Answers.Body: <p>Gradient clipping just clips the magnitude of the gradient, it has no effect on the values themselves (if you have x=3 and your gradient as g=4, then if you do x - g, you will get -1. You can clip the gradient within [-2, 2], but if you perform two updates, you will still get -1 in the end).</p>
  <p>Hence, if you have a constraint optimization problem, then you definitely need to clip the values. Another approach is to use mirror descent or reparametrization.
  See the following blog post which compares all three approaches: [1]: <a href="https://vene.ro/blog/mirror-descent.html#projected-gradient" rel="nofollow noreferrer">https://vene.ro/blog/mirror-descent.html#projected-gradient</a></p>
  <p>I think that in general, when you need gradient clipping, this means that your learning rate scheduling is improper. In fact, gradient clipping can be seen as an adaptive learning rate method. Therefore, I do not think there is any problem in using it to solve large gradient norms (which are dependent on the objective function shape, not on the gradient descent algorithm. So, unless you can change the objective function - which can also be another potential solution -, gradient clipping is perfectly fine!)</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-07-26T05:04:23.300Z
Answers.LastActivityDate: 2024-07-26T05:04:23.300Z
Answers.UserDisplayName: Ahnel
Body: <p>I'm exploring machine learning and currently studying calculus, specifically gradient descent. To practice, I am using the function: <span class="math-container">$$ f(x,y)= x^{2}y $$</span> <br />
  I have implemented the gradient descent algorithm and plotted the 3D function along with the path towards the global minimum.</p>
  <p>Context:</p>
  <p><strong>Initial Problem</strong>: I initially encountered issues where the values computed during gradient descent became too large to be represented.</p>
  <ul>
  <li>Solution: I added gradient clipping to ensure that the norm of the gradient doesn't exceed the threshold.</li>
  </ul>
  <p><strong>Local Minima</strong>: My gradient descent was getting stuck in suboptimal local minima.</p>
  <ul>
  <li>Solution: I incorporated momentum to help the algorithm avoid local minima and saddle points.</li>
  </ul>
  <p><strong>Bounds Issue</strong>: Although I successfully found the global minimum, the path of my gradient descent was going out of the function's bounds. To address this, I used <code>np.clip()</code> on both <code>x</code> and <code>y</code> to ensure they remained within the function's bounds.</p>
  <p>I am unsure whether using both gradient clipping and value clipping (with <code>np.clip()</code>) is the correct approach. Is gradient clipping alone sufficient to handle out-of-bounds values, or is it necessary to also clip the variable values? Additionally, could clipping the variables potentially mask underlying problems?</p>
  <p>Output with Gradient and Variable Clipping (desired output):
  <a href="https://i.sstatic.net/2ZSzDxM6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2ZSzDxM6.png" alt="3D plot with gradiend + variable clipping" /></a></p>
  <p>Output only with Gradient Clipping:</p>
  <p><a href="https://i.sstatic.net/kESxD3Zb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kESxD3Zb.png" alt="3D plot with gradient clipping" /></a></p>
  <hr />
  <p>Here is the code for my implementation:</p>
  <pre><code>import numpy as np
  import matplotlib.pyplot as plt
  
  def normGrad(v, max_norm=1):
      norm = np.linalg.norm(v)
      if norm &gt; max_norm:
          v = v * max_norm / norm # Scale the vector
      return v
  
  def GradientDescent(x_start, y_start, n_iterations, momentum=0.8, learning_rate=0.1, tol_threshold=1e-6, clip_threshold=10):
  
      '''
      learning rate: control the step size during updates
      '''
      x_path, y_path = [x_start], [y_start]
      x,y = x_start, y_start
      X_change = 0.0
      Y_change = 0.0
  
      for i in range(n_iterations):
          x_grad = Fx(x,y)
          y_grad = Fy(x,y)
         
          grad = normGrad(np.array([x_grad, y_grad]), max_norm=10)
          x_grad, y_grad = grad[0], grad[1]
  
