How I’d build an intelligent machine: a position paper

Finally, after weeks of work, the position paper is written! It contains a hopefully clear presentation of my ideas about AGI and a tentative outline how to build it. Franz (2015) Artificial general intelligence through recursive data compression and grounded reasoning

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Hierarchical epsilon machine reconstruction

Having read this long paper by James P. Crutchfield (1994) “Calculi of emergence”, I have to admit, that it is very inspiring. Let’s think about the implications.

The first achievement of the paper is the definition of statistical complexity. Note that Kolmogorov complexity is a misnomer, since it rather measures randomness than complexity. Complexity on the other hand, seems to be somewhere between simplicity and randomness. Crutchfield measures it by taking a string and mapping each entry on a set of states where two entries have the same state and the predict the same distribution of futures conditioned on the past. Given that definition, the string is mapped essentially on a Markov chain. Since Markov chains possess a marginal distribution of states after running for a long time, the entropy of this distribution is the definition of statistical complexity.

However, this definition is only appropriate, if we get a finite description. In many cases however, a Markov chain or finite automaton, is not sufficient. Essentially, when trying to increase modeling precision, such as looking at ever longer past histories and future predictions, the number of states grows without bounds. When it grows to large, the system is pressed to “invent” a new computational class by grouping the states together to a new automaton, e.g. one with a register or other forms of memory. And, it is suggested to do that until a finite description is achieved. The statistical complexity is defined as the Shannon entropy of the state distribution in this highest description (lowest level finite description to be precise).

The cool part is that it somehow achieves filtering out the random part, somehow even including the pseudo-random part. After all, a theme running through the paper is also the following. The dichotomy between simplicity and randomness is somehow paralleled by the dichotomy between order and chaos. This parallel has been intriguing me for a long time already. The last time was when I asked myself, which strings are compressible, but are not incrementally compressible, i.e. that don’t have any properties. Another reason is that throughout the experience in machine learning, computation seems to happen best at the edge between order and chaos. The failure of Kolmogorov “complexity”, let’s call it (algorithmic) entropy, is evident here as well: chaotic sequences often have a low algorithmic entropy, even though they appear random. They appear so that convincingly that we use chaotic sequences in order to generate pseudo-random numbers in our computers. Somehow statistical complexity gets rid of those complications, slices away the random part and uncovers the “true” computational aspects of the system.

Another aspect of the discussion is the interesting interplay between different perspectives: an observer modeling a natural system / data set, the AI aspiration of automatizing this very modeling as a means to implement universal induction and nature’s creation of living beings that model their environments while getting ever more complex. The latter aspect is quite mesmerizing because of the interplay between order and chaos in nature, since nature seems to be a processes of self-organized criticality which means that the edge/border between order and chaos is an attractor, i.e. a stable state. If that state is exactly where innovation happens then it could explain why living beings become increasingly complex. And here is another contribution of that paper to that aspect: it is at the edge of chaos where the number of states tends to diverge into infinity. After all, both orderly/regular and chaotic regimes seem to lead to finite descriptions. But it is the divergence of state numbers that brings the modeling to its resource limits, forcing it to switch to a higher representation, a more powerful machine, a higher complexity level.

A yet another inspiring aspect of the paper is the parallel between “state grouping”, which is referred to as some processing that models the symmetries of the data, and detecting features in my incremental compression (IC) theory. I mentioned above that I suspect that it may be precisely the complex strings that are incrementally compressible. Given that hierarchical epsilon machine reconstruction is declared as a process the incrementally detects symmetries, the parallel is fairly strong. Here, in the questions about incremental/hierarchical construction, in the interplay between order and chaos, is a deep mystery to be uncovered, about evolution, the origin of complexity in nature and the essence of intelligence. I am quite confident about this.

