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by Vadim Kosoy 30 days ago | link | parent

Your confusion is because you are thinking about regret in an anytime setting. In an anytime setting, there is a fixed policy \(\pi\), we measure the expected reward of \(\pi\) over a time interval \(t\) and compare it to the optimal expected reward over the same time interval. If \(\pi\) has probability \(p > 0\) to walk into a trap, regret has the linear lower bound \(\Omega(pt)\).

On other hand, I am talking about policies \(\pi_t\) that explicitly depend on the parameter \(t\) (I call this a “metapolicy”). Both the advisor and the agent policies are like that. As \(t\) goes to \(\infty\), the probability \(p(t)\) to walk into a trap goes to \(0\), so \(p(t)t\) is a sublinear function.

A second difference with the usual definition of regret is that I use an infinite sum of rewards with geometric time discount \(e^{-1/t}\) instead of a step function time discount that cuts off at \(t\). However, this second difference is entirely inessential, and all the theorems work about the same with step function time discount.



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