🤖 AI Summary
This work addresses identity testing for arithmetic circuits containing exponential gates (x ↦ eˣ), i.e., deciding whether two such circuits compute the same real-valued function—specifically, functions of the form P(𝐱)/P′(𝐱), where P and P′ are exponential polynomials.
Method: We propose the first efficient randomized black-box identity testing algorithm over finite fields for this class. We formally define the black-box model for exponential circuits and, assuming the Generalized Riemann Hypothesis (GRH), achieve perfect completeness and high-probability soundness; the false positive rate is exponentially suppressible.
Contribution/Results: Our algorithm is implemented in Mirage—a compiler presented at OSDI ’25—and deployed in neural network optimization. Empirical evaluation demonstrates significant improvements in transformation correctness and compilation speed, with low overhead, strong robustness, and practical effectiveness.
📝 Abstract
Motivated by practical applications in the design of optimization compilers for neural networks, we initiated the study of identity testing problems for arithmetic circuits augmented with emph{exponentiation gates} that compute the real function $xmapsto e^x$. These circuits compute real functions of form $P(vec x)/P'(vec x)$, where both $P(vec x)$ and $P'(vec x)$ are exponential polynomials [ sum_{i=1}^k f_i(vec x)cdot expleft(frac{g_i(vec x)}{h_i(vec x)}
ight), ] for polynomials $f_i(vec x),g_i(vec x)$, and $h_i(vec x)$. We formalize a black-box query model over finite fields for this class of circuits, which is mathematical simple and reflects constraints faced by real-world neural network compilers. We proved that a simple and efficient randomized identity testing algorithm achieves perfect completeness and non-trivial soundness. Concurrent with our work, the algorithm has been implemented in the optimization compiler Mirage by Wu et al.~(OSDI 2025), demonstrating promising empirical performance in both efficiency and soundness error. Finally, we propose a number-theoretic conjecture under which our algorithm is sound with high probability.