Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers (2107.13163v3)

Abstract: A common lens to theoretically study neural net architectures is to analyze the functions they can approximate. However, constructions from approximation theory may be unrealistic and therefore less meaningful. For example, a common unrealistic trick is to encode target function values using infinite precision. To address these issues, this work proposes a formal definition of statistically meaningful (SM) approximation which requires the approximating network to exhibit good statistical learnability. We study SM approximation for two function classes: boolean circuits and Turing machines. We show that overparameterized feedforward neural nets can SM approximate boolean circuits with sample complexity depending only polynomially on the circuit size, not the size of the network. In addition, we show that transformers can SM approximate Turing machines with computation time bounded by $T$ with sample complexity polynomial in the alphabet size, state space size, and $\log (T)$. We also introduce new tools for analyzing generalization which provide much tighter sample complexities than the typical VC-dimension or norm-based bounds, which may be of independent interest.

• The paper introduces a novel Statistically Meaningful (SM) approximation framework that balances expressivity with statistical learnability.
• It demonstrates that overparameterized feedforward networks can approximate Boolean circuits with sample complexity polynomial in the intrinsic circuit size.
• Transformers are shown to SM-approximate Turing machines with sample complexity polynomial in the logarithm of computation time, paving the way for practical deep learning designs.

Statistically Meaningful Approximation of Turing Machines and Boolean Circuits

In this paper, the authors propose a novel framework for examining neural network architectures known as Statistically Meaningful (SM) approximation. Distinguishing itself from classical approximation theory, SM approximation addresses practical concerns about expressivity and statistical learnability. Specifically, it entails that the approximating network must exhibit satisfactory statistical learnability, taking into consideration not only function expressivity but also sample complexity aspects related to optimization and generalization.

The paper focuses on two case studies to demonstrate the efficacy of the SM approximation framework. The first paper involves overparameterized feedforward neural networks, showing that these networks can SM-approximate boolean circuits with a sample complexity polynomial in the intrinsic circuit size. This result does not depend on network size, addressing issues encountered in classical approximation where sample complexities often balloon with network size, particularly in overparameterized settings. By using a function class $\mathcal{F}$ that incorporates novel sample complexity tools, such as all-layer margin methods, the authors achieve this dimension-independent approximation.

In the second case paper, the authors demonstrate the SM approximation of Turing machines using transformer architectures. Notably, they achieve this with a sample complexity that's polynomial in the logarithm of the Turing machine’s computation time ($T$) and other relevant parameters, such as alphabet size and state space. This is a substantial improvement over previous constructions that would have required linear dependencies on $T$, resulting in much less feasible sample complexity levels.

The concept introduced in the paper bears substantial implications both in theory and application. Practically, SM approximation provides a robust framework for designing neural network architectures with assured sample complexities that align more closely with realistic learning and optimization scenarios. Theoretically, it challenges existing approximation paradigms by critiquing unrealistic conventions like models requiring infinite precision, paving the way for more meaningful assessments of expressivity in deep learning models.

The work acknowledges certain limitations, particularly concerning optimization analysis, which remains unresolved for even basic neural network constructs. It suggests potential future research directions, including exploring SM approximation bounds for a wider class of functions and architectures, aiming to enhance our understanding of how statistical learnability can be systematically ensured.

In conclusion, the paper makes significant strides toward refining the theoretical understanding of neural networks through the lens of statistical learnability. By proving SM approximation capabilities with strong sample complexity bounds for both boolean circuits and Turing machines, it opens new avenues for constructing neural architectures that are credible both in terms of expressivity and sample learnability.