Descriptive complexity for neural networks via Boolean networks (2308.06277v4)

Abstract: We investigate the expressive power of neural networks from the point of view of descriptive complexity. We study neural networks that use floating-point numbers and piecewise polynomial activation functions from two perspectives: 1) the general scenario where neural networks run for an unlimited number of rounds and have unrestricted topologies, and 2) classical feedforward neural networks that have the topology of layered acyclic graphs and run for only a constant number of rounds. We characterize these neural networks via Boolean networks formalized via a recursive rule-based logic. In particular, we show that the sizes of the neural networks and the corresponding Boolean rule formulae are polynomially related. In fact, in the direction from Boolean rules to neural networks, the blow-up is only linear. Our translations result in a time delay, which is the number of rounds that it takes to simulate a single computation step. In the translation from neural networks to Boolean rules, the time delay of the resulting formula is polylogarithmic in the size of the neural network. In the converse translation, the time delay of the neural network is linear in the formula size. Ultimately, we obtain translations between neural networks, Boolean networks, the diamond-free fragment of modal substitution calculus, and a class of recursive Boolean circuits. Our translations offer a method, for almost any activation function F, of translating any neural network in our setting into an equivalent neural network that uses F at each node. This even includes linear activation functions, which is possible due to using floats rather than actual reals!

• The paper formulates Boolean network logic to relate neural network sizes with Boolean rule formulae through polynomial transformations and linear blow-up bounds.
• It demonstrates efficient two-way translations: neural networks convert to Boolean representations in polylogarithmic time, while the reverse process is polynomially bounded.
• The research bridges computational logic and neural models, offering actionable insights for algorithm design, hardware performance, and transparent AI diagnostics.

Descriptive Complexity for Neural Networks via Boolean Networks

The paper entitled "Descriptive complexity for neural networks via Boolean networks" by Veeti Ahvonen, Damian Heiman, and Antti Kuusisto investigates the descriptive complexity of neural networks through the lens of Boolean network logic (BNL). The work offers a logical characterisation of a broad class of neural networks, primarily focusing on networks with unrestricted topologies and piecewise polynomial activation functions. Here, the discretisation and Boolean transformations form the core analytical constructs. This essay will summarise the key contributions, strong numerical results, and implications of this research, both theoretical and practical, and posit on the future directions for Neural Networks (NN) and AI.

Overview and Contributions

The research pursues a comprehensive characterisation of neural networks using rule-based logic typical in Boolean networks. It demonstrates how the sizes of neural networks and their corresponding Boolean rule formulae are polynomially related, noting that the blow-up from Boolean rules to neural networks is merely linear. Conversely, translating from neural networks to Boolean rules induces a polylogarithmic time delay relative to the neural network size and linearly dependent on time, offering robust scalability.

Key contributions include:

1. Formulation of Boolean Network Logic (BNL): The authors extend typical Boolean networks to BNL, including terminal clauses and characterising various computational settings.
2. Polynomial Translations: Establish polynomial relations between NN sizes and BNL program sizes, showing both directions of translations.
3. Descriptive Analogues: Corroborate BNL's correlation with the diamond-free fragment of modal substitution calculus (SC) and self-feeding circuits. This contributes to understanding how traditional fixed-point logics can be substituted into neural network-based computations, drawing an accessible bridge between discrete computational theories and continuous neural paradigms.

Numerical Results and Verification

Translation Efficiency:

• Given a neural network $N$ in a floating-point system $S(p, q, \beta)$ with nodes $N$, degree $\Delta$, piece-size $P$, and order $\Omega$, its corresponding BNL program $\Lambda$ has size: $$O(N (\Delta + P \Omega^{2}) (r^{4} + r^{3} \beta^{2} + r \beta^{4}))$$, where $r = \max \{p, q\}$.
• Time complexity for $\Lambda$ to simulate $N$ is $$O((\log(\Omega) + 1)(\log(r) + \log(\beta)) + \log(\Delta))$$.

Reversals from BNL to NNs:

• The conversion from a BNL program $\Lambda$ comprising size $s$ and depth $d$ to a general neural network $N$ guarantees a polynomial relationship in both size and time. Specifically, producing $N$:
  • Size: ≤ $s$,
  • Degree: ≤ 2,
  • Activation function: $\mathrm{ReLU} (x) = \max\{0, x\}$.

Implications and Future Work

Theoretical Impact:

• Boundaries of Logic and Learnability: The blend of logic-based and non-symbolic methods reaffirms that symbolic representations play a crucial underpinning in understanding the capacities and limits of neural networks. The theoretical contacts established between BNL and NN models encompass application to recursively enumerable languages through finite and non-finite input spaces.
• Randomisation and Extensions: Enhancing these networks to support randomness and other arithmetic forms like fixed-point computing could foster new insights and abstract capabilities inclusive of AI learning dynamics.

Practical Relevance:

• Performance Metrics: Knowing bounds on translation delays and fixed time overheads has practical implications in hardware advances, directly influencing how neural processors like TPUs (Tensor Processing Units) are designed.
• Algorithmic Insights: The polynomial bounds direct efforts towards constructing effective algorithms that leverage the logic-awareness of our models, possibly purifying iteration methodologies in context-heavy neural evaluations such as visual arts or complex linguistic assessments.

Speculations on AI:

• Advanced Diagnoses and Interpretability: The ability to revert neural expressions to Boolean logic paves the way for more transparent, interpretable, AI tools with built-in diagnostic capacities, possibly aiding fields like healthcare, finance, or autonomous systems.
• Ethical AI: Logical underpinnings ensure a stronger ethical frontend, with controllable, verifiable AI actions. Comprehension of these networks through BNL and logical calculi pronounces a step towards mitigated bias and reproducible fairness in decision-making models.

Conclusion

By exploring the descriptive complexity of neural networks via Boolean networks, the researchers elucidate a significant relationship between different computational models. The synthesis of logic-based and neural frameworks underscores the comprehensive adaptability of neural networks, setting trajectories for future advancements in computational logic and artificial intelligence. This interdisciplinary approach accentuates both practical and theoretical angles, poised to influence forthcoming innovations in AI research and application.