In LLM alignment and many other ML applications, one often faces the Multi-Objective Fine-Tuning (MOFT) problem, i.e. fine-tuning an existing model with datasets labeled w.r.t. different objectives simultaneously. To address the challenge, we propose the HyperDPO framework, a conditioned one-shot fine-tuning approach that extends the Direct Preference Optimization (DPO) technique, originally developed for efficient LLM alignment with preference data, to accommodate the MOFT settings. By substituting the Bradley-Terry-Luce model in DPO with the Plackett-Luce model, our framework is capable of handling a wide range of MOFT tasks that involve listwise ranking datasets. Compared with previous approaches, HyperDPO enjoys an efficient one-shot training process for profiling the Pareto front of auxiliary objectives, and offers post-training control over trade-offs. Additionally, we propose a novel Hyper Prompt Tuning design, that conveys continuous importance weight across objectives to transformer-based models without altering their architecture, and investigate the potential of temperature-conditioned networks for enhancing the flexibility of post-training control. We demonstrate the effectiveness and efficiency of the HyperDPO framework through its applications to various tasks, including Learning-to-Rank (LTR) and LLM alignment, highlighting its viability for large-scale ML deployments.

Entropic optimal transport (OT) and the Sinkhorn algorithm have made it practical for machine learning practitioners to perform the fundamental task of calculating transport distance between statistical distributions. In this work, we focus on a general class of OT problems under a combination of equality and inequality constraints. We derive the corresponding entropy regularization formulation and introduce a Sinkhorn-type algorithm for such constrained OT problems supported by theoretical guarantees. We first bound the approximation error when solving the problem through entropic regularization, which reduces exponentially with the increase of the regularization parameter. Furthermore, we prove a sublinear first-order convergence rate of the proposed Sinkhorn-type algorithm in the dual space by characterizing the optimization procedure with a Lyapunov function. To achieve fast and higher-order convergence under weak entropy regularization, we augment the Sinkhorn-type algorithm with dynamic regularization scheduling and second-order acceleration. Overall, this work systematically combines recent theoretical and numerical advances in entropic optimal transport with the constrained case, allowing practitioners to derive approximate transport plans in complex scenarios.

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast $O(n^2)$ per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein $W_1, W_2$ distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

Multi-objective optimization (MOO) aims to optimize multiple, possibly conflicting objectives with widespread applications. We introduce a novel interacting particle method for MOO inspired by molecular dynamics simulations. Our approach combines overdamped Langevin and birth-death dynamics, incorporating a "dominance potential" to steer particles toward global Pareto optimality. In contrast to previous methods, our method is able to relocate dominated particles, making it particularly adept at managing Pareto fronts of complicated geometries. Our method is also theoretically grounded as a Wasserstein-Fisher-Rao gradient flow with convergence guarantees. Extensive experiments confirm that our approach outperforms state-of-the-art methods on challenging synthetic and real-world datasets.

We consider the online path learning problem in a graph with non-additive gains/losses. Various settings of full information, semi-bandit, and full bandit are explored. We give an efficient implementation of EXP3 algorithm for the full bandit setting with any (non-additive) gain. Then, focusing on the large family of non-additive count-based gains, we construct an intermediate graph which has equivalent gains that are additive. By operating on this intermediate graph, we are able to use algorithms like Component Hedge and ComBand for the first time for non-additive gains. Finally, we apply our methods to the important application of ensemble structured prediction.

We consider the problem of repeatedly solving a variant of the same dynamic programming problem in successive trials. An instance of the type of problems we consider is to find a good binary search tree in a changing environment. At the beginning of each trial, the learner probabilistically chooses a tree with the n keys at the internal nodes and the n + 1 gaps between keys at the leaves. The learner is then told the frequencies of the keys and gaps and is charged by the average search cost for the chosen tree. The problem is online because the frequencies can change between trials. The goal is to develop algorithms with the property that their total average search cost (loss) in all trials is close to the total loss of the best tree chosen in hindsight for all trials. The challenge, of course, is that the algorithm has to deal with exponential number of trees. We develop a general methodology for tackling such problems for a wide class of dynamic programming algorithms. Our framework allows us to extend online learning algorithms like Hedge and Component Hedge to a significantly wider class of combinatorial objects than was possible before.