There seems to be a deep parallel between the evolutionary incremental innovation process and the incremental process of induction. In the paper, innovation seems to consist of two parts. One is state grouping, which is a well-defined procedure, independently of the data. The other part is the “true novelty” which is found by random search in a small space, which results from the state grouping. In IC-theory no such random search is necessary once the regularity/feature/state grouping is found. Another aspect that comes to my mind is that this hierarchical epsilon machine process will often lead to a high hierarchy for natural data, since there is structure on all scales, hence there we have this open-ended hierarchical construction process. This reveals the role of the modeling precision: the more precisely we model, the more we sort-of zoom in on the data, the higher our hierarchies will have to become. Hence, it is not just the living beings that become ever more complex during evolution, but it is the structure of matter itself that organizes itself in such a way that it can only be modeled by a high hierarchy. The author views living beings as “dual” to their environment (matter), which is what it means to fill a “niche” in the environment. Thus it seems that this self-organizing criticality (SOC) process of nature forces matter into complex states which forces living beings to increase in complexity as well if they want to model those states properly in order to predict their environment and survive. Unfortunately, SOC is not sufficient for complexity, as evidenced by the simplicity of the showcase SOC model: the sand pile. Therefore, we don’t really know how nature forces matter into complexity.

Another interesting aspect of the paper is the following. The process of hierarchical epsilon machine reconstruction demands to build a machine by grouping those histories that entail the same future distributions. By increasing the precision, i.e. decreasing epsilon, the length of the look-back and look-ahead sequence increases which increases the number of reconstructed states. If this number grows without bounds as precision increases then it is advised to switch to a more powerful machine by grouping the new states again or some other innovation and do so repeatingly until a finite representation is reached.

This process reminds me of the “universal approximation” property of neural networks, see previous post on my ideas on that. The number of neurons generally also grows without bounds with increasing precision. Could it also be due to the simple fact that the present representation is not powerful enough? Maybe the particular neural network has to switch to some more powerful computation class? After all, my observations in that post implied that universal approximation is not the same as algorithmic completeness. Crutchfield’s paper helps to flesh out the difference: a simple representation can often approximate the data arbitrarily precisely, but at the expense of a large number of parameters, while more powerful computational classes can find a finite representation. The paper suggests a way of doing this, but I’m afraid that this “state grouping” procedure is too specific. Can we formally and more precisely describe what happens here?

Now that I think about it, there are several ways, in which hierarchical epsilon machine reconstruction (HEMR) differs from IC theory. First, although the author claims that the goal of state grouping is to find symmetries (features) of the data, this is not what happens. Rather, the attempt is to find a full description of the data in a single step in the hierarchy by using a not-so-powerful machine. After all, the goal of HEMR is to separate the random part from the “intrinsic computation” part, i.e. signal from noise. In IC theory, however, one tries to decompose the signal part into various features one of which will turn out to be noise which is carried through the parameters up to the highest layer since it doesn’t have features and is not compressible. Thus, IC decomposes the signal into crisp separate parts using functions/features from a Turing-complete set right at the beginning, while HEMR tries to model the whole signal at once but uses a not-so-powerful machine and steps up in the computation hierarchy if the machine doesn’t do it. If we combine IC and HEMR in a smart way, we might get a procedure that decomposes a string into parts which can be looked for by increasingly powerful machines. This would alleviate the IC-problem of having to search for features in a Turing complete space and the HEMR-problem of having to find a full description in a single step. Could we posit HEMR on the firm grounds of algorithmic information theory?

And what is the fundamental reason for the statistical complexity to peak at intermediate entropies? It seems that this peak corresponds to the most complex intrinsic structure which has to be modeled by the more powerful machines.

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Curious abstractions: how do we know that circles don’t have corners?

A simple but deep question. Why do we want to know this?

In order to speed up compression we have mainly considered two ideas, exploiting the compositionality of functions that generate data (incremental compression) and exploiting power laws in natural data (hierarchical compression). When looking at the human visual perception hierarchy we realize that in the lowest level of the hierarchy a very narrow set of functions is used. We mainly have edge detectors for various orientations and moving speeds and a few others like centre-surround cells, end-stopped cells, grid cells etc., in order words, a set of functions far too narrow for processing a wide range of data.

What’s the trick? Having a narrow set of functions definitely increases the ability to reduce data dimension if it is possible to generate the data by those functions. Let’s say, we only have edge detectors and we have to perceive a circle. How do we do that? It looks like our visual system is not particularly prepared to recognize arcs of various curvature. Then only way that comes to mind is to approximate the circle by a set of lines, which form a regular polygon. However, every such polygon has corners. So how do we know that a circle doesn’t have corners when all we can perceive are things made up of straight lines? After all, we know the difference between a regular 100-gon and a circle, although our representations can not detect a difference.

