Inspired by Solomonoffs theory of inductive inference, we propose a prior based on circuit complexity. There are several advantages to this approach. First, it relies on a complexity measure that does not depend on the choice of UTM. There is one universal definition for Boolean circuits involving an universal operation such as nand with simple conversions to alternative definitions such as and, or, and not. Second, there is no analogue of the halting problem. The output value of a circuit can be calculated recursively by computer in time proportional to the number of gates, while a short program may run for a very long time. Our prior assumes that a Boolean function, or equivalently, Boolean string of fixed length, is generated by some Bayesian mixture of circuits. This model is appropriate for learning Boolean functions from partial information, a problem often encountered within machine learning as "binary classification." We argue that an inductive bias towards simple explanations as measured by circuit complexity is appropriate for this problem.

Bayesian inference provides a uniquely rigorous approach to obtain principled justification for uncertainty in predictions, yet it is difficult to articulate suitably general prior belief in the machine learning context, where computational architectures are pure abstractions subject to frequent modifications by practitioners attempting to improve results. Parsimonious inference is an information-theoretic formulation of inference over arbitrary architectures that formalizes Occam's Razor; we prefer simple and sufficient explanations. Our universal hyperprior assigns plausibility to prior descriptions, encoded as sequences of symbols, by expanding on the core relationships between program length, Kolmogorov complexity, and Solomonoff's algorithmic probability. We then cast learning as information minimization over our composite change in belief when an architecture is specified, training data are observed, and model parameters are inferred. By distinguishing model complexity from prediction information, our framework also quantifies the phenomenon of memorization. Although our theory is general, it is most critical when datasets are limited, e.g. small or skewed. We develop novel algorithms for polynomial regression and random forests that are suitable for such data, as demonstrated by our experiments. Our approaches combine efficient encodings with prudent sampling strategies to construct predictive ensembles without cross-validation, thus addressing a fundamental challenge in how to efficiently obtain predictions from data.

Like any field of empirical science, AI may be approached axiomatically. We formulate requirements for a general-purpose, human-level AI system in terms of postulates. We review the methodology of deep learning, examining the explicit and tacit assumptions in deep learning research. Deep Learning methodology seeks to overcome limitations in traditional machine learning research as it combines facets of model richness, generality, and practical applicability. The methodology so far has produced outstanding results due to a productive synergy of function approximation, under plausible assumptions of irreducibility and the efficiency of back-propagation family of algorithms. We examine these winning traits of deep learning, and also observe the various known failure modes of deep learning. We conclude by giving recommendations on how to extend deep learning methodology to cover the postulates of general-purpose AI including modularity, and cognitive architecture. We also relate deep learning to advances in theoretical neuroscience research.

According to Dennett, the same system may be described using a `physical' (mechanical) explanatory stance, or using an `intentional' (belief- and goal-based) explanatory stance. Humans tend to find the physical stance more helpful for certain systems, such as planets orbiting a star, and the intentional stance for others, such as living animals. We define a formal counterpart of physical and intentional stances within computational theory: a description of a system as either a device, or an agent, with the key difference being that `devices' are directly described in terms of an input-output mapping, while `agents' are described in terms of the function they optimise. Bayes' rule can then be applied to calculate the subjective probability of a system being a device or an agent, based only on its behaviour. We illustrate this using the trajectories of an object in a toy grid-world domain.

Ray Solomonoff, a physicist who was one of the founders of the field of artificial intelligence, died on Dec. 7 in Boston. He was 83 and had homes in New Ipswich, N.H., and Cambridge, Mass. The cause was a ruptured brain aneurysm, said his wife, Grace. As a child Mr. Solomonoff developed what would become a lifelong passion for mathematical theorems, and as a teenager he became captivated with idea of creating machines that could learn and ultimately think. In 1952 he met Marvin Minsky, a cognitive scientist who was also exploring the idea of machine learning, and John McCarthy, a young mathematician.

We continue our analysis of volume and energy measures that are appropriate for quantifying inductive inference systems. We extend logical depth and conceptual jump size measures in AIT to stochastic problems, and physical measures that involve volume and energy. We introduce a graphical model of computational complexity that we believe to be appropriate for intelligent machines. We show several asymptotic relations between energy, logical depth and volume of computation for inductive inference. In particular, we arrive at a "black-hole equation" of inductive inference, which relates energy, volume, space, and algorithmic information for an optimal inductive inference solution. We introduce energy-bounded algorithmic entropy. We briefly apply our ideas to the physical limits of intelligent computation in our universe.

We propose that Solomonoff induction is complete in the physical sense via several strong physical arguments. We also argue that Solomonoff induction is fully applicable to quantum mechanics. We show how to choose an objective reference machine for universal induction by defining a physical message complexity and physical message probability, and argue that this choice dissolves some well-known objections to universal induction. We also introduce many more variants of physical message complexity based on energy and action, and discuss the ramifications of our proposals. "If you wish to make an apple pie from scratch, you must first invent the universe."

We show these capacity measures count the number of hypotheses about a dataset that a learning algorithm falsifies when it finds the classifier in its repertoire minimizing empirical risk. It then follows from that the future performance of predictors on unseen data is controlled in part by how many hypotheses the learner falsifies. As a corollary we show that empirical VC-entropy quantifies the message length of the true hypothesis in the optimal code of a particular probability distribution, the so-called actual repertoire.

We reminisce and discuss applications of algorithmic probability to a wide range of problems in artificial intelligence, philosophy and technological society. We propose that Solomonoff has effectively axiomatized the field of artificial intelligence, therefore establishing it as a rigorous scientific discipline. We also relate to our own work in incremental machine learning and philosophy of complexity.

Could there exist an elegant and universal theory of sequence pre diction? Solomonoff's model of induction rapidly learns to make optimal predict ions for any computable sequence, including probabilistic ones [13, 14]. In deed the problem of sequence prediction could well be considered solved [9, 8], if it were not for the fact that Solomonoff's theoretical model is incomputab le. Among computable theories there exist powerful general predict ors, such as the Lempel-Ziv algorithm [5] and Context Tree Weighting [18], that can learn to predict some complex sequences, but not others. Some prediction methods, such as the Minimum Description Length principle [12] and the Minimum Messa ge Length principle [17], can even be viewed as computable approximation s to Solomonoff induction [10]. However in practice their power and genera lity are limited by the power of compression and coding methods employed, as well as having a significantly reduced data efficiency as compared to Solom onoff induction [11]. This work was supported by SNF grant 200020-107616.