          X_change = learning_rate * x_grad + momentum * X_change
          Y_change = learning_rate * y_grad + momentum * Y_change
  
          x = x - X_change
          y = y - Y_change
  
          # x = np.clip(x, -10, 10)
          # y = np.clip(y, -10, 10)
  
          # Check for convergence
          # This can help detect if the values are diverging.
          if i &gt; 0 and np.linalg.norm([x, y] - np.array([x_path[-1], y_path[-1]])) &lt; tol_threshold:
              break
  
          x_path.append(x) # append to path
          y_path.append(y)
  
  
      return x_path, y_path
  
  def f(x,y):
      return x**2*y
  
  def Fx(x,y):
      return 2*x*y
  
  def Fy(x,y):
      return x**2 
  
  x = np.arange(-10,10,0.1)
  y = np.arange(-10,10,0.1)
  X,Y = np.meshgrid(x,y) # array of values containing all value pairs (xi, yi)
  Z = f(X,Y)
  
  s_x,s_y = 5,5
  n_iter = 100
  x_path, y_path = GradientDescent(s_x, s_y, n_iter)
  
  fig, ax = plt.subplots(subplot_kw={&quot;projection&quot;:&quot;3d&quot;})
  ax.plot_surface(X, Y, Z,
                  cmap=&quot;jet&quot;, rstride=1, cstride=1, alpha=0.6)
  ax.contour(X, Y, Z, 10,
             offset=np.min(Z), cmap=&quot;jet&quot;) # offset=min(Z) set the countor on the xy plane
  z_path = f(np.array(x_path), np.array(y_path))
  ax.plot(x_path, y_path, z_path, 'k.--')
  
  ax.set_xlim([-10, 10])
  ax.set_ylim([-10, 10])
  ax.set_zlim([np.min(Z), np.max(Z)])
  ax.set_xlabel('x')
  ax.set_ylabel('y')
  ax.set_zlabel('z')
  plt.show()
  </code></pre>
  
Tags: <machine-learning><gradient-descent><calculus>
Title: Why is gradient clipping not preventing my gradient descent from going out of bounds?
UserDisplayName: Ian Aragão
ThreadText: # Why is gradient clipping not preventing my gradient descent from going out of bounds?
  
  (Asked by: username_0 on 2024-07-26)
  
  I'm exploring machine learning and currently studying calculus, specifically gradient descent. To practice, I am using the function: $$ f(x,y)= x^{2}y $$   
  I have implemented the gradient descent algorithm and plotted the 3D function along with the path towards the global minimum.
  
  Context:
  
  **Initial Problem**: I initially encountered issues where the values computed during gradient descent became too large to be represented.
  
  * Solution: I added gradient clipping to ensure that the norm of the gradient doesn't exceed the threshold.
  
  **Local Minima**: My gradient descent was getting stuck in suboptimal local minima.
  
  * Solution: I incorporated momentum to help the algorithm avoid local minima and saddle points.
  
  **Bounds Issue**: Although I successfully found the global minimum, the path of my gradient descent was going out of the function's bounds. To address this, I used `np.clip()` on both `x` and `y` to ensure they remained within the function's bounds.
  
  I am unsure whether using both gradient clipping and value clipping (with `np.clip()`) is the correct approach. Is gradient clipping alone sufficient to handle out-of-bounds values, or is it necessary to also clip the variable values? Additionally, could clipping the variables potentially mask underlying problems?
  