The reason is probably that there are many ways to approximate a circle with a polygon. The existence of edges and their position and quantity directly depend on the choice of approximation. They are a property of the approximation, not of the world. In practice, the system would try to fit a polygon and will realize that there are many possibilities to fit a polygon with arbitrary numbers of corners and lengths of the connecting lines. Also, the system will realize that those are not arbitrary polygons but quite regular ones, whose corners always have the same distance to the centre, which is an achievement of compression. Nevertheless, it is not exactly clear to me, how the “cornerlessness” property is inferred.

Solving this class of problems seems important to me since it would allow to achieve quite general induction abilities while using a narrow set of detectors are lower levels of processing that can reduce the dimensionality of the input significantly.

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Universal approximators vs. algorithmic completeness

Finally, it has dawned on me. A problem that I had troubles conceptualizing is the following. On the one hand, for the purposes of universal induction, it is necessary to search in an algorithmically complete space. This is currently not possible because of the huge number of possible programs to search through. My mind then automatically wandered toward algorithms that don’t go through the trouble of searching through the whole search space available to them. Any parametrized machine learning algorithm, be it neural networks or linear regression, finds solutions by gradient descent or even simpler, but none bothers to loop through all weight combinations.

The next thought was that this kind of efficient search is only possible within a fixed algorithmically incomplete representation. However, it is well-known that neural networks are universal function approximators, which sounds like algorithmic completeness, does it not? After all, Turing machines merely compute a integer functions. Why then would it be so hard for neural networks to represent some structure not optimally suitable for them? For example, a circle can be represented by the equation x^2+y^2=const, which is difficult to represent with, say, sigmoidal neurons. I attributed this difficulty to the fact that there are different reference machines after all and some things are easy to represent with one machine but difficult in another and vice versa.

But now I realize that there is something fundamentally different between universal function approximators and algorithmically complete languages. After all, polynoms are also universal approximators, too! But ever tried to fit a polynom to an exponential function? It will severely overfit and never generalize.

So, what is the difference between an algorithmically incomplete universal function approximator and a sub-optimal algorithmically complete reference machine? While the latter differs from another alg. complete machine merely by the constant length needed to encode a compiler between those machines, the former requires ever more resources when the approximation precision is increased. For example, every alg. complete language will be able to represent the exponential function exactly; some languages will require longer, some shorter expressions, but the representation will be exact. And the additional length in some languages is merely due to the fact that some functions are cumbersome to define in them, but once done, your function will be represented exactly. Conversely, a universal non-complete approximator will require ever longer representations when you increase precision and goes even to infinity when precision requirements are highest. More precisely, if \epsilon is the maximum approximation error that you allow and N is the size of the network N(\epsilon) increases as \epsilon decreases. My point is that it should not depend on \epsilon. For example, if you can represent the data x perfectly in some representation with M bits, then your representation should take N = M + \mbox{const.} bits where the constant does not depend on x (i.e. const is just the length of the compiler that translates between the two representations). And this is just not the case for neural networks. And that is also why the Taylor expansion of the exponential function is an infinite degree polynomial — polynomials are simply not suited to represent the exponential function.

But why should precision matter? Why do we care about this effect? Should we not rather care about the generalization abilities? For example, imagine a family of exponentials e^{-x/d}, where d is the width of the exponential. If I fit a polynom to it, I would need a different set of parameters of the polynom for each value of d! Therefore, polynoms don’t really capture the structure of the function. On the other hand, an alg. complete language will always have such a single parameter available and once the representation is constructed, the whole family is covered. That’s actually the point. In order to generalize, it is necessary to recognize the sample as an example of a family that it belongs to and to parametrize that family. And neither neural networks nor polynoms can parametrize the exponential family, without picking completely different infinite (not single!) parameter sets for the respective polynomials. An alg. complete language, however, can do this.

I think, essentially, the criterion is: if your data is drawn from a class which is parametrizable with n parameters in some Turing complete representation, then your representation should be able to express that same data with not (much) more than n parameters. Otherwise, it is “just an approximation” which may fit the data well, but will certainly overfit, since it will not capture the structure of the data. This follows from universal induction. If your data x=yz is computed by a program p, U(p)=x, then taking a longer program q that computes some prefix y of x, U(q)=yw, will hardly make q able to predict the rest of x: w will not equal z. And it is well known, that the shortest program computing x is the best predictor with high probability, so that every bit wasted in program length reduces your prediction abilities exponentially, due to the 2^{-l(p)} terms in the Solomonoff Prior!

