LIME: Learning Inductive Bias for Primitives of Mathematical Reasoning

While designing inductive bias in neural architectures has been widely studied, we hypothesize that transformer networks are flexible enough to learn inductive bias from suitable generic tasks. Here, we replace architecture engineering by encoding inductive bias in the form of datasets. Inspired by Peirce's view that deduction, induction, and abduction form an irreducible set of reasoning primitives, we design three synthetic tasks that are intended to require the model to have these three abilities. We specifically design these synthetic tasks in a way that they are devoid of mathematical knowledge to ensure that only the fundamental reasoning biases can be learned from these tasks. This defines a new pre-training methodology called "LIME" (Learning Inductive bias for Mathematical rEasoning). Models trained with LIME significantly outperform vanilla transformers on three very different large mathematical reasoning benchmarks. Unlike dominating the computation cost as traditional pre-training approaches, LIME requires only a small fraction of the computation cost of the typical downstream task. Inductive bias is essential for successful neural network learning. Many of the breakthroughs in machine learning are accompanied by new neural architectures with better inductive biases, such as locality bias in convolutional neural networks (LeCun et al., 1999), recurrence and memory in LSTMs (Hochreiter and Schmidhuber, 1997), and structural bias in graph neural networks (Scarselli et al., 2008). However, existing designs of inductive biases need to be explicitly encoded in neural architecture. This is sometimes difficult as one may not know the exact mechanism for an abstract ability, in order to describe the architectural bias explicitly. In particular, designing proper inductive bias for abstract concepts such as mathematical reasoning becomes an extremely challenging task. Moreover, attempts to design elaborate architectures for reasoning often fall short of the performance of more generic transformer architecture.