Christos Papadimitriou

Daniel Mitropolsky, Christos Papadimitriou

Despite tremendous progress in neuroscience, we do not have a compelling narrative for the precise way whereby the spiking of neurons in our brain results in high-level cognitive phenomena such as planning and language. We introduce a simple mathematical formulation of six basic and broadly accepted principles of neuroscience: excitatory neurons, brain areas, random synapses, Hebbian plasticity, local inhibition, and inter-area inhibition. We implement a simulated neuromorphic system based on this formalism, which is capable of basic language acquisition: Starting from a tabula rasa, the system learns, in any language, the semantics of words, their syntactic role (verb versus noun), and the word order of the language, including the ability to generate novel sentences, through the exposure to a modest number of grounded sentences in the same language. We discuss several possible extensions and implications of this result.

Oliver Biggar, Christos Papadimitriou

Characterizing the limit behavior -- that is, the attractors -- of learning dynamics is one of the most fundamental open questions in game theory. In recent work on this front, it was conjectured that the attractors of the replicator dynamic are in one-to-one correspondence with the sink equilibria of the game -- the sink strongly connected components of a game's preference graph -- , and it was established that they do stand in at least one-to-many correspondence with them. Here, we show that the one-to-one conjecture is false. We disprove this conjecture over the course of three theorems: the first disproves a stronger form of the conjecture, while the weaker form is disproved separately in the two-player and $N$-player ($N>2$) cases. By showing how the conjecture fails, we lay out the obstacles that lie ahead for characterizing attractors of the replicator, and introduce new ideas with which to tackle them. All three counterexamples derive from an object called a local source -- a point lying within the sink equilibrium, and yet which is `locally repelling'; we prove that the absence of local sources is necessary, but not sufficient, for the one-to-one property to be true. We complement this with a sufficient condition: we introduce a local property of a sink equilibrium called pseudoconvexity, and establish that when the sink equilibria of a two-player game are pseudoconvex then they precisely define the attractors. Pseudoconvexity generalizes the previous cases -- such as zero-sum games and potential games -- where this conjecture was known to hold, and reformulates these cases in terms of a simple graph property.

William Brown, Christos Papadimitriou, Tim Roughgarden

In repeated interaction problems with adaptive agents, our objective often requires anticipating and optimizing over the space of possible agent responses. We show that many problems of this form can be cast as instances of online (nonlinear) control which satisfy \textit{local controllability}, with convex losses over a bounded state space which encodes agent behavior, and we introduce a unified algorithmic framework for tractable regret minimization in such cases. When the instance dynamics are known but otherwise arbitrary, we obtain oracle-efficient $O(\sqrt{T})$ regret by reduction to online convex optimization, which can be made computationally efficient if dynamics are locally \textit{action-linear}. In the presence of adversarial disturbances to the state, we give tight bounds in terms of either the cumulative or per-round disturbance magnitude (for \textit{strongly} or \textit{weakly} locally controllable dynamics, respectively). Additionally, we give sublinear regret results for the cases of unknown locally action-linear dynamics as well as for the bandit feedback setting. Finally, we demonstrate applications of our framework to well-studied problems including performative prediction, recommendations for adaptive agents, adaptive pricing of real-valued goods, and repeated gameplay against no-regret learners, directly yielding extensions beyond prior results in each case.