We study the online load balancing problem on unrelated machines, with the objective of minimizing the square of the $\ell_2$ norm of the loads on the machines. The greedy algorithm of Awerbuch et al. (STOC'95) is optimal for deterministic algorithms and achieves a competitive ratio of $3 + 2 \sqrt{2} \approx 5.828$, and an improved $5$-competitive randomized algorithm based on independent rounding has been shown by Caragiannis (SODA'08). In this work, we present the first algorithm breaking the barrier of $5$ on the competitive ratio, achieving a bound of $5-c$ for some $c \geq 0.0157$. To obtain this result, we use a new primal-dual framework to analyze this problem based on a natural semidefinite programming relaxation, together with an online implementation of a correlated randomized rounding procedure of Im and Shadloo (SODA'20). This novel primal-dual framework also yields new, simple and unified proofs of the competitive ratio of the $(3 + 2 \sqrt{2})$-competitive greedy algorithm, the $5$-competitive randomized independent rounding algorithm, and that of a new $4$-competitive optimal fractional algorithm. We also provide lower bounds showing that the previous best randomized algorithm is optimal among independent rounding algorithms, that our new fractional algorithm is optimal, and that a simple greedy algorithm is optimal for the closely related online scheduling problem $R || \sum w_j C_j$.

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