# Rigorous proof of the good regulator theorem

As an applied mathematician with an interest in control theory, I have read several research papers that explicitly use the good regulator theorem of Conant and Ashby 1 which states that:

Every good regulator of a system must be a model of that system.

As pointed out by the authors of [3], the importance and generality of this theorem in control theory makes it comparable in importance to Einstein's $$E=mc^2$$ for physics. However, as John C. Baez carefully argues in a blog post titled The Internal Model Principle it's not clear that Conant and Ashby's paper demonstrates what it sets out to prove. I'd like to add that many other researchers, besides myself, share John C. Baez' perspective.

At present, I think an information-theoretic approach may be used to demonstrate a general version of the good regulator theorem and in the near future I will probably attempt such a demonstration. Meanwhile, might there be control theorists on the MathOverflow that know of a proof of the good regulator theorem that is both clear and rigorous?

Such a publication, if it exists, would be of interest to applied mathematicians working on control-theoretic problems, researchers in the area of behavioural neuroscience as well as artificial intelligence researchers.

References:

1. Roger C. Conant and W. Ross Ashby, Every good regulator of a system must be a model of that system), International Journal of Systems Science 1 (1970), 89–97.
2. B. A. Francis and W. M. Wonham, The internal model principle of control theory, Automatica 12 (1976) 457–465
3. Daniel L. Scholten, Every good key must be a model of the lock it opens (the Conant & Ashby Theorem revisited), 2010.
• Have you made any progress on devising "an information-theoretic approach ... to demonstrate a general version of the good regulator theorem"? May 19, 2021 at 22:46
• @FrankH there is the work of Karl Friston which appears to lead to useful algorithmic abstractions in an information-theoretic setting: nba.uth.tmc.edu/homepage/cnjclub/2010Spring/Friston%202009.pdf May 20, 2021 at 0:01
• @FrankH you might also be interested in recent work pursued by AI researchers at Google that builds upon Friston's framework: arxiv.org/pdf/2009.01791.pdf May 20, 2021 at 0:03
• Thanks for the links, they look interesting... Do you think that these are the equivalent of the good regulator theorem? May 28, 2021 at 6:42
• @FrankH that's a good question. I wrote a response in the form of an answer because it is a bit too long for a comment. May 29, 2021 at 13:07

There are two principled information-theoretic approaches to identifying a good regulator for an organism's environment. Within the context of reinforcement learning, both approaches are unsupervised in the sense that organisms don't need an explicit reward function. Instead of maximising a reward function, the organism's objective is to construct a suitable model of their environment for decision-making and planning.

## Empowerment:

The empowerment approach, developed by Daniel Polani and his collaborators consists of approximating the channel capacity in the perception-action loop:

$$$$C = \max_{p(x)} I(X;Y)$$$$

where $$X$$ represents the space of possible actions and $$Y$$ represents the sensory space(i.e. percepts). One practical challenge facing this approach is that it generally requires identifying an environment's state-transition function which is intractable for large and complex state-spaces. That said, scientists at Deep Mind have recently demonstrated that variational approximations of the channel capacity $$C$$ address certain scalability issues [2].

## Free Energy Principle:

The Free Energy Principle, developed by Karl Friston rests upon the ergodic assumption that self-organising biological agents minimise the expected surprise or the entropy of their sensory states:

$$$$H(y) = - \int p(y|m) \ln p(y|m) dy = \lim_{T \to \infty} \frac{1}{T} \int_{0}^T - \ln p(y|m) dt$$$$

where $$m$$ represents the model used by the agent and $$y$$ represents its sensory input.

Some scientists have pointed out that Friston's theory is a theory of cognitive dissonance minimisation which leaves it vulnerable to the dark room problem. However, scientists at Google Brain and Google DeepMind recently argued that the Free Energy Principle is a theory of niche construction which forces organisms' to find the ecological niche they are most suited for [5].

## Challenges facing a definitive proof:

An important challenge facing any mathematical formalism for good regulators is that the practical consequences of such a formalism in complex environments requires experimentation which escapes the analytical process.

## References:

1. Christoph Salge, Cornelius Glackin and Daniel Polani. Empowerment — An Introduction. 2013.

2. Shakir Mohamed and Danilo J. Rezende. Variational Information Maximisation for Intrinsically Motivated Reinforcement Learning. 2015.

3. Karl Friston. The free-energy principle: a rough guide to the brain? Cell press. 2009.

4. Zekun Sun & Chaz Firestone. The Dark Room Problem. Cell Press. 2020.

5. Danijar Hafner et al. Action and Perception as Divergence Minimization. 2020.