arXiv subjects

Andrew Critch

Publications and source records attributed to Andrew Critch.

24 records · Page 2Linked to original sources

arXiv2017

Servant of Many Masters: Shifting priorities in Pareto-optimal sequential decision-making

It is often argued that an agent making decisions on behalf of two or more principals who have different utility functions should adopt a {\em Pareto-optimal} policy, i.e., a policy that cannot be improved upon for one agent without making sacrifices for another. A famous theorem of Harsanyi shows that, when the principals have a common prior on the outcome distributions of all policies, a Pareto-optimal policy for the agent is one that maximizes a fixed, weighted linear combination of the principals' utilities. In this paper, we show that Harsanyi's theorem does not hold for principals with different priors, and derive a more precise generalization which does hold, which constitutes our main result. In this more general case, the relative weight given to each principal's utility should evolve over time according to how well the agent's observations conform with that principal's prior. The result has implications for the design of contracts, treaties, joint ventures, and robots.

cs.AI↗

arXiv2017

A Formal Approach to the Problem of Logical Non-Omniscience

We present the logical induction criterion for computable algorithms that assign probabilities to every logical statement in a given formal language, and refine those probabilities over time. The criterion is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence phi is associated with a stock that is worth $1 per share if phi is true and nothing otherwise, and we interpret the belief-state of a logically uncertain reasoner as a set of market prices, where pt_N(phi)=50% means that on day N, shares of phi may be bought or sold from the reasoner for 50%. A market is then called a logical inductor if (very roughly) there is no polynomial-time computable trading strategy with finite risk tolerance that earns unbounded profits in that market over time. We then describe how this single criterion implies a number of desirable properties of bounded reasoners; for example, logical inductors outpace their underlying deductive process, perform universal empirical induction given enough time to think, and place strong trust in their own reasoning process.

cs.LO↗

arXiv2017

Toward negotiable reinforcement learning: shifting priorities in Pareto optimal sequential decision-making

Existing multi-objective reinforcement learning (MORL) algorithms do not account for objectives that arise from players with differing beliefs. Concretely, consider two players with different beliefs and utility functions who may cooperate to build a machine that takes actions on their behalf. A representation is needed for how much the machine's policy will prioritize each player's interests over time. Assuming the players have reached common knowledge of their situation, this paper derives a recursion that any Pareto optimal policy must satisfy. Two qualitative observations can be made from the recursion: the machine must (1) use each player's own beliefs in evaluating how well an action will serve that player's utility function, and (2) shift the relative priority it assigns to each player's expected utilities over time, by a factor proportional to how well that player's beliefs predict the machine's inputs. Observation (2) represents a substantial divergence from naïve linear utility aggregation (as in Harsanyi's utilitarian theorem, and existing MORL algorithms), which is shown here to be inadequate for Pareto optimal sequential decision-making on behalf of players with different beliefs.

cs.AI↗

arXiv2016

Parametric Bounded Löb's Theorem and Robust Cooperation of Bounded Agents

Löb's theorem and Gödel's theorems make predictions about the behavior of systems capable of self-reference with unbounded computational resources with which to write and evaluate proofs. However, in the real world, systems capable of self-reference will have limited memory and processing speed, so in this paper we introduce an effective version of Löb's theorem which is applicable given such bounded resources. These results have powerful implications for the game theory of bounded agents who are able to write proofs about themselves and one another, including the capacity to out-perform classical Nash equilibria and correlated equilibria, attaining mutually cooperative program equilibrium in the Prisoner's Dilemma. Previous cooperative program equilibria studied by Tennenholtz (2004) and Fortnow (2009) have depended on tests for program equality, a fragile condition, whereas "Löbian" cooperation is much more robust and agnostic of the opponent's implementation.

cs.GT↗

arXiv2014

Algebraic Geometry of Matrix Product States

We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions. Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS. We make four conjectures on the identifiability of MPS parameters.

quant-ph↗

arXiv2012

A note on the proportionality between some consistency indices in the AHP

Analyzing the consistency of preferences is an important step in decision making with pairwise comparison matrices, and several indices have been proposed in order to estimate it. In this paper we prove the proportionality between some consistency indices in the framework of the Analytic Hierarchy Process. Knowing such equivalences eliminates redundancy in the consideration of evidence for consistent preferences.

stat.ME↗