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Kolmogorov complexity makes reward learning worse
discussion post by Stuart Armstrong 18 days ago | discuss

A putative new idea for AI control; index here.

In a previous post, I argued that Kolmogorov complexity/simplicity priors do not help when learning human values - that some extreme versions of the reward or planners were of roughly equal complexity.

Here I’ll demonstrate that it’s even worse than that: the extreme versions are likely simpler than a “reasonable” one would be.

Of course, as with any statement about Kolmogorov complexity, this is dependent on the computer language used. But I’ll aim to show that for a “reasonable” language, the result holds.

So let \((p, R)\) be a reasonable pair that encodes what we want to encode in human rationality and reward. It is compatible with the human policy \(\pi_H\), in that \(p(R)=\pi_H\).

Let \((p_r, R_r)\) be the compatible pair where \(p_r\) is the rational Bayesian expected reward maximiser, with \(R_r\) the corresponding reward so that \(p_r(R_r)=\pi_H\).

Let \((p_i, 0)\) be the indifferent planner (indifferent to the choice of reward), chosen so that \(p_i(R')=\pi_H\) for all \(R'\). The reward \(0\) is the trivial reward.

Information content present in each pair

The planer \(p_i\) is simply a map to \(\pi_H\), so the only information in \(p_i\) (and \((p_i, 0)\)) is the definition of \(\pi_H\).

The policy \(\pi_H\) and the brief definition of an expected reward maximiser \(p_r\) are the only information content in \((p_r, R_r)\).

On the other hand, \((p, R)\) defines not only \(\pi_H\), but, at every action, it defines the bias or inefficiency of \(\pi_H\), as the difference between the value of \(\pi_H\) and the ideal \(R\)-maximising policy \(\pi_R\). This is a large amount of information, including, for instance, every single human bias and example of bounded rationality.

None of the other pairs have this information (there’s no such thing as bias for the flat reward \(0\), nor for the expected reward maximiser \(p_r\)), so \((p, R)\) contains a lot more information than the other pairs, so we expect it to have higher Kolmogorov complexity.



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