Simon Segert

Taylor W. Webb, Steven M. Frankland, Awni Altabaa et al.

A central challenge for cognitive science is to explain how abstract concepts are acquired from limited experience. This has often been framed in terms of a dichotomy between connectionist and symbolic cognitive models. Here, we highlight a recently emerging line of work that suggests a novel reconciliation of these approaches, by exploiting an inductive bias that we term the relational bottleneck. In that approach, neural networks are constrained via their architecture to focus on relations between perceptual inputs, rather than the attributes of individual inputs. We review a family of models that employ this approach to induce abstractions in a data-efficient manner, emphasizing their potential as candidate models for the acquisition of abstract concepts in the human mind and brain.

Simon Segert, Jonathan Cohen

The ability to learn and predict simple functions is a key aspect of human intelligence. Recent works have started to explore this ability using transformer architectures, however it remains unclear whether this is sufficient to recapitulate the extrapolation abilities of people in this domain. Here, we propose to address this gap by augmenting the transformer architecture with two simple inductive learning biases, that are directly adapted from recent models of abstract reasoning in cognitive science. The results we report demonstrate that these biases are helpful in the context of large neural network models, as well as shed light on the types of inductive learning biases that may contribute to human abilities in extrapolation.

Simon Segert

We consider the problem of linear estimation, and establish an extension of the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero but bounded with respect to a matrix norm of Schatten type. We derive simple and explicit formulas for the optimal estimator in the cases of Nuclear and Spectral norms (with the Frobenius case recovering ridge regression). Additionally, we analytically derive the generalization error in multiple random matrix ensembles, and compare with Ridge regression. Finally, we conduct an extensive simulation study, in which we show that the cross-validated Nuclear and Spectral regressors can outperform Ridge in several circumstances.

Simon Segert, Nathan Wycoff

Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $\Vert \cdot \Vert_*$. However, despite the assortment of computational methods for such problems, there is a surprising lack of understanding of the underlying probability distributions being referred to. In this article, we study the distribution with density $f(X)\propto e^{-λ\Vert X\Vert_*}$, finding many of its fundamental attributes to be analytically tractable via differential geometry. We use these facts to design an improved MCMC algorithm for low rank Bayesian inference as well as to learn the penalty parameter $λ$, obviating the need for hyperparameter tuning when this is difficult or impossible. Finally, we deploy these to improve the accuracy and efficiency of low rank Bayesian matrix denoising and completion algorithms in numerical experiments.

Cutter Dawes, Simon Segert, Kamesh Krishnamurthy et al.

A major challenge in the use of neural networks both for modeling human cognitive function and for artificial intelligence is the design of systems with the capacity to efficiently learn functions that support radical generalization. At the roots of this is the capacity to discover and implement symmetry functions. In this paper, we investigate a paradigmatic example of radical generalization through the use of symmetry: base addition. We present a group theoretic analysis of base addition, a fundamental and defining characteristic of which is the carry function -- the transfer of the remainder, when a sum exceeds the base modulus, to the next significant place. Our analysis exposes a range of alternative carry functions for a given base, and we introduce quantitative measures to characterize these. We then exploit differences in carry functions to probe the inductive biases of neural networks in symmetry learning, by training neural networks to carry out base addition using different carries, and comparing efficacy and rate of learning as a function of their structure. We find that even simple neural networks can achieve radical generalization with the right input format and carry function, and that learnability is closely correlated with carry function structure. We then discuss the relevance this has for cognitive science and machine learning.