Constrained MDPs and the reward hypothesis

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It's been a looong ago that I posted on this blog. But this should not mean the blog is dead. Slow and steady wins the race, right? Anyhow, I am back and today I want to write about constrained Markovian Decision Process (CMDPs). The post is prompted by a recent visit of Eugene Feinberg , a pioneer of CMDPs, of our department, and also by a growing interest in CMPDs in the RL community (see this , this , or this paper). For impatient readers, a CMDP is like an MDP except that there are multiple reward functions, one of which is used to set the optimization objective, while the others are used to restrict what policies can do. Now, it seems to me that more often than not the problems we want to solve are easiest to specify using multiple objectives (in fact, this is a borderline tautology!). An example, which given our current sad situation is hard to escape, is deciding what interventions a government should apply to limit the spread of a virus while maintaining economic
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A notion of function compression

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The following compressibility concept is introduced by Harnik and Naor in their recent paper: Given a function $f$ over some domain, a compression algorithm for $f$ should efficiently compress an input $x$ in a way that will preserve the information needed to compute $f(x)$. Note that if $f$ is available (and efficiently computable) then compression is trivial as then $y=f(x)$ will serve the purpose of the compact representation of $x$. Actually, the concept was originally studied in the framework of NP decision problems where unless $P=NP$ $f$ is not efficiently computable, hence the trivial solution is not available.

I am wondering if this compressibility notion could be used in learning theory or function approximation? Consider e.g. classification problems so that the output of $f$ is $\{0,1\}$. In order to prevent the trivial solution we may require that $x$ be compressed to some $y$ such that for some fixed (efficiently computable) function $g$, $f(g(y))=f(x)$. We do not require that $g(y)=x$, though if $g$ satisfies this then we certainly get a solution: if a compact representation of $\{ x| f(x)=1\}$ is available then $f$ can be well-compressed. We might allow of course for some error and study approximation rates. Of course, the notation applies to real-valued or more general functions, too. What is the class of functions that can be efficiently compressed? How is this class related to classical smoothness spaces (in the case of real-valued functions)? One obvious thought is that the smoother the target function is, the better the obvious discretization approach would be. It would be interesting to find out whether this new approch could lead to new regularity descriptions by varying the restrictions on the compression.

Another related thought is to compress the training samples in a learning problem such that the solution is kept (roughly) the same.

Comments

  1. Christos Dimitrakakis17 September 2013 at 08:18

    I think this question has been answered in some ways in algorithmic complexity/probability and at a more concrete instantiation, the minimum description length literature.

    There is an additional element here, since in MDL you typically just want to compress $x \in X$ (or a sequence of $x_i$). However, there might be additional gains in our case, since we only care about preserving those aspects of $x$ required to approximate $f$.

    Then I suppose that the question becomes in what sense you want to approximate $f$ - if you want something that works uniformly over $X$ then why should what the original input matter at all?

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