Inf Norm giving nan gradient via autograd

[Jason Brown]

Hello.

I am trying to use pymanopt to minimize an infinity norm linear algebra problem. I am paraphrasing some code but essentially I generate some matrix R and want my problem setup to be 

import pymanopt

import autograd as ag

manifold = Sphere(500)

def cost(X): return ag.linalg.norm(R @ X, ord=np.inf)

problem = Problem(manifold=manifold, cost=cost)

    

solver = SteepestDescent()

Xopt = solver.solve(problem)

    

Running any code with the ord = np.inf immediately gives a nan for the first gradient and there is clearly no convergence. I am unsure if this is a user error or rather an error with autograd, but it works as intended for ord = 2 and ag.sum(). I have tried installing tensorflow but I suspect it may be interfering with my packages as after installing all the ones I need, I am unable to install Spyder to actually run the code. Theano also gave an error as it seems it is not compatible with Python 3.9 from the error message.

Any help would be appreciated as I would think the infinity norm would be a rather standard for cost functions yet I seem unable to reach any convergence.

Thanks - 

[Jason Brown]

To be more specific, the following code works when ord = 2, but breaks when ord = np.inf and I do not understand why. 

import autograd.numpy as ag

from pymanopt.manifolds import Sphere

from pymanopt.manifolds import Euclidean

from pymanopt import Problem

from pymanopt.solvers import SteepestDescent

import numpy.random as random

import numpy as np

R = random.randn(10,10)

manifold = Euclidean(R.shape[1])

def cost(X): return ag.linalg.norm( R @ X, ord = 2)

problem = Problem(manifold=manifold, cost=cost)

solver = SteepestDescent()

    # let Pymanopt do the rest

Xopt = solver.solve(problem)

[Jason Brown]

Update: I believe I'm able to workaround this by simply using ag.amax( ag.abs( R @ X)), but I will leave the question up because it might help for others or for future use.

[Nicolas Boumal]

Hello,

As both the 2-norm and in infinity-norm are nondifferentiable at some points, it's not unreasonable that automatic differentiation behaves erratically. This cannot be truly fixed, except by smoothing the norms (e.g., soft-max for the infinity norm).

Glad to hear that your workaround mostly does the trick! After all, the norm is differentiable almost everywhere, so it does make sense that the computation would go through most of the time, depending on how the computations are arranged.

Best,

Nicolas