Algorithmic Probability-Guided Machine Learning on Non-Differentiable Spaces.

We show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this model-driven approach may require less training data and can potentially be more generalizable as it shows greater resilience to random attacks. In an algorithmic space the order of its element is given by its algorithmic probability, which arises naturally from computable processes. We investigate the shape of a discrete algorithmic space when performing regression or classification using a loss function parametrized by algorithmic complexity, demonstrating that the property of differentiation is not required to achieve results similar to those obtained using differentiable programming approaches such as deep learning. In doing so we use examples which enable the two approaches to be compared (small, given the computational power required for estimations of algorithmic complexity). We find and report that 1) machine learning can successfully be performed on a non-smooth surface using algorithmic complexity; 2) that solutions can be found using an algorithmic-probability classifier, establishing a bridge between a fundamentally discrete theory of computability and a fundamentally continuous mathematical theory of optimization methods; 3) a formulation of an algorithmically directed search technique in non-smooth manifolds can be defined and conducted; 4) exploitation techniques and numerical methods for algorithmic search to navigate these discrete non-differentiable spaces can be performed; in application of the (a) identification of generative rules from data observations; (b) solutions to image classification problems more resilient against pixel attacks compared to neural networks; (c) identification of equation parameters from a small data-set in the presence of noise in continuous ODE system problem, (d) classification of Boolean NK networks by (1) network topology, (2) underlying Boolean function, and (3) number of incoming edges.

Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity.

Natural selection explains how life has evolved over millions of years from more primitive forms. The speed at which this happens, however, has sometimes defied formal explanations when based on random (uniformly distributed) mutations. Here, we investigate the application of a simplicity bias based on a natural but algorithmic distribution of mutations (no recombination) in various examples, particularly binary matrices, in order to compare evolutionary convergence rates. Results both on synthetic and on small biological examples indicate an accelerated rate when mutations are not statistically uniform but algorithmically uniform. We show that algorithmic distributions can evolve modularity and genetic memory by preservation of structures when they first occur sometimes leading to an accelerated production of diversity but also to population extinctions, possibly explaining naturally occurring phenomena such as diversity explosions (e.g. the Cambrian) and massive extinctions (e.g. the End Triassic) whose causes are currently a cause for debate. The natural approach introduced here appears to be a better approximation to biological evolution than models based exclusively upon random uniform mutations, and it also approaches a formal version of open-ended evolution based on previous formal results. These results validate some suggestions in the direction that computation may be an equally important driver of evolution. We also show that inducing the method on problems of optimization, such as genetic algorithms, has the potential to accelerate convergence of artificial evolutionary algorithms.

Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence.

Is undecidability a requirement for open-ended evolution (OEE)? Using methods derived from algorithmic complexity theory, we propose robust computational definitions of open-ended evolution and the adaptability of computable dynamical systems. Within this framework, we show that decidability imposes absolute limits on the stable growth of complexity in computable dynamical systems. Conversely, systems that exhibit (strong) open-ended evolution must be undecidable, establishing undecidability as a requirement for such systems. Complexity is assessed in terms of three measures: sophistication, coarse sophistication, and busy beaver logical depth. These three complexity measures assign low complexity values to random (incompressible) objects. As time grows, the stated complexity measures allow for the existence of complex states during the evolution of a computable dynamical system. We show, however, that finding these states involves undecidable computations. We conjecture that for similar complexity measures that assign low complexity values, decidability imposes comparable limits on the stable growth of complexity, and that such behavior is necessary for nontrivial evolutionary systems. We show that the undecidability of adapted states imposes novel and unpredictable behavior on the individuals or populations being modeled. Such behavior is irreducible. Finally, we offer an example of a system, first proposed by Chaitin, that exhibits strong OEE.

A Decomposition Method for Global Evaluation of Shannon Entropy and Local Estimations of Algorithmic Complexity.

We investigate the properties of a Block Decomposition Method (BDM), which extends the power of a Coding Theorem Method (CTM) that approximates local estimations of algorithmic complexity based on Solomonoff-Levin's theory of algorithmic probability providing a closer connection to algorithmic complexity than previous attempts based on statistical regularities such as popular lossless compression schemes. The strategy behind BDM is to find small computer programs that produce the components of a larger, decomposed object. The set of short computer programs can then be artfully arranged in sequence so as to produce the original object. We show that the method provides efficient estimations of algorithmic complexity but that it performs like Shannon entropy when it loses accuracy. We estimate errors and study the behaviour of BDM for different boundary conditions, all of which are compared and assessed in detail. The measure may be adapted for use with more multi-dimensional objects than strings, objects such as arrays and tensors. To test the measure we demonstrate the power of CTM on low algorithmic-randomness objects that are assigned maximal entropy (e.g., π ) but whose numerical approximations are closer to the theoretical low algorithmic-randomness expectation. We also test the measure on larger objects including dual, isomorphic and cospectral graphs for which we know that algorithmic randomness is low. We also release implementations of the methods in most major programming languages-Wolfram Language (Mathematica), Matlab, R, Perl, Python, Pascal, C++, and Haskell-and an online algorithmic complexity calculator.