axtreme.utils.gradient

Helper for gradient assessment.

NOTE: currently just used as a helper in tests. Could move their if we do not consider it useful to users

Functions

is_smooth_1d(x, y[, d1_threshold, ...])

Helper to warn if 1d function is likely not smooth (twice differntiable).
axtreme.utils.gradient.is_smooth_1d(x: Tensor, y: Tensor, d1_threshold: float = 3.0, d2_threshold: float = 150, *, plot: bool = False, test: bool = True) → None | Figure

Helper to warn if 1d function is likely not smooth (twice differntiable).

Parameters:

  • x – (n,) points representing the x (input) values of a function

  • y – (n,) points representing the y (output) values of a function

  • d1_threshold – Maximum step size allowed in 1st derivative function to be considered smooth.

  • d2_threshold – Maximum step size allowed in 1st derivative function to be considered smooth.

  • plot – If true, will plot the first and second derivative functions.

  • test – is assert statments should be run

Details
Smoothness: (defined up to K, the Kth derivative you can take that give a contious funciton over domain)

  • C_0: set of continous functions

  • C_1 (once differentiable):

    • C_0 can be differentiated

    • AND the resulting function is continuous (No steps or holes)

  • …

We want C_2 (twice differentiable) for optimisation. As such here we test.
  • if f’(x) is continous.

    • Check if there are not step = not big changes between points.

    • Knowing the appropriate step threshold is hard, the gradient is easier to think about.

      • Easy way to do this is check if max slope is not exceeded (derivate 2nd)

  • f’’(x) is continous.

    • Check if there are not step = not big changes between points

    • Easy way to do this is check if max slope is not exceeded (derivate 2nd)

NOTE: The thresholds have been set heuristically. It is recomended to plot your function and establish smoothness, the use this for regression testing once suitable values have been determined.

NOTE: The smaller the step size, the better the estimate of gradient (large and small). See y = sim(50 * x)` with torch.linspace(0,1,100)` and torch.linspace(0,1,1000)