Another point is that algorithmic completeness may not even be necessary, if the data created by Nature comes from an incomplete representation (if the world consisted only of circles, then you only need to parametrize circles in order to represent the data and generalize from them perfectly). That this is the case is not even improbable, since physics imposes locality and criticality (power law distributed mutual information in natural data), which probably makes the data compressible by hierarchies (as reflected by my theory of incremental compression. However, this class is still quite large, and we should be able to capture its structure appropriately.

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My best paper

I have presented this paper in the AGI conference in New York this year.

Some theorems on incremental compression

It presents a general way of speeding up the search of short descriptions of data that is made up of features – which is what our world is made of. This may well lead to a tractable solution of the problem of universal inductive inference.

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The merits of indefinite regress

The whole field of machine learning, and artificial intelligence in general, is plagued by a particular problem: the well known curse of dimensionality. In a nutshell, this curse means that whenever we try to increase the dimension of our search spaces in order to make our models more expressive we run into an exponentially increasing number of search points to visit until a satisfactory solution is found.

In practice, people are then forced to use smaller models with manageable parameter spaces to search through which sets up the whole problem of “narrow AI”: we end up being able to solve merely narrowly defined specific tasks.

We should know better though than searching more or less blindly through vast search spaces. Both decades of practice in machine learning, our philosophical thinking, scientific theories and the theory of computer science have demonstrated undeniably the validity of a well-known principle, Occam’s razor: “Entities must not be multiplied beyond necessity” (Non sunt multiplicanda entia sine necessitate). Especially in recent decades, Solomonoff’s theory of universal induction has shown that it is supremely important to find simple solutions to problems, i.e. that simple solutions are a priori exponentially more probable than complex ones.

Of course, we have known that before already and have been sincerely trying to reduce the size of our models and the number of their parameters. We have introduced various information criteria like the Akaike Information Criterion, the Bayesian Information Criterion, various penalty terms for model sizes. However, this is by far not good enough, as I will argue and will suggest a solution to it.

The problem with introducing a simplicity bias comes from the fact that we don’t know how to compute simplicity or “complexity” of any data set. Fortunately, we do have a formal definition of “complexity” – the Kolmogorov complexity, which I will call algorithmic entropy (since it actually is more related to entropy than to complexity, after all randomness is usually not considered complex in the usual way we use the word). The algorithmic entropy of a data string x is given by

C(x)=\mbox{min}(l(p):\; U(p)=x)

which is the length l(p) of the shortest program p being able to compute that data set on a universal Turing machine U. In practice, in order to know the algorithmic entropy of a data set, we’d have to find the shortest program being able to generate it and measure its length (the number of bits it takes to specify the program).

However, finding a short description is the reason why we started this whole thing in the first place: we need a the algorithmic entropy in order to evaluate the appropriateness of candidate solutions but we need the solutions in order to compute their entropy – a hen-egg problem. And as most hen-egg problems in computer science, this one may also be solvable in an iterative way.

Consider a typical set up of a machine learning problem. We have chosen some model and want to fit its parameters to the data using some cost function. The size of the model is defined by the number of parameters which we can vary. For example, a the size of a neural network can be regulated by the number of neurons involved which affects the number of weights that need to be fitted.

Ideally, we do not want to go through the parameter space blindly but to try first “simple”, low entropy, parameter settings. For example, a convolutional layer of a neural network consists of batches of a small number of weights per neuron where each neuron has got the same set of weights. “Simple” means that the program describing the weights will be short since it only needs to specify the small number of weights of a single neuron. By going through the simple first we ensure to fulfil the Occam’s razor principle and stop as soon as we find a solution that fits the data well.

The problem is, however, the following. For a fixed size of the parameter set, which can be represented as a binary string of a fixed length, the fraction of simple strings is very low. After all, a simple counting argument shows that there are at most 2^{n-c+1}-1 programs of length at most n-c, i.e. that are able to compress the string by at least c bits. Since there are 2^n bits of length n, the fraction of programs that compress a string by c bits goes like 2^{-c+1}. There are two things to learn from this. First, the fraction of simple parameter descriptions does not depend on the length of the parameter set n, i.e. the number of parameters. And second, the fraction drops very fast, exponentially with every compressed bit. Hence, the number of simple descriptions is very low.

