Sometimes less is more: Investigating inductive biases of MLPs and Transformers on noisy k-parity features

Transformers are remarkably versatile and their design is largely consistent across a variety of applications. But are they optimal for any given task or dataset? The answer may be key for pushing AI beyond merely scaling current designs. Method. We present a method to optimize a transformer architecture for a given dataset, which we use as a tool to study optimal task-specific inductive biases. This method replaces the most important non-linearities (GeLUs,;softmax) with functions learned on held-out data. We then train the resulting architectures on other datasets, as a way to evaluate the compatibility between pairs of tasks. Findings. On algorithmic toy tasks, we identify new architectures with dramatic improvements in learning speed, in- and out-of-distribution generalization, and stability across seeds. The new designs prove very task-specific however, and indicate that these tasks require inductive biases very different from those of standard transformers. On code and language modeling datasets, we also find architectures with consistent, yet smaller improvements. These designs transfer much better across datasets and domains (English & computer code). Implications. Our results show that standard transformers are rarely a local optimum in the space of architectures. Simple alternatives can perform much better but sacrifice universality. This suggests that there may be room for improved architectures that better support multiple capabilities simultaneously, such as fluency and robust reasoning.

Several formal theories have been developed to characterize the capabilities and limitations of the transformer architecture, each employing different abstractions that capture different features of the architecture. Due to the differences in their underlying assumptions and conceptual frameworks, these theories can disagree in their predictions about what tasks transformers find easy or hard. We map out seven theories of transformer capability (Star-free, $\mathsf{AC}^0$, $\mathsf{TC}^0$, C-RASP, MPC, BAPO, and Sensitivity) and translate their results into a common matrix of predictions across a suite of tasks covering different computational structures. We provide new theoretical results to complete this matrix. We then evaluate theoretical predictions through controlled experiments measuring in-distribution sample complexity, length generalization, and pretrained language model performance. We find that no single theory explains the observed pattern of successes and failures. When a theory does not capture the transformer behaviors relevant to a task, its predictions break down; for instance, we show that the $k$-hop induction task is predicted hard by C-RASP because it does not model content-dependent attention, but we train small transformers on this task that length-generalize. Similarly, theories assuming hard attention like $\mathsf{AC}^0$ and BAPO predict binary majority to be hard, for which we also find success in practice. Together, our results point to several specific features that a theory of transformers must capture: soft rather than hard attention, dynamic content-dependent retrieval, bounded computation at each position, an account of learnability and inductive biases, and a combination of upper and lower bounds.

A central question in the debate over large language models is whether transformers can learn rules they have never seen, or whether they can only interpolate: predict new cases from their similarity to training examples. We test this in a controlled setting where interpolation provably fails, so success can only come from computation beyond interpolation. We train small transformers to predict the rollout of a cellular automaton whose update rule is pure XOR, and remove one entry of the rule's truth table from all direct supervision. The missing entry's output is never shown to the model; its only trace is indirect, as wrong values corrupt visible predictions at later timesteps. Because XOR parity flips whenever one input bit is changed, every one-bit neighbour of the missing entry carries the opposite label, and we prove that similarity-based predictors, including nearest-neighbour, kernel, and Gaussian-process methods, are forced to the wrong answer. A two-layer transformer can nevertheless recover the missing entry, and circuit extraction confirms it computes XOR exactly. Ablations show the recovery depends on gradient signal propagating through multi-step prediction, and a second, structurally unrelated benchmark on symbolic operator chains exhibits the same capacity under ordinary autoregressive training. Together with a constructive proof that a standard transformer block can implement exact local Boolean rules, these results provide an existence proof that transformers can learn rule structure not directly observed in training and express it explicitly. This rules out the strongest architectural form of the interpolation-only account, the claim that transformers cannot in principle discover and communicate unseen rules, while leaving open when such behaviour arises in large-scale language training.