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Rapport Année : 2022

On the Multidimensional Random Subset Sum Problem

Résumé

In the Random Subset Sum Problem, given $n$ i.i.d.\ random variables $X_1, ..., X_n$, we wish to approximate any point $z \in [-1,1]$ as the sum of a suitable subset $X_{i_1(z)}, ..., X_{i_s(z)}$ of them, up to error $\varepsilon$. Despite its simple statement, this problem is of fundamental interest to both theoretical computer science and statistical mechanics. More recently, it gained renewed attention for its implications in the theory of Artificial Neural Networks. An obvious multidimensional generalisation of the problem is to consider $n$ i.i.d.\ $d$-dimensional random vectors, with the objective of approximating every point $\mathbf{z} \in [-1,1]^d$. Rather surprisingly, after Lueker's 1998 proof that, in the one-dimensional setting, $n=O(\log \frac 1\varepsilon)$ samples guarantee the approximation property with high probability, little progress has been made on achieving the above generalisation. In this work, we prove that, in $d$ dimensions, $n = O(d^3\log \frac 1\varepsilon \cdot (\log \frac 1\varepsilon + \log d))$ samples suffice for the approximation property to hold with high probability. As an application highlighting the potential interest of this result, we prove that a recently proposed neural network model exhibits \emph{universality}: with high probability, the model can approximate any neural network within a polynomial overhead in the number of parameters.
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Dates et versions

hal-03738204 , version 1 (25-07-2022)
hal-03738204 , version 2 (26-07-2022)
hal-03738204 , version 3 (15-11-2022)

Identifiants

  • HAL Id : hal-03738204 , version 1

Citer

Luca Becchetti, Arthur Carvalho Walraven da Cuhna, Andrea Clementi, Francesco d'Amore, Hicham Lesfari, et al.. On the Multidimensional Random Subset Sum Problem. [Research Report] Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France; Sapienza Università di Roma, Rome, Italy; Università Bocconi, Milan, Italy; Università di Roma Tor Vergata, Rome, Italy. 2022. ⟨hal-03738204v1⟩
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