Therefore, if a successful search of the parameter space with appropriate bias toward simple parameter sets is to be performed, we’d rather find a way to fish out those very few simple sets out of a vast number of random ones. Here, we also note that merely reducing the number of parameters does not help much to find the simple ones: the fraction of simple ones remains constant for any number of parameters.

To get an intuition about the problem consider a parameter set in the form of a 100 bit string. Searching through all 2^{100} is tedious, since it is a large number. We would like to go through all strings with an algorithmic entropy less than 50 bits. Running up to 2^{50} 50 bit programs that print 100-bit strings is now manageable and we majorly reduce our parameter search space while giving priority to simpler parameter sets. However, there are two problems: more complex parameter sets cannot be represented that way and the 50 bit programs are themselves not ordered by their entropy. Thus, only a small fraction of them is simple and we end up searching blindly through a mostly random program space – it is the same problem as with the parameters. Now, if we represent the programs by a model with its parameters, we can proceed the same way as before: build another model on top of it that generates simple parameter sets. In essence, in order to achieve a simplicity biased search through the parameter space, it seems like a good idea to build a meta-model and a meta-meta-model and so on – an indefinite regress of models!

One may ask oneself why building a 50 bit meta-model to print 100 bit parameters of the first-level model to explain data? Why not taking simply 50 bit parameters without a meta-model which corresponds to the same entropy of the whole model pyramid? Since 100 bit parameters are simply more expressive, even though a small fraction, only 2^{50} sets are considered.

How could this Occam biased search be performed? Here is an outline:

  1. Take model M with a variable set of parameters and the data set \vec{p}_0=\vec{x} to be generated. Set l=0.
  2. Set l\leftarrow l+1, n_l=1 and define a new meta-model that estimates the input: \hat{\vec{p}}_{l-1}=M(\vec{p}_l)
  3. Use \vec{p}_l to compute an estimate of the input \hat{\vec{x}}=M(\vec{p}_1)=M(M(\vec{p}_2))=M(\cdots M(\vec{p}_l)) and use your favourite search algorithm to minimize the objective function E(\vec{p}_l, \vec{x}) by searching through the n_l-dimensional parameter space (n_l=\mbox{dim}(\vec{p}_l)).
  4. If a termination criterion satisfied, then break and return the best parameter set.
  5. Else, set n_l\leftarrow n_l+1. If n_l>N, go to (2), else go to (3).

This get’s the main idea across. The limit number of parameters N should be chosen fairly small, pretty much as soon as there is a significant difference between simple and complex parameter sets.

Obviously, this approach requires a model that can describe itself, its own parameters. And at each step in the hierarchy has to compress its input somewhat. This approach is reminiscent of deep neural networks – where the model is simple, a sigmoid neuron – but the parameter space can be huge, which is the space where the weights live. Inputs are described by hidden neurons which are in turn described by yet other hidden neurons etc. However, note that there is a significant restriction usually in deep networks: usually the activation of hidden neurons is taken as representation/description of the input while the entropy in the weights is neglected. In order for this approach to work well, both the entropy in the hidden units and in the weights has to be taken into account.

Further note, that the meta-models will tend to be ever smaller and that the overall description length of the models is limited by the entropy of the input (there is no point in creating a hierarchy of models with higher entropy than the inputs, since it won’t generalize). Overall, the approach resembles a growing pyramid of models stacked onto each other as we go from simpler to more complex descriptions.

What can we hope to gain from that approach? Consider again the 100-bit parameter string example. What if the solution is a simple 100-bit parameter string? In such a case, the situation has three properties:

  1. The search space 2^{100} is too large to be searched through – exhaustively or any other way. That’s the curse of dimensionality.
  2. The solution is however contained in that space.
  3. The solution is algorithmically simple since it has a high prior probability of actually occurring.

If you think about it, this is a quite common situation. The classical solution to that problem was to consider narrow tasks where the search space is small enough to afford an non-Occam-biased search. Or a simple model has been picked such as linear regression, where a strong assumption of linearity allows an efficient search in a high-dimensional parameter space.

