Gradient Descent with the same initial point and step-size. The question is whether we can prove that
Vulture1991
Consider a function ##f\in\mathcal{C}^2## with Lipschitz continuous gradient (with constant ##L##)- we also assume the function is lowerbounded and has at least one minimum. Let ##\{x^k\}_k## be the sequence generated by Gradient Descent algorithm with initial point $x^0$ and step-size ##0<\alpha<2/L##:

x^{k+1}=x^k-\alpha\nabla f (x^k).

We know that the sequence will converge to a critical point.

Now consider the new function ##\tilde{f}(x)=f(x)+x'Ax## with some ##A\succeq\mathbf{0}##. Let ##\{\tilde{x}^k\}_k## be the sequence generated by Gradient Descent algorithm with **the same** initial point ##\tilde{x}^0=x^0## and step-size ##0<\alpha<2/L##:

\tilde{x}^{k+1}=\tilde{x}^k-\alpha\nabla \tilde{f} (\tilde{x}^k).

**Can we prove that ##\mathrm{dist}\left(\{\tilde{x}^k\}_k,\{{x}^k\}_k\right)## is uniformly bounded, independent of $A$ and step-size $\alpha$?**

I tried to prove it by assuming existence of a compact sublevel set ##\mathcal{L}=\{x:f(x)\leq f(x^0)\}## and using the fact that Gradient Descent generates a decreasing sequence of objective values (implying that the sequence remains in the compact sublevel set). However I cannot prove existence of a set independent of both $A$ and $\alpha$.

Do you have a reference where people can look up the algorithm in order to understand your question?

What is ##A## here, a matrix?

## 1. What is the GD algorithm and how does it relate to uniform boundedness?

The GD algorithm, or Gradient Descent algorithm, is a popular optimization method used in machine learning and other fields to find the minimum value of a function. It involves iteratively updating the parameters of a model in the direction of the steepest descent of the cost function. Uniform boundedness refers to the property of the GD algorithm to keep the parameters within a certain range, preventing them from becoming too large or too small.

## 2. Why is uniform boundedness important in the GD algorithm?

Uniform boundedness is important in the GD algorithm because it ensures that the parameters of the model do not diverge and become unstable. If the parameters become too large or too small, the model may not be able to accurately represent the data and may lead to poor performance.

## 3. How is uniform boundedness achieved in the GD algorithm?

Uniform boundedness is achieved in the GD algorithm by using a learning rate, which controls the size of the parameter updates. If the learning rate is too large, the parameters may become too large and lead to instability. If it is too small, the algorithm may take a long time to converge. By choosing an appropriate learning rate, the parameters can be kept within a reasonable range.

## 4. Can the GD algorithm still work without uniform boundedness?

Yes, the GD algorithm can still work without uniform boundedness, but it may not always converge or may take a longer time to converge. If the parameters become too large or too small, the algorithm may not be able to find the minimum value of the cost function and may get stuck in a local minimum instead of the global minimum.

## 5. Are there any drawbacks to enforcing uniform boundedness in the GD algorithm?

Enforcing uniform boundedness in the GD algorithm may lead to slower convergence, as the learning rate needs to be carefully chosen to prevent the parameters from becoming too large or too small. Additionally, in some cases, the optimal solution may require parameters that are outside of the bounded range, which may not be achievable with uniform boundedness.

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