Effect of Perturbation on Gradient Descent Sequence

In summary: A$ and $\alpha$, as desired. Therefore, we have proven that $\mathrm{dist}\left(\{\tilde{x}^k\}_k,\{{x}^k\}_k\right)$ is uniformly bounded, independent of $A$ and step-size $\alpha$.In summary, we can prove that the distance between the sequences generated by Gradient Descent for two different functions, one with a Lipschitz continuous gradient and the other with a Lipschitz continuous gradient plus a quadratic term, is uniformly bounded. This is done by considering the Lipschitz continuity of the gradient of the original function and using this to bound the distance between the points in the two sequences. This bound is independent of the specific function and
  • #1
Vulture1991
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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 $0<\alpha<2/L$:
\begin{equation}
x^{k+1}=x^k-\alpha\nabla f (x^k).
\end{equation}
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 $0<\alpha<2/L$:
\begin{equation}
\tilde{x}^{k+1}=\tilde{x}^k-\alpha\nabla \tilde{f} (\tilde{x}^k).
\end{equation}

**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 $\alpha$.
 
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  • #2


I would like to point out that the statement as it is currently written is not entirely clear. It is not specified what is meant by the distance between the two sequences, and it is not clear what is meant by "uniformly bounded". Is it meant to be bounded by a constant, or by a function of $k$? Furthermore, the notation used for the two sequences is inconsistent, as one uses $\{x^k\}_k$ and the other uses $\{\tilde{x}^k\}_k$.

Assuming that the distance between the two sequences refers to the Euclidean distance between the points in the sequences, and that "uniformly bounded" means bounded by a constant, I would approach this problem by considering the Lipschitz continuity of the gradient of the original function $f$. Since $f$ has a Lipschitz continuous gradient with constant $L$, we know that for any two points $x,y$, the following inequality holds:
\begin{equation}
|\nabla f(x)-\nabla f(y)| \leq L|x-y|.
\end{equation}
Using this, we can show that the distance between the points in the two sequences is bounded by a constant. Let $d_k=|x^k-\tilde{x}^k|$, then we have:
\begin{align}
d_{k+1} &= |x^{k+1}-\tilde{x}^{k+1}| \\
&= |x^k-\alpha\nabla f(x^k)-\tilde{x}^k+\alpha\nabla \tilde{f}(\tilde{x}^k)| \\
&= |d_k-\alpha(\nabla f(x^k)-\nabla \tilde{f}(\tilde{x}^k))| \\
&\leq |d_k|+\alpha|\nabla f(x^k)-\nabla \tilde{f}(\tilde{x}^k)| \\
&\leq |d_k|+\alpha L|x^k-\tilde{x}^k| \\
&= (1+\alpha L)d_k.
\end{align}
Thus, we have shown that the distance between the points in the two sequences is bounded by a constant, specifically $(1+\alpha L)^k$. This bound is independent of $
 
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FAQ: Effect of Perturbation on Gradient Descent Sequence

1. What is the "Effect of Perturbation" on the Gradient Descent Sequence?

The "Effect of Perturbation" on the Gradient Descent Sequence refers to how small changes or disturbances in the input data can affect the performance and convergence of the gradient descent algorithm. This can lead to changes in the direction and magnitude of the gradient, which can impact the speed and accuracy of the algorithm's updates.

2. How does Perturbation affect the Convergence of Gradient Descent?

Perturbation can affect the convergence of gradient descent in several ways. If the perturbation is too large, it can cause the algorithm to overshoot the minimum and prevent it from converging. On the other hand, if the perturbation is too small, it can slow down the convergence process. Additionally, perturbation can also lead to oscillations and instability in the gradient descent sequence, making it difficult to reach the optimal solution.

3. Can Perturbation improve the Performance of Gradient Descent?

Perturbation can have both positive and negative effects on the performance of gradient descent. In some cases, small perturbations can help the algorithm escape local minima and find a better solution. However, in general, perturbation is more likely to hinder the performance of gradient descent by causing instability and slowing down the convergence process.

4. How can we mitigate the Negative Effects of Perturbation on Gradient Descent?

There are several ways to mitigate the negative effects of perturbation on gradient descent. One approach is to use adaptive learning rates, where the size of the updates is adjusted based on the magnitude of the perturbation. Another method is to use momentum, which helps smooth out the effects of perturbation by taking into account the previous updates. Additionally, regularization techniques can also help prevent overfitting and improve the stability of the gradient descent sequence.

5. Is there a trade-off between Perturbation and Convergence in Gradient Descent?

Yes, there is often a trade-off between perturbation and convergence in gradient descent. A larger perturbation can help the algorithm escape local minima and find a better solution, but it may also hinder the convergence process. On the other hand, a smaller perturbation may lead to a more stable and accurate convergence, but it may also take longer to reach the optimal solution. Finding the right balance between perturbation and convergence is crucial for the success of gradient descent.

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