A Theory of Bounded Inductive Rationality

The dominant theories of rational choice assume logical omniscience. That is, they assume that when facing a decision problem, an agent can perform all relevant computations and determine the truth value of all relevant logical/mathematical claims. This assumption is unrealistic when, for example, we offer bets on remote digits of pi or when an agent faces a computationally intractable planning problem. Furthermore, the assumption of logical omniscience creates contradictions in cases where the environment can contain descriptions of the agent itself. Importantly, strategic interactions as studied in game theory are decision problems in which a rational agent is predicted by its environment (the other players). In this paper, we develop a theory of rational decision making that does not assume logical omniscience. We consider agents who repeatedly face decision problems (including ones like betting on digits of pi or games against other agents). The main contribution of this paper is to provide a sensible theory of rationality for such agents. Roughly, we require that a boundedly rational inductive agent tests each efficiently computable hypothesis infinitely often and follows those hypotheses that keep their promises of high rewards. We then prove that agents that are rational in this sense have other desirable properties. For example, they learn to value random and pseudo-random lotteries at their expected reward. Finally, we consider strategic interactions between different agents and prove a folk theorem for what strategies bounded rational inductive agents can converge to.

Embedded Agency

Traditional models of rational action treat the agent as though it is cleanly separated from its environment, and can act on that environment from the outside. Such agents have a known functional relationship with their environment, can model their environment in every detail, and do not need to reason about themselves or their internal parts. We provide an informal survey of obstacles to formalizing good reasoning for agents embedded in their environment. Such agents must optimize an environment that is not of type ``function''; they must rely on models that fit within the modeled environment; and they must reason about themselves as just another physical system, made of parts that can be modified and that can work at cross purposes.

Inductive Coherence

While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool for reasoning about computer programs, we consider this problem from the perspective of integrating probability and logic. Recent work on assigning probabilities to mathematical statements has used the concept of coherent distributions, which satisfy logical constraints such as the probability of a sentence and its negation summing to one. Although there are algorithms which converge to a coherent probability distribution in the limit, this yields only weak guarantees about finite approximations of these distributions. In our setting, this is a significant limitation: Coherent distributions assign probability one to all statements provable in a specific logical theory, such as Peano Arithmetic, which can prove what the output of any terminating computation is; thus, a coherent distribution must assign probability one to the output of any terminating computation. To model uncertainty about computations, we propose to work with approximations to coherent distributions. We introduce inductive coherence, a strengthening of coherence that provides appropriate constraints on finite approximations, and propose an algorithm which satisfies this criterion.

Asymptotic Logical Uncertainty and The Benford Test

We give an algorithm A which assigns probabilities to logical sentences. For any simple infinite sequence of sentences whose truth-values appear indistinguishable from a biased coin that outputs "true" with probability p, we have that the sequence of probabilities that A assigns to these sentences converges to p.

Outperforming Word2Vec on Analogy Tasks with Random Projections

We present a distributed vector representation based on a simplification of the BEAGLE system, designed in the context of the Sigma cognitive architecture. Our method does not require gradient-based training of neural networks, matrix decompositions as with LSA, or convolutions as with BEAGLE. All that is involved is a sum of random vectors and their pointwise products. Despite the simplicity of this technique, it gives state-of-the-art results on analogy problems, in most cases better than Word2Vec. To explain this success, we interpret it as a dimension reduction via random projection.