Daily Papers

Many modern deep learning applications require balancing multiple objectives that are often conflicting. Examples include multi-task learning, fairness-aware learning, and the alignment of Large Language Models (LLMs). This leads to multi-objective deep learning, which tries to find optimal trade-offs or Pareto-optimal solutions by adapting mathematical principles from the field of Multi-Objective Optimization (MOO). However, directly applying gradient-based MOO techniques to deep neural networks presents unique challenges, including high computational costs, optimization instability, and the difficulty of effectively incorporating user preferences. This paper provides a comprehensive survey of gradient-based techniques for multi-objective deep learning. We systematically categorize existing algorithms based on their outputs: (i) methods that find a single, well-balanced solution, (ii) methods that generate a finite set of diverse Pareto-optimal solutions, and (iii) methods that learn a continuous Pareto set of solutions. In addition to this taxonomy, the survey covers theoretical analyses, key applications, practical resources, and highlights open challenges and promising directions for future research. A comprehensive list of multi-objective deep learning algorithms is available at https://github.com/Baijiong-Lin/Awesome-Multi-Objective-Deep-Learning.

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

In Multi-Task Learning (MTL), tasks may compete and limit the performance achieved on each other, rather than guiding the optimization to a solution, superior to all its single-task trained counterparts. Since there is often not a unique solution optimal for all tasks, practitioners have to balance tradeoffs between tasks' performance, and resort to optimality in the Pareto sense. Most MTL methodologies either completely neglect this aspect, and instead of aiming at learning a Pareto Front, produce one solution predefined by their optimization schemes, or produce diverse but discrete solutions. Recent approaches parameterize the Pareto Front via neural networks, leading to complex mappings from tradeoff to objective space. In this paper, we conjecture that the Pareto Front admits a linear parameterization in parameter space, which leads us to propose Pareto Manifold Learning, an ensembling method in weight space. Our approach produces a continuous Pareto Front in a single training run, that allows to modulate the performance on each task during inference. Experiments on multi-task learning benchmarks, ranging from image classification to tabular datasets and scene understanding, show that Pareto Manifold Learning outperforms state-of-the-art single-point algorithms, while learning a better Pareto parameterization than multi-point baselines.

Expensive multi-objective optimization problems (EMOPs) are common in real-world scenarios where evaluating objective functions is costly and involves extensive computations or physical experiments. Current Pareto set learning methods for such problems often rely on surrogate models like Gaussian processes to approximate the objective functions. These surrogate models can become fragmented, resulting in numerous small uncertain regions between explored solutions. When using acquisition functions such as the Lower Confidence Bound (LCB), these uncertain regions can turn into pseudo-local optima, complicating the search for globally optimal solutions. To address these challenges, we propose a novel approach called SVH-PSL, which integrates Stein Variational Gradient Descent (SVGD) with Hypernetworks for efficient Pareto set learning. Our method addresses the issues of fragmented surrogate models and pseudo-local optima by collectively moving particles in a manner that smooths out the solution space. The particles interact with each other through a kernel function, which helps maintain diversity and encourages the exploration of underexplored regions. This kernel-based interaction prevents particles from clustering around pseudo-local optima and promotes convergence towards globally optimal solutions. Our approach aims to establish robust relationships between trade-off reference vectors and their corresponding true Pareto solutions, overcoming the limitations of existing methods. Through extensive experiments across both synthetic and real-world MOO benchmarks, we demonstrate that SVH-PSL significantly improves the quality of the learned Pareto set, offering a promising solution for expensive multi-objective optimization problems.

Like generic multi-task learning, continual learning has the nature of multi-objective optimization, and therefore faces a trade-off between the performance of different tasks. That is, to optimize for the current task distribution, it may need to compromise performance on some previous tasks. This means that there exist multiple models that are Pareto-optimal at different times, each addressing a distinct task performance trade-off. Researchers have discussed how to train particular models to address specific trade-off preferences. However, existing algorithms require training overheads proportional to the number of preferences -- a large burden when there are multiple, possibly infinitely many, preferences. As a response, we propose Imprecise Bayesian Continual Learning (IBCL). Upon a new task, IBCL (1) updates a knowledge base in the form of a convex hull of model parameter distributions and (2) obtains particular models to address task trade-off preferences with zero-shot. That is, IBCL does not require any additional training overhead to generate preference-addressing models from its knowledge base. We show that models obtained by IBCL have guarantees in identifying the Pareto optimal parameters. Moreover, experiments on standard image classification and NLP tasks support this guarantee. Statistically, IBCL improves average per-task accuracy by at most 23% and peak per-task accuracy by at most 15% with respect to the baseline methods, with steadily near-zero or positive backward transfer. Most importantly, IBCL significantly reduces the training overhead from training 1 model per preference to at most 3 models for all preferences.