Here, I presented a way to go beyond those restrictions. Who knows, maybe a not-too-bad search would thus be possible in a Turing compete search space. It looks like this is what you have to do, in order to perform an efficient and Occam-bias-optimal induction. Interestingly, it entails models described by meta-models which are themselves described by other and yet other meta-models – in an indefinitely deep regress. Doing this in a Turing complete language requires homoiconicity – the property of being able to use programs as data, of which LISP is such a prominent example.

Does this idea finally break out of the narrow AI paradigm? I have come up with a simple test whether a system is narrow or not. Let a model have a set of n parameters each taking D bits to specify. Can I think of an input best fitted by a set of m>n parameters whose description is smaller than n\;D? Then we have found an input whose parameters can be described in a simple way albeit not in the present representation even though it allows the required entropy. For example, consider the space of 3-dimensional polynomials trying to fit points along an 6-dimensional polynomial of the form: 6x^6+5x^5+4x^4+3x^3+2x^2+1x^1+0x^0. Of course, this is not possible. Let each parameter take integer values from -8 to +7, i.e. we need 4 bits to specify it. Then specifying a 3-dimensional polynomial takes 3\times 4=12 bits. However, it takes much less information to specify that particular 6-dimensional polynomial since it is so regular. As of now, I have not met a single practical machine learning technique or model, for which a simple example could not be found that is outside the scope of the model.

How is my indefinite regress perform in that respect? If M= polynomials, limiting the entropy to 12 bits will include simple high-degree polynomials like the 6-dimensional above since it takes a 1-dim polynomial, a line, to specify the decline of numbers 6,5,4,3,2,1,0 and the 1-dim polynomial is easily in the scope of the 12 bit description. Thus, we seem to be holding an approach in our hands, that has the potential to be truly general and to break the curse of narrow AI.

On a more general note, the approach is reminiscent of the infinite regress of thought, where we can become aware or a thought, then become aware of that awareness and of that awareness etc. It suggests the thought provoking hypothesis that it may be the necessity of doing proper Occam-bias-optimal inference that has made awareness and self-awareness possible during the course of the history of human mind.

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Using features for the specialization of algorithms

A widespread sickness of present “narrow AI” approaches is the almost irresistible urge to set up rigid algorithms that find solutions in an as large as possible search space. This always leads to a narrow search space containing very complex solutions. As I have pointed out many times, we need an algorithm that covers a wide range of simple solutions, rather than going deep into complexity.

This approach however, requires to ability to construct new algorithms that are appropriate for the given task on the fly, since general intelligence means to be able to solve problems without knowing them in advance when the system is built. This on the fly construction is called specialization. I have referred to a similar process of data-dependent search space expansion. After all, if the algorithm at hand covers a narrow search space and the solution lies outside of it, the algorithm has to be modified such that its search space expands toward the solution and possibly narrows down in the opposite direction.

I suggest to use features. After all, enumerating an objects features leads to an incremental narrowing of the set of objects that still fall into the description. For example, if I say, the object is yellow, I have narrowed down the set of possible objects significantly, but there are still quite many. If I add that the object is of the size of a cat, the set is reduced to cat-sized yellow objects, and so on. Similarly, the set can be expanded by releasing some feature constraints.

Imagine a high-dimensional search space and the solution is part of a small subspace. For example, in probabilistic generative models, often a constraint is given by the data, such as X+Y=10, and a solution may be X=3, Y=7. To find the solution, it is much more efficient to parametrize the line represented by the constraint and search through the one-dimensional space rather than the two dimensional one.

When the way we travel through the search space is not fixed, it seems like we land in the field of policy learning like in reinforcement learning. Together with feature induction this reminds very much of AIXI.

The setting is the following. Let a generative model in form of a function f(X,Y)=X+Y be given and some loss function L(X,Y)=\left(f(X,Y)-10\right)^2. We can always sample from X and Y. Of course, we could now perform gradient descent or some other stupid, wannabe-general optimization algorithm. Instead, we want to system to construct it’s own algorithm that is well suited for this particular function. And it shall construct a different algorithm for a different function, for example g(X,Y)=\sqrt(X^2+Y^2). The first algorithm would ideally parametrize a straight line, the second a circle. This is what is meant by specialization.

