Shahar Gottlieb

Ofir Razon, Yoav Harris, Shahar Gottlieb et al.

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for $e, e^2, tan(1)$, and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.

Niv Giladi, Shahar Gottlieb, Moran Shkolnik et al.

Background: Distributed training is essential for large scale training of deep neural networks (DNNs). The dominant methods for large scale DNN training are synchronous (e.g. All-Reduce), but these require waiting for all workers in each step. Thus, these methods are limited by the delays caused by straggling workers. Results: We study a typical scenario in which workers are straggling due to variability in compute time. We find an analytical relation between compute time properties and scalability limitations, caused by such straggling workers. With these findings, we propose a simple yet effective decentralized method to reduce the variation among workers and thus improve the robustness of synchronous training. This method can be integrated with the widely used All-Reduce. Our findings are validated on large-scale training tasks using 200 Gaudi Accelerators.

Gal Raayoni, Shahar Gottlieb, George Pisha et al.

Fundamental mathematical constants like $e$ and $π$ are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating fundamental constants have been scarce and usually discovered sporadically. Here we propose a novel and systematic approach that leverages algorithms for deriving mathematical formulas for fundamental constants and help reveal their underlying structure. Our algorithms find dozens of well-known as well as previously unknown continued fraction representations of $π$, $e$, Catalan's constant, and values of the Riemann zeta function. Two example conjectures found by our algorithm and so far unproven are: \begin{equation*} \frac{24}{π^2} = 2 + 7\cdot 0\cdot 1+ \frac{8\cdot1^4}{2 + 7\cdot 1\cdot 2 + \frac{8\cdot2^4}{2 + 7\cdot 2\cdot 3 + \frac{8\cdot3^4}{2 + 7\cdot 3\cdot 4 + \frac{8\cdot4^4}{..}}}} \quad\quad,\quad\quad \frac{8}{7 ζ(3)} = 1\cdot 1 - \frac{1^6}{3\cdot 7 - \frac{2^6}{5\cdot 19 - \frac{3^6}{7\cdot 37 - \frac{4^6}{..}}}} \end{equation*} We present two algorithms that proved useful in finding conjectures: a Meet-In-The-Middle (MITM) algorithm and a Gradient Descent (GD) tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values and thus they conjecture formulas without providing proofs and without requiring prior knowledge on any underlying mathematical structure. This approach is especially attractive for constants for which no mathematical structure is known, as it reverses the conventional approach of sequential logic in formal proofs. Instead, our work supports a different approach for research: algorithms utilizing numerical data to unveil mathematical structures, thus trying to play the role of intuition of great mathematicians of the past, providing leads to new mathematical research.