  Output with Gradient and Variable Clipping (desired output):
  [![3D plot with gradiend + variable clipping](https://i.sstatic.net/2ZSzDxM6.png)](https://i.sstatic.net/2ZSzDxM6.png)
  
  Output only with Gradient Clipping:
  
  [![3D plot with gradient clipping](https://i.sstatic.net/kESxD3Zb.png)](https://i.sstatic.net/kESxD3Zb.png)
  
  ---
  
  Here is the code for my implementation:
  
  ```
  import numpy as np
  import matplotlib.pyplot as plt
  
  def normGrad(v, max_norm=1):
      norm = np.linalg.norm(v)
      if norm > max_norm:
          v = v * max_norm / norm # Scale the vector
      return v
  
  def GradientDescent(x_start, y_start, n_iterations, momentum=0.8, learning_rate=0.1, tol_threshold=1e-6, clip_threshold=10):
  
      '''
      learning rate: control the step size during updates
      '''
      x_path, y_path = [x_start], [y_start]
      x,y = x_start, y_start
      X_change = 0.0
      Y_change = 0.0
  
      for i in range(n_iterations):
          x_grad = Fx(x,y)
          y_grad = Fy(x,y)
         
          grad = normGrad(np.array([x_grad, y_grad]), max_norm=10)
          x_grad, y_grad = grad[0], grad[1]
  
          X_change = learning_rate * x_grad + momentum * X_change
          Y_change = learning_rate * y_grad + momentum * Y_change
  
          x = x - X_change
          y = y - Y_change
  
          # x = np.clip(x, -10, 10)
          # y = np.clip(y, -10, 10)
  
          # Check for convergence
          # This can help detect if the values are diverging.
          if i > 0 and np.linalg.norm([x, y] - np.array([x_path[-1], y_path[-1]])) < tol_threshold:
              break
  
          x_path.append(x) # append to path
          y_path.append(y)
  
      return x_path, y_path
  
  def f(x,y):
      return x**2*y
  
  def Fx(x,y):
      return 2*x*y
  
  def Fy(x,y):
      return x**2 
  
  x = np.arange(-10,10,0.1)
  y = np.arange(-10,10,0.1)
  X,Y = np.meshgrid(x,y) # array of values containing all value pairs (xi, yi)
  Z = f(X,Y)
  
  s_x,s_y = 5,5
  n_iter = 100
  x_path, y_path = GradientDescent(s_x, s_y, n_iter)
  
  fig, ax = plt.subplots(subplot_kw={"projection":"3d"})
  ax.plot_surface(X, Y, Z,
                  cmap="jet", rstride=1, cstride=1, alpha=0.6)
  ax.contour(X, Y, Z, 10,
             offset=np.min(Z), cmap="jet") # offset=min(Z) set the countor on the xy plane
  z_path = f(np.array(x_path), np.array(y_path))
  ax.plot(x_path, y_path, z_path, 'k.--')
  
  ax.set_xlim([-10, 10])
  ax.set_ylim([-10, 10])
  ax.set_zlim([np.min(Z), np.max(Z)])
  ax.set_xlabel('x')
  ax.set_ylabel('y')
  ax.set_zlabel('z')
  plt.show()
  
  ```
  
  **Tags:** machine-learning, gradient-descent, calculus
  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2024-07-26. Score: 0 | Accepted answer
  
  Gradient clipping just clips the magnitude of the gradient, it has no effect on the values themselves (if you have x=3 and your gradient as g=4, then if you do x - g, you will get -1. You can clip the gradient within [-2, 2], but if you perform two updates, you will still get -1 in the end).
  
  Hence, if you have a constraint optimization problem, then you definitely need to clip the values. Another approach is to use mirror descent or reparametrization.
  See the following blog post which compares all three approaches: [1]: <https://vene.ro/blog/mirror-descent.html#projected-gradient>
  
  I think that in general, when you need gradient clipping, this means that your learning rate scheduling is improper. In fact, gradient clipping can be seen as an adaptive learning rate method. Therefore, I do not think there is any problem in using it to solve large gradient norms (which are dependent on the objective function shape, not on the gradient descent algorithm. So, unless you can change the objective function - which can also be another potential solution -, gradient clipping is perfectly fine!)
  
  
  ---
  
  