Multiobjective combinatorial optimization (MOCO) problems can be found in many real-world applications. However, exactly solving these problems would be very challenging, particularly when they are NP-hard. Many handcrafted heuristic methods have been proposed to tackle different MOCO problems over the past decades. In this work, we generalize the idea of neural combinatorial optimization, and develop a learning-based approach to approximate the whole Pareto set for a given MOCO problem without further search procedure. We propose a single preference-conditioned model to directly generate approximate Pareto solutions for any trade-off preference, and design an efficient multiobjective reinforcement learning algorithm to train this model. Our proposed method can be treated as a learning-based extension for the widely-used decomposition-based multiobjective evolutionary algorithm (MOEA/D). It uses a single model to accommodate all the possible preferences, whereas other methods use a finite number of solutions to approximate the Pareto set. Experimental results show that our proposed method significantly outperforms some other methods on the multiobjective traveling salesman problem, multiobjective vehicle routing problem, and multiobjective knapsack problem in terms of solution quality, speed, and model efficiency.

In many applications of machine learning, like drug discovery and material design, the goal is to generate candidates that simultaneously maximize a set of objectives. As these objectives are often conflicting, there is no single candidate that simultaneously maximizes all objectives, but rather a set of Pareto-optimal candidates where one objective cannot be improved without worsening another. Moreover, in practice, these objectives are often under-specified, making the diversity of candidates a key consideration. The existing multi-objective optimization methods focus predominantly on covering the Pareto front, failing to capture diversity in the space of candidates. Motivated by the success of GFlowNets for generation of diverse candidates in a single objective setting, in this paper we consider Multi-Objective GFlowNets (MOGFNs). MOGFNs consist of a novel Conditional GFlowNet which models a family of single-objective sub-problems derived by decomposing the multi-objective optimization problem. Our work is the first to empirically demonstrate conditional GFlowNets. Through a series of experiments on synthetic and benchmark tasks, we empirically demonstrate that MOGFNs outperform existing methods in terms of Hypervolume, R2-distance and candidate diversity. We also demonstrate the effectiveness of MOGFNs over existing methods in active learning settings. Finally, we supplement our empirical results with a careful analysis of each component of MOGFNs.

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which n component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter L. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is mathcal{O}( n + nLvarepsilon^{-1}), which improves upon existing methods by a factor up to n. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

Building world models that accurately and comprehensively represent the real world is the utmost aspiration for conditional image generative models as it would enable their use as world simulators. For these models to be successful world models, they should not only excel at image quality and prompt-image consistency but also ensure high representation diversity. However, current research in generative models mostly focuses on creative applications that are predominantly concerned with human preferences of image quality and aesthetics. We note that generative models have inference time mechanisms - or knobs - that allow the control of generation consistency, quality, and diversity. In this paper, we use state-of-the-art text-to-image and image-and-text-to-image models and their knobs to draw consistency-diversity-realism Pareto fronts that provide a holistic view on consistency-diversity-realism multi-objective. Our experiments suggest that realism and consistency can both be improved simultaneously; however there exists a clear tradeoff between realism/consistency and diversity. By looking at Pareto optimal points, we note that earlier models are better at representation diversity and worse in consistency/realism, and more recent models excel in consistency/realism while decreasing significantly the representation diversity. By computing Pareto fronts on a geodiverse dataset, we find that the first version of latent diffusion models tends to perform better than more recent models in all axes of evaluation, and there exist pronounced consistency-diversity-realism disparities between geographical regions. Overall, our analysis clearly shows that there is no best model and the choice of model should be determined by the downstream application. With this analysis, we invite the research community to consider Pareto fronts as an analytical tool to measure progress towards world models.

We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics poczos13aistats, to anomaly detection in piezometer data of embankment dams Jung15Exploration, to cosmology Ntampaka16Dynamical,Ravanbakhsh16ICML1. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

We consider a fair division model in which agents have positive, zero and negative utilities for items. For this model, we analyse one existing fairness property - EFX - and three new and related properties - EFX_0, EFX^3 and EF1^3 - in combination with Pareto-optimality. With general utilities, we give a modified version of an existing algorithm for computing an EF1^3 allocation. With -alpha/0/alpha utilities, this algorithm returns an EFX^3 and PO allocation. With absolute identical utilities, we give a new algorithm for an EFX and PO allocation. With -alpha/0/beta utilities, this algorithm also returns such an allocation. We report some new impossibility results as well.

Multi-objective reinforcement learning (MORL) approaches have emerged to tackle many real-world problems with multiple conflicting objectives by maximizing a joint objective function weighted by a preference vector. These approaches find fixed customized policies corresponding to preference vectors specified during training. However, the design constraints and objectives typically change dynamically in real-life scenarios. Furthermore, storing a policy for each potential preference is not scalable. Hence, obtaining a set of Pareto front solutions for the entire preference space in a given domain with a single training is critical. To this end, we propose a novel MORL algorithm that trains a single universal network to cover the entire preference space scalable to continuous robotic tasks. The proposed approach, Preference-Driven MORL (PD-MORL), utilizes the preferences as guidance to update the network parameters. It also employs a novel parallelization approach to increase sample efficiency. We show that PD-MORL achieves up to 25% larger hypervolume for challenging continuous control tasks and uses an order of magnitude fewer trainable parameters compared to prior approaches.

We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. To understand the importance of this structure in optimization problems we consider the problem of maximizing function value under various types of constraints. To demonstrate the modeling power of submodular order functions we show applications in two different settings. First, we apply our results to the extensively