We give a new algorithm for finding balanced separators in minor-free and induced-minor-free graphs. A key technical idea is a new connection to online learning: we formulate a continuous, non-convex optimization problem and solve it using the Follow the Regularized Leader (FTRL) framework. Our implementation of each FTRL step yields a purely combinatorial algorithm that assigns and adjusts vertex weights over multiple rounds. For graphs with $n$ vertices and $m$ edges, we obtain the following results. (1) For $H$-induced-minor-free graphs, we find a balanced separator of size $O(\log(|V(H)|)\sqrt{|V(H)|+|E(H)|}\sqrt{m})$ in expected linear time for fixed $H$. As a corollary, we find balanced separators of size $O(\sqrt{m})$ in string graphs in expected linear time, without requiring string representations. (2) For $K_h$-minor-free graphs, we find a balanced separator of size $O(h\log h\sqrt{n})$ in expected linear time for fixed $h$. Our algorithm unifies two long lines of work on separators in minor-free graphs, dating back to Alon, Seymour, and Thomas (1990): one optimizing separator size and the other optimizing running time. It also gives a natural interpretation of the vertex weighting scheme in a recent linear-time algorithm by Bonnet, Korhonen, Le, Li, and Masarik (2026). The separator bounds for both graph classes are flow-optimal: techniques based on flow-cut duality, including ours, cannot yield balanced separators of asymptotically smaller size.

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