Provable Failure of Language Models in Learning Majority Boolean Logic via Gradient Descent (2504.04702v1)

Abstract: Recent advancements in Transformer-based architectures have led to impressive breakthroughs in natural language processing tasks, with models such as GPT-4, Claude, and Gemini demonstrating human-level reasoning abilities. However, despite their high performance, concerns remain about the inherent limitations of these models, especially when it comes to learning basic logical functions. While complexity-theoretic analyses indicate that Transformers can represent simple logic functions (e.g., $\mathsf{AND}$, $\mathsf{OR}$, and majority gates) by its nature of belonging to the $\mathsf{TC}^0$ class, these results assume ideal parameter settings and do not account for the constraints imposed by gradient descent-based training methods. In this work, we investigate whether Transformers can truly learn simple majority functions when trained using gradient-based methods. We focus on a simplified variant of the Transformer architecture and consider both $n=\mathrm{poly}(d)$ and $n=\exp(\Omega(d))$ number of training samples, where each sample is a $d$-size binary string paired with the output of a basic majority function. Our analysis demonstrates that even after $\mathrm{poly}(d)$ gradient queries, the generalization error of the Transformer model still remains substantially large, growing exponentially with $d$. This work highlights fundamental optimization challenges in training Transformers for the simplest logical reasoning tasks and provides new insights into their theoretical limitations.

Understanding the Limitations of Gradient Descent in Training Transformers for Boolean Logic Tasks

The paper "Provable Failure of LLMs in Learning Majority Boolean Logic via Gradient Descent" presents a rigorous analytical framework for evaluating the capacity of Transformer models, trained via gradient descent methods, to learn basic logical functions — specifically, the majority function. Despite the architectural advancements and practical success of Transformer-based models such as GPT-4, Claude, and Gemini in complex reasoning tasks, this paper surfaces inherent limitations when these models encounter fundamental logic functions.

Key Findings

The authors delve into the mathematical foundations of Transformer architectures, exploring their placement within the computational complexity class $\mathsf{TC}^0$. This class implies that, theoretically, Transformers should be capable of representing simple Boolean logic functions like $\mathsf{AND}$, $\mathsf{OR}$, and majority gates under ideal conditions. However, these theoretical results often overlook the dynamics introduced by gradient descent-based training methods applied in realistic settings.

The paper investigates the efficacy of a simplified Transformer variant in learning the majority function using two distinct data regimes: polynomial and exponential sample complexities. Contrary to the optimistic theoretical backdrop, the results demonstrate significant limitations:

1. Hardness in Polynomial-Sample Regimes: It is proven that for polynomial samples ($n=\mathrm{poly}(d)$) with $d$ being the dimension of input binary vectors, the generalization error of a Transformer remains notably high, underscoring the optimization obstacles encountered during the learning process. The error grows exponentially with dimension $d$, reflecting an inability to generalize beyond the training dataset adequately.
2. Hardness in Exponential-Sample Regimes: Even as the number of training samples increases to exponential levels ($n=\exp(\Omega(d))$), the paper finds the error bounds remain unexpectedly large. This suggests an intrinsic complexity within the gradient descent optimization path that precludes the attainment of an efficient representation for even the simplest majority logic tasks.

Methodological Contributions

The theoretical advancement in this paper lies in establishing lower bounds for generalization errors and providing a formal framework for analyzing optimization challenges posed by gradient descent methods in learning tasks. Two key theorems highlight the models' failure even under expansive training sets:

• The main result concerning polynomial samples demonstrates the persistent and substantial generalization error.
• The analogous result for exponential samples extends this idea to more extensive datasets, still concluding with high generalization errors.

Prominent is the innovative use of combinatorial and probabilistic tools to handle the binomial coefficient computations associated with these Boolean tasks, allowing the authors to derive precise characterizations of gradient variance and bounds on learning capabilities.

Implications and Future Work

These findings have significant implications for both theoretical and practical AI research. The demonstrated restrictions suggest that achieving generalized logical reasoning in Transformers might require architectural changes or alternative training frameworks beyond standard gradient descent. This revelation mandates future explorations into developing novel training algorithms or architectures that can act upon these foundational limitations.

Theoretically, this paper opens avenues for evaluating other basic logical functions and extending these principles to more complex constructs such as nested Boolean operations or arithmetic evaluations. Additionally, the exploration of alternative means of training neural networks, possibly incorporating symbolic reasoning elements or hybrid models, appears compelling.

In conclusion, while Transformers excel in many domains, this paper highlights their stark challenges with gradient descent in learning elementary logic functions, emphasizing the complexity and non-trivial nature of such seemingly simple tasks within the neural computation landscape. This calls for continued investigation into optimizing methods and architectures to bridge this gap in AI models' reasoning capabilities.