Now, the point is not to do function inversion, e.g. the analytic or numeric construction of f^{-1} or g^{-1}, since first, it is difficult, and second generally intractable. After all, humans seem to be able to move efficiently around search spaces that are not easily representable in an analytic way. Or am I too confident about not using inverse functions? After all, they are the ones that exhibit the highest compression, and it is compression that I want to be guided by. Further, my incremental compression scheme involves the search for inverse functions. Let’s keep that question open.

Here is how features can come into play. Suppose I have found a few samples with zero or close to zero loss. For example, X=2,4,9 and Y=8,6,1, which form a sequence. Then one could search for an inverse function 10-X to compute the Y values in the parameters while have f as the feature function. However, this is a bad example, since the blind search for a feature inverse is just as hard as inverting the whole function, since this problem seems to have a single feature.

This looks much more difficult than it seemed at first glance.


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The physics of structure formation

The entropy in equilibrium thermodynamics is defined as S \sim \ln(\Omega), which always increases in closed systems. It is clearly a special case of Shannon entropy H = -\sum_i p_i \ln(p_i). If the probabilities are uniform, p_i = const, then Shannon entropy boils down to thermodynamic entropy.

A system can be called structured, if some states are predicted with higher probability than others, which leads to lower entropy. As I have argued in a different post, the loss of the ability to predict the system is diagnosed by increasing entropy. In the extreme case, if microstates transition randomly with equal probability, chances are high to get to an unstructured state than to a structured one.

In order to describe both the structure and the transition laws, the concept of algorithmic complexity is needed. If the microstate i is described by a set of numbers, say the speed and position of N particles, then this set of numbers can be written as a sequence. Then a Kolmogorov complexity can be assigned to state i: K(i). Since K(i) is quite high for most sequences, starting out with low K and transiting randomly will increase K. Therefore the Kolmogorov complexity of the system will increase with time.

Interestingly, a low K allows a better (spatial) prediction of the sequence. However, one may have a model, a “law” if you wish, that governs the time evolution of the system. One may not restrict oneself with a microscopic law, but one about some emergent variables of the system. Even macroscopic ones that we have in classical thermodynamics. The “length” of such laws is, in some sense the Kolmogorov complexity of the system, neglecting all the micro- and mesoscopic details.

What happens when structures occur such as living systems or galaxies etc.? The system evolves into a low complexity state. Moreover, it looks like there is scale-free structure in the universe. Why is it not possible nowadays to plug-in the laws of physics, and see chemistry and biology evolve? Because, even with todays computers things would become too complicated.


Interestingly, the minimal entropy of a system corresponds roughly to the algorithmic complexity. Vitanyi and Li write on p. 187: “the interpretation is that H(X) bits are on average sufficient to describe an outcome x. Algorithmic complexity says that an object x has complexity, or algorithmic information, C(x) equal to the minimum length of a binary program for x.”

What would be the holy grail?

If we could update the theory of statistical physics to phenomena that create low complexity, i.e. structures, a theory of self-organization, really. If we could state the condition under which complexity of a subsystem will drop below some bound, then it would be a great thing. This may even become a theory of life.

The even holier grail would be to explain the emergence of intelligent life. To me, it seems sufficient to explain the emergence of subsystems that can develop compressed representations of some of its surroundings. For example, a frog that predicts the flight trajectory of a fly has achieved some degree of intelligence since it compresses the trajectory in its “mind”, which enables prediction in the first place.

What does that mean in physical terms? What is representation? In what sense is representation “about” something else? Somehow, it is the ability to create, to unpack, our representation into an image of the represented object, which is what it means to imagine something. Even if the frog does not imagine the future trajectory, it acts as if it knew how it will continue. Essentially, successful goal-directed action is possible only if you predict, that is only if you compress. However, actions and goals are still not part of physical vocabulary.

It will turn out that in order to maintain a low complexity state, the system will have to be open and exchange energy with the environment. After all, in a closed system, entropy and therefore complexity must increase. In order words, the animal has to eat and to shit 🙂

If you extract the energy from your environment in such a way that you maintain your simple state, does it not imply intelligent action already? Don’t you have to be fairly selective of what kind of energy you take and what you reject? If a large stone flies toward you, you may want to avoid collision: that type of energy transfer is not welcome, since it does not help to maintain a low complexity state. However, a crystal also maintains its simplicity. Probably because it becomes so firm that after a while it just does not desintegrate from the influence of the environment. It any case, it does not represent anything about it’s surroundings. It does not react to the environment either.


