SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front

We consider a stochastic multi-objective bandit problem where, at each round, the agent selects a slate of $k$ arms and observes their $d$-dimensional reward vectors under semi-bandit feedback. We do not aim at identifying a single optimal arm; instead, we consider the problem of maintaining a small set of actions that jointly approximate the Pareto frontier. We formalize this objective through the dominated hypervolume induced by the selected subset of arms, and define an $α$-approximate hypervolume regret with respect to the best size-$k$ subset achievable in hindsight, where $α= 1 - 1/e$ reflects the approximation guarantee of greedy maximization for monotone submodular functions. To address this problem, we introduce \textit{THV-UCB}, an optimistic algorithm that selects arms greedily based on optimistic estimates of their marginal hypervolume contributions. We establish a gap-free regret bound $\tilde{O}(d\sqrt{nkT})$ that holds on every instance, together with a gap-dependent bound $\tilde{O}(nk^{2.5}/Δ_{\min})$ that becomes polylogarithmic in $T$ once the arms are sufficiently well separated. Our results provide theoretical support for using small subsets to approximate Pareto fronts in various multi-objective applications.

Preference-conditioned multi-objective reinforcement learning aims to learn a single policy that captures trade-offs across preferences, but under nonlinear scalarization the uniqueness and continuity of the preference-to-solution correspondence remain unclear. We study this problem in tabular multi-objective Markov decision processes (MDPs) using smooth Tchebycheff scalarization as a monotone utility. Under mild interior conditions on the preference set, we prove that each preference induces a unique Pareto-optimal return vector and that this vector depends Lipschitz-continuously on the preference, providing a principled foundation for preference sweeping toward dense Pareto-front coverage. To compute these targets, we formulate the problem over occupancy measures and derive Concave Mirror Descent Policy Iteration (CMDPI), which achieves an $O(1/k)$ objective-suboptimality rate. We further show that each update is equivalent to solving a Kullback-Leibler-regularized MDP with the previous policy as reference, yielding a policy-iteration interpretation and finite-iterate policy continuity across preferences. We instantiate the update as a deep actor-critic algorithm preserving previous-policy regularization. On eight MO-Gymnasium tasks, it achieves the best average hypervolume rank among recent baselines and strong expected-utility performance. Continuous-control experiments indicate gains beyond the discrete-action setting.

Offline multi-objective optimization requires not only moving the objective vectors of candidate designs toward the Pareto front, but also distributing them effectively along it. Generative methods have recently emerged as a natural approach because they learn a distribution over feasible designs while allowing generation to be steered toward promising designs. Existing methods, however, largely retain classical sample-wise guidance strategies, leaving the distribution-level modeling capability of generative methods underused. We propose ParetoTransport, a training-free guidance method for pre-trained flow-matching models that explicitly specifies and refines a population-level distribution in objective space. ParetoTransport guides a flow-matching sampler to iteratively transport the empirical offline distribution toward the Pareto front, with Wasserstein matching to intermediate proxy distributions. This directly controls distributional displacement and mass allocation along the front. We establish a convergence result and demonstrate state-of-the-art performance on standard offline MOO benchmarks, extending recent evaluations beyond hypervolume to generational distance, inverted generational distance, and Wasserstein distance.