The Mean-Field Limit of Online Stochastic Vector Balancing
We study an online vector balancing problem, in which $n$ independent Gaussian random vectors $\boldsymbol{\zeta}(1),\dots,\boldsymbol{\zeta}(n) \sim \mathcal{N}(0, I_n)$, each of dimension $n$, arrive one at a time. The goal is to choose signs $\varepsilon(1),\dots,\varepsilon(n) \in \{\pm 1\}$ with $\varepsilon(k)$ depending only on $\boldsymbol{\zeta}(1),\dots,\boldsymbol{\zeta}(k)$, so as to minimize the expected $\ell^{\infty}$ norm of the signed sum $\frac{1}{\sqrt{n}}\sum_{k = 1}^n \varepsilon(k) \boldsymbol{\zeta}(k)$. Prior work showed that the optimal value $V^n$ is $O(1)$, at least for Rademacher $\boldsymbol{\zeta}(k)$'s, by constructing specific algorithms. Our main contribution is to determine the exact limit $V^{\infty} = \lim_{n\to\infty} V^n$ as the value of a nonstandard stochastic control problem of mean-field type: find the narrowest terminal interval into which a Brownian motion can be adaptively steered under a uniform-in-time $L^2$ constraint on the drift. The proof of the lower bound $V^{\infty} \leq \liminf_{n \to \infty} V^n$ uses probabilistic compactness arguments, and is very flexible. In fact, we show that the lower bound is universal, in that it holds as long as the entries of the $\boldsymbol{\zeta}(k)$ vectors are i.i.d. with mean zero, variance 1, and finite fourth moment. The proof of the upper bound $\limsup_{n \to \infty} V^n \leq V^{\infty}$ is more delicate, relying on dynamic programming principles and a priori bounds obtained from a coupling procedure involving the F\"ollmer drift, which makes explicit use of the Gaussian structure. In addition to our main convergence result, we provide some analysis and asymptotics for the limiting mean-field control problem.