Transformers Learn Shortcuts to Automata (2210.10749v2)

Abstract: Algorithmic reasoning requires capabilities which are most naturally understood through recurrent models of computation, like the Turing machine. However, Transformer models, while lacking recurrence, are able to perform such reasoning using far fewer layers than the number of reasoning steps. This raises the question: what solutions are learned by these shallow and non-recurrent models? We find that a low-depth Transformer can represent the computations of any finite-state automaton (thus, any bounded-memory algorithm), by hierarchically reparameterizing its recurrent dynamics. Our theoretical results characterize shortcut solutions, whereby a Transformer with $o(T)$ layers can exactly replicate the computation of an automaton on an input sequence of length $T$. We find that polynomial-sized $O(\log T)$-depth solutions always exist; furthermore, $O(1)$-depth simulators are surprisingly common, and can be understood using tools from Krohn-Rhodes theory and circuit complexity. Empirically, we perform synthetic experiments by training Transformers to simulate a wide variety of automata, and show that shortcut solutions can be learned via standard training. We further investigate the brittleness of these solutions and propose potential mitigations.

Shortcuts in Computational Simulation with Transformers

The paper of deep learning models has increasingly focused on understanding how these models perform complex tasks requiring algorithmic reasoning. In exploring this domain, the paper "Transformers Learn Shortcuts to Automata" investigates how Transformers—deep learning architectures known for their parallelizable, non-recurrent nature—can efficiently simulate the computations of finite-state automata. This is intriguing, given the classical association of algorithmic reasoning and sequential computation with recurrent models, such as Turing machines.

Central Hypothesis and Approach

Transformers typically operate with fewer layers than their recurrent counterparts, which raises a question about how they address tasks traditionally thought to need iterative, sequential computations. The central hypothesis of the paper is that Transformers leverage "shortcut" solutions to simulate automata computations. These shortcuts essentially bypass the need for depth proportional to the length of the input sequence.

The authors show theoretically that a shallow Transformer can represent finite-state automata through hierarchical reparameterization of automata's recursive dynamics. Specifically, they demonstrate that for any semiautomaton with state space $Q$ and input alphabet $\Sigma$, shallow Transformers can simulate its operation using a computation depth that is logarithmic in the input sequence length, $T$.

Key Contributions and Results

The paper presents several important theoretical and empirical findings:

1. Existence of Shortcuts: A pivotal result is that for any semiautomaton, one can construct a Transformer simulating it with depth $O(\log T)$. This is significantly shallower than a naive $O(T)$ depth setup that a recurrent solution would require.
2. Beyond Logarithmic Depth: Remarkably, for semiautomata classified as solvable (having no non-solvable permutation groups within their transformation semigroups), constant-depth solutions exist. The authors leverage the Krohn-Rhodes decomposition, a deep result in algebraic automata theory, showing that these semiautomata can be simulated by depth-$O(1)$ Transformers.
3. Experimental Validations: The paper reports extensive experiments where Transformers are able to learn these shortcut solutions across diverse sets of automata. However, the results also note statistical brittleness—transformers struggle with out-of-distribution generalization when relying on shortcuts.
4. Implications for Complexity Theory: The findings highlight a practical approximation of the computational complexity challenge. Specifically, they show that improving these results for non-solvable automata, thereby decreasing the depth further, ties into the unresolved question of whether $\mathsf{TC}^0 = \mathsf{NC}^1$.

Implications and Future Directions

From a theoretical standpoint, these results enrich our understanding of how Transformers might be leveraging underlying algebraic structures, offering a broader perspective on their capacity for algorithmic abstraction. Practically, these findings suggest exciting opportunities for developing more computationally efficient deep learning architectures by deliberately embedding and exploiting such algebraic properties.

However, the practical realization of these findings into robust general-purpose architectures remains a challenge, particularly given the noted brittleness of shortcut-based models. Future efforts may include refining architectural designs and training protocols to better generalize these shortcuts, or combining them with more traditional recurrent methods to balance efficiency and stability.

In conclusion, this paper pushes the boundary of how we understand and harness neural networks' capabilities in algorithmic reasoning. It opens potential avenues for designing architectures that blend theoretical insights from automata and algebra with practical deep learning workflows, offering a fascinating intersection of theoretical computer science and modern machine learning.