Efficient Transfer Learning via Causal Bounds

Transfer learning seeks to accelerate sequential decision-making by leveraging offline data from related agents. However, data from heterogeneous sources that differ in observed features, distributions, or unobserved confounders often render causal effects non-identifiable and bias naive estimators. We address this by forming ambiguity sets of structural causal models defined via integral constraints on their joint densities. Optimizing any causal effect over these sets leads to generally non-convex programs whose solutions tightly bound the range of possible effects under heterogeneity or confounding. To solve these programs efficiently, we develop a hit-and-run sampler that explores the entire ambiguity set and, when paired with a local optimization oracle, produces causal bound estimates that converge almost surely to the true limits. We further accommodate estimation error by relaxing the ambiguity set and exploit the Lipschitz continuity of causal effects to establish precise error propagation guarantees. These causal bounds are then embedded into bandit algorithms via arm elimination and truncated UCB indices, yielding optimal gap-dependent and minimax regret bounds. To handle estimation error, we also develop a safe algorithm for incorporating noisy causal bounds. In the contextual-bandit setting with function approximation, our method uses causal bounds to prune both the function class and the per-context action set, achieving matching upper and lower regret bounds with only logarithmic dependence on function-class complexity. Our analysis precisely characterizes when and how causal side-information accelerates online learning, and experiments on synthetic benchmarks confirm substantial regret reductions in data-scarce or confounded regimes.