From Jeremy England (2013).

Irreversible systems increase the entropy (and the heat) of the heat bath.

\mbox{Heat production} + \mbox{internal entropy change} - \mbox{irreversibility} \ge 0  (8)

If we want internal structure (dS shall be negative) and high irreversibility, then a lot of heat would be released into the bath. Can this result be transformed into an expression with algorithmic complexity? If yes, and if we figure out how to construct a system such that it does create that, then we have figured out, how to create structure.

We can also increase beta, which is done by lowering the temperature. Thus, unsurprisingly, freezing leads to structure formation. But that’s not enough for life. Freezing is also fairly irreversible. So, maybe, structure formation is not enough. What we need is structure representation! What does it mean to represent and to predict in physical terms?

Let’s say a particle travels along a straight line. If a living organism can predict it, it means that it has somehow internally found a short program that, when executed, can create the trajectory and also expand it further in time. It can compute points and moments in future where the particle is going to be. It is the birth of intentionality, of “aboutness”. If there is an ensemble of particles with all their microstates, how can they be “about” some other external particle?

The funny thing is, you need such representations, in order to decrease entropy. After all, the more you compress, the less degrees of freedom are remaining, hence the state space is reduced and the entropy decreases therefore. There can also be hierarchical representations within a system, which means that there is an “internal aboutness” as well. Thus the internal entropy decreases once an ensemble of particles at one level is held in a macrostate determined by a higher level ensemble! Hence, predicting the “outer” world may simply be a special case of predicting the “inner” world, you own macrostates. Thus, in order to decrease the state space in such a way, a few high level macrostates have to physically determine all the microstate at lower levels. For example, in an autoencoder the hidden layer compresses and recreates the inputs at the input layer. In the nervous system neurons get active or not active and therefore take up a large part of the entropy of the brain. The physical determination happens through the propagation of an electric potential though the axons and dendrites of neurons. But, especially in the beginning of life, things have to work out without a nervous system.

Instead of thinking about a practical implementation, I could think of a theoretical description, such as the formula (8). I imagine all microstates of a level being partitioned with a macrostate assigned to each subset of the partition. Those macrostates would correspond to the microstates of the level above. Now, there is a highly structured outside world, which means that the entropy is low and there are much less states than are theoretically possible. If you have got sensors, some part of the outer world activates them. Their probability distribution is the one to be compressed. Which means, if the world has been created by running a short program, your job as a living being is to find that program. And why should that happen? It would be very cool to show that our world is made in a way such that subsystem will emerge that try to represent it and ultimately find out the way it has been made. Predicting food trajectories may be a start to do exactly that.

So, it is not just the goal of decreasing internal entropy, but to do it in such a way that it represents the outer world, the entropy of which is already decreased by the laws of nature. And what does represent mean in that sense? In the internal sense it means to physically determine ones own internal states. And for the outer world it means to have sensors somehow such that the states of the outer world are reflected by the states of your sensors. So, we can imagine the lowest layer/level to encode the states of the outer world, at least a part of it. And it does so in a non-compressing way, hence it is a one to one map of a part of the world. Can we show, that under such circumstances, compression in terms of algorithmic complexity is the best thing to be done? Basically, it means that some part of the animal is driven by some outer influences and therefore can not be changed, hence contributes majorly to the entropy of the animal. In order to decrease the entropy nevertheless, the animal has to find a way to recreate those same inputs.

Now, I have to clarify, what probability distributions are meant, when we compute the entropy. In a deterministic world, probabilities are always the reflection of our – the scientists’ – lack of knowledge. Those probabilities are different from the probabilities assigned by the animal: those are the animal’s knowledge. We should treat them as the same.

A way to reduce internal entropy is to couple all remaining internal states to the sensor inputs. Which does not necessarily mean to compress. Well, it does decrease the entropy, since only the sensor entropy remains, but it does not decrease it even further! In order to decrease it even further, the sensor entropy has to depend on internal states, which should be fewer in number. They have to GENERATE the microstates of the sensors.

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