1
$\begingroup$

Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*$ (assuming that $T$ is strictly contractive so that the fixed point exists and is unique): $$ f^* = \mathcal{T}f^*. $$ One motivating example is Q-iteration in Markov decision process, where $$ f(s, a) \leftarrow \mathcal{T} f(s,a): =r(s) + \gamma \mathbf{E}_{s'\sim p(\cdot | s, a)} \left\{\max_{a'\in \mathcal{A}}f(s', a')\right\}\quad (\gamma<1). $$ Here $s\in \mathbb{R}^d$ and $a \in \mathcal{A}$ denote the state and action, where $\mathcal{A}$ is a discrete set. Meanwhile, $r(s): \mathbb{R}^d \rightarrow \mathbb{R}$ is the reward function, $p(\cdot | s, a)$ is the transition kernel, that is, a conditional distribution of the next state $s'$ given the current state and action, and $a'$ is the action taken at the next state $a'$.

Without the max operator, the above fixed-point iteration corresponds to a linear operator. In such a simplified case, what can we say about the properties of the fixed point $f^*$, assuming properties of the operator, which depends on the transition kernel and reward function? (For example, the smoothness of $f^*$, or more generally, its Sobolev- or Hölder-norm.)

What about the case with the max operator? More generally, what about general strictly contractive nonlinear operators?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.