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I have two friendly functions $g,h:\mathcal{X}\subseteq\mathbb{R}^N\to\mathbb{R}$ whose exact properties I'm somewhat flexible on. Maybe for starters they are lower semi-continuous and convex. Their proximal operator with respect to a norm $\Vert\cdot\Vert_\ell$ is given by:

$$ \textrm{prox}_g^{\ell}(x_0) = \underset{x\in\mathcal{X}}{\textrm{argmin}} \,\, \frac{\Vert x-x_0\Vert^2_\ell}{2} + g(x)\,, $$ and likewise for $\textrm{prox}_h^{\ell}$.

Let $\Vert a\Vert_s^2 = s\Vert a \Vert_2^2$ and $\Vert a\Vert_{\mathbf{C}}^2 = a^\top\mathbf{C}a \,,$ where $\mathbf{C}$ is a positive definite matrix.

Say I have a bound on $\Vert \textrm{prox}_g^{s}(x_0) - \textrm{prox}_h^{s}(x_0) \Vert_2$, i.e. the difference between proximal operator actions on $x_0$ with scaled Euclidean norm. Can I exchange it for a bound on $\Vert \textrm{prox}_g^{\mathbf{C}}(x_0) - \textrm{prox}_h^{\mathbf{C}}(x_0) \Vert_2$, i.e. the proximal operators with $\mathbf{C}$-norm?

That is, I'm interested in some bound of the form: $$ \Vert \textrm{prox}_g^{\mathbf{C}}(x_0) - \textrm{prox}_h^{\mathbf{C}}(x_0) \Vert_2 \leq \rho(C,s) \Vert \textrm{prox}_g^{s}(x_0) - \textrm{prox}_h^{s}(x_0) \Vert_2 \,. $$ I have used the notation $\rho$ to suggest something reminiscent of a condition number.

Do you know of such a bound, or a good reason why we should not expect one? Does the answer change if $s$ is some function of $\mathbf{C}$, for instance, its largest eigenvalue?

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  • $\begingroup$ @Daniel Shapero in an extended comment suggested an interesting proof strategy for a subproblem. I have nevertheless placed a bounty on this question because I suspect that there is some community out there to which the answer is already well known; I further suspect that we can get the answer using primarily convex-analytic tools (but I'm flexible on this). $\endgroup$ Commented Jul 1 at 18:42

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This is just a guess and maybe doesn't cover your whole question but it's too long for a comment. If I had to take a crack at this I'd try the following. Define the functional

$$L(x, K) = \frac{1}{2}\langle K^{-1}(x - x_0), x - x_0\rangle + g(x) - \alpha\log\det K.$$

Then

$$\text{prox}_g^C(x_0) = \text{argmin}_xL(x, C^{-1}).$$

The extra log-determinant term doesn't affect the prox computation. I am... mostly? confident that, provided $g$ is convex, then $L(x, K)$ is jointly convex on the product of the space $\mathcal X$ and the space $\mathcal S$ of all symmetric and strictly positive-definite matrices. The inverse is crucial for the joint convexity. There's a natural metric on $\mathcal S$ which is invariant under inversion.

Now you can imagine seeing what happens along a path $K_\lambda = (1 - \lambda)K_1 + \lambda K_2$ between two matrices $K_1$ and $K_2$. Finally take $K_1 = I$ and $K_2 = C^{-1}$. Getting what you want is probably going to involve using the implicit function theorem and a homotopy continuation between $I$ and $C^{-1}$.

Wish I could give you more, hope this is useful.

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    $\begingroup$ very interesting thanks; i'll see if I can squeeze any juice out of this $\endgroup$ Commented Jun 30 at 23:41

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