What Algorithms can Transformers Learn? A Study in Length Generalization
(2310.16028)Abstract
Large language models exhibit surprising emergent generalization properties, yet also struggle on many simple reasoning tasks such as arithmetic and parity. This raises the question of if and when Transformer models can learn the true algorithm for solving a task. We study the scope of Transformers' abilities in the specific setting of length generalization on algorithmic tasks. Here, we propose a unifying framework to understand when and how Transformers can exhibit strong length generalization on a given task. Specifically, we leverage RASP (Weiss et al., 2021) -- a programming language designed for the computational model of a Transformer -- and introduce the RASP-Generalization Conjecture: Transformers tend to length generalize on a task if the task can be solved by a short RASP program which works for all input lengths. This simple conjecture remarkably captures most known instances of length generalization on algorithmic tasks. Moreover, we leverage our insights to drastically improve generalization performance on traditionally hard tasks (such as parity and addition). On the theoretical side, we give a simple example where the "min-degree-interpolator" model of learning from Abbe et al. (2023) does not correctly predict Transformers' out-of-distribution behavior, but our conjecture does. Overall, our work provides a novel perspective on the mechanisms of compositional generalization and the algorithmic capabilities of Transformers.
Overview
-
The paper investigates the ability of Transformer architectures to generalize specific algorithms beyond the lengths seen during training, focusing on length generalization in algorithmic tasks.
-
It introduces the RASP-Generalization Conjecture and demonstrates that Transformers can generalize well on tasks solvable by concise RASP programs, but struggle with more complex operations like parity and addition.
-
The study provides empirical evidence highlighting how reformulating tasks and incorporating intermediate steps can enhance the length generalization ability of Transformers.
A Study in Length Generalization: What Algorithms can Transformers Learn?
Abstract: LLMs have demonstrated significant promise in a variety of NLP tasks but often struggle with straightforward algorithmic reasoning tasks like arithmetic and parity. This variability in performance engenders crucial questions about the inherent capabilities of Transformer architectures to learn and accurately generalize specific algorithms, particularly when generalizing beyond lengths seen during training. This essay provides a comprehensive overview of a paper that investigates these questions. Focusing on the length generalization of algorithmic tasks, the paper introduces a conceptual framework and empirically tests when and how Transformers display robust length generalization.
Introduction
LLMs have shown impressive skill in diverse tasks ranging from code synthesis to commonsense reasoning. However, they often falter on algorithmic tasks, particularly when these tasks demand out-of-distribution generalization. This dichotomy in performance suggests there is a nuanced complexity in the way Transformers learn tasks and the kinds of tasks they can inherently generalize on. The core research question addressed by the paper is: under what conditions can Transformers learn the underlying true algorithm of a task and generalize out-of-distribution in terms of input sequence length?
RASP and the RASP-Generalization Conjecture
To address this question, the authors leverage the Restricted Assembly for Sequence Processing (RASP), a programming language tailored for transforming models akin to Transformers. They introduce the RASP-Generalization Conjecture, positing that Transformers tend to generalize in length on tasks if these tasks can be solved by short RASP programs operable across all input lengths. This conjecture, although simplistic, explains many known instances of length generalization failures and successes and guides experiments on various algorithmic tasks.
Empirical Findings
The empirical investigation spans multiple tasks:
- Count: The Transformer model successfully generalized to sequences much longer than those encountered during training. The model accurately transitioned from training lengths up to 50 tokens to testing on lengths up to 150 tokens, provided the training sequences were varied and diverse.
- Mode and Copy (with unique tokens): Transformers also showed robust length generalization abilities in tasks such as determining the mode of a sequence and copying sequences with unique tokens. These tasks can be solved using straightforward RASP programs.
- Sort: Although slightly more complex than the counting task, sorting a sequence, which can also be expressed through a concise RASP program, exhibited strong length generalization.
- Hard Tasks (Parity and Addition): Length generalization failed for tasks like parity and addition when formulated in their standard formats. These tasks require more intricate and non-uniform operations that are not easily representable in a short RASP program.
Enhancing Generalization via Task Reformulation
To explore the boundaries of the RASP-Generalization Conjecture further, the research shifted to reformulate difficult tasks through scratchpad mechanisms and task-specific reformulations:
- Addition with Index Hints: By introducing index hints and reversing the order of output digits, length generalization for addition drastically improved. This simplified the operations into constituent steps that are approachable via RASP programs.
- Parity with Scratchpads: Similarly, the inclusion of intermediate scratchpad-like outputs allowed the parity task, traditionally viewed as challenging for Transformers, to generalize well when trained up to suitable sequence lengths.
Theoretical Implications and Limitations
The theoretical aspect of this work lies in its conjecture that the simplicity of representing algorithms in a Transformer-friendly domain like RASP correlates with their learnability by these models. Furthermore, the study discusses why some commonly used architectural choices might fail in providing the desired generalization.
Notably:
- Realizability: There are inherent constraints on what can be achieved through a single, fixed Transformer configuration.
- Simplicity and Diversity: The paper highlights how diversity in the training set is crucial in preventing simplistic, non-generalizing programs from being learned.
Comparison to Prior Work
Comparatively, this study diverges from the perspective that Transformers operate based on analogical pattern matching alone, as put forth by \citet{FF}. Instead, it substantiates that Transformers can systematically learn algorithms subject to the tasks being representable through ‘simple’ RASP programs. The counterexamples demonstrated underline the limitations of the minimal-degree-interpolator hypothesis from \citet{abbe2023generalization}, offering a more architecture-appropriate measure of complexity and simplicity for Transformers.
Conclusion
The research sheds light on the conditions under which Transformers can exhibit strong length generalization, offering a grounded perspective on their algorithmic capabilities. It emphasizes how suitable task reformulation and adequate training data complexity can push the boundaries of what these models can achieve. Moving forward, the study sets a foundation for further investigating the mechanisms underlying compositional generalization in AI, both theoretically and practically.
Future Directions
Future work might involve refining the notion of "simplicity" in Transformer-specific contexts. Exploring more sophisticated architectures or training regimes that naturally extend to diverse algorithmic patterns will also be pivotal. Further, efforts could be directed at formally characterizing the class of algorithms that Transformers can efficiently learn, thereby potentially advancing our understanding of their inductive biases and limitations.
The insights gathered from this study are instrumental in advancing the field of scalable and reliable AI, setting the stage for developing models that can robustly generalize over a more comprehensive set of tasks.
Create an account to read this summary for free: