A mathematical theory of relational generalization in transitive inference.

Humans and animals routinely infer relations between different items or events and generalize these relations to novel combinations of items. This allows them to respond appropriately to radically novel circumstances and is fundamental to advanced cognition. However, how learning systems (including the brain) can implement the necessary inductive biases has been unclear. We investigated transitive inference (TI), a classic relational task paradigm in which subjects must learn a relation ([Formula: see text] and [Formula: see text]) and generalize it to new combinations of items ([Formula: see text]). Through mathematical analysis, we found that a broad range of biologically relevant learning models (e.g. gradient flow or ridge regression) perform TI successfully and recapitulate signature behavioral patterns long observed in living subjects. First, we found that models with item-wise additive representations automatically encode transitive relations. Second, for more general representations, a single scalar "conjunctivity factor" determines model behavior on TI and, further, the principle of norm minimization (a standard statistical inductive bias) enables models with fixed, partly conjunctive representations to generalize transitively. Finally, neural networks in the "rich regime," which enables representation learning and improves generalization on many tasks, unexpectedly show poor generalization and anomalous behavior on TI. We find that such networks implement a form of norm minimization (over hidden weights) that yields a local encoding mechanism lacking transitivity. Our findings show how minimal statistical learning principles give rise to a classical relational inductive bias (transitivity), explain empirically observed behaviors, and establish a formal approach to understanding the neural basis of relational abstraction.

Hierarchical clustering optimizes the tradeoff between compositionality and expressivity of task structures for flexible reinforcement learning.

A hallmark of human intelligence, but challenging for reinforcement learning (RL) agents, is the ability to compositionally generalise, that is, to recompose familiar knowledge components in novel ways to solve new problems. For instance, when navigating in a city, one needs to know the location of the destination and how to operate a vehicle to get there, whether it be pedalling a bike or operating a car. In RL, these correspond to the reward function and transition function, respectively. To compositionally generalize, these two components need to be transferable independently of each other: multiple modes of transport can reach the same goal, and any given mode can be used to reach multiple destinations. Yet there are also instances where it can be helpful to learn and transfer entire structures, jointly representing goals and transitions, particularly whenever these recur in natural tasks (e.g., given a suggestion to get ice cream, one might prefer to bike, even in new towns). Prior theoretical work has explored how, in model-based RL, agents can learn and generalize task components (transition and reward functions). But a satisfactory account for how a single agent can simultaneously satisfy the two competing demands is still lacking. Here, we propose a hierarchical RL agent that learns and transfers individual task components as well as entire structures (particular compositions of components) by inferring both through a non-parametric Bayesian model of the task. It maintains a factorised representation of task components through a hierarchical Dirichlet process, but it also represents different possible covariances between these components through a standard Dirichlet process. We validate our approach on a variety of navigation tasks covering a wide range of statistical correlations between task components and show that it can also improve generalisation and transfer in more complex, hierarchical tasks with goal/subgoal structures. Finally, we end with a discussion of our work including how this clustering algorithm could conceivably be implemented by cortico-striatal gating circuits in the brain.