Backpropagation is a leaky abstraction (2016)

Karpathy suggests the following error:

  def clipped_error(x): 
    return tf.select(tf.abs(x) < 1.0, 
                   0.5 * tf.square(x), 
                   tf.abs(x) - 0.5) # condition, true, false

Following the same principles that he outlines in this post, the "- 0.5" part is unnecessary since the gradient of 0.5 is 0, therefore -0.5 doesn't change the backpropagated gradient. In addition, a nicer formula that achieves the same goal as the above is √(x²+1)

If we don't subtract from the second branch, there will be a discontinuity around x = 1, so the derivative will not be well-defined. Also the value of the loss will jump at this value, which will make it hard to inspect the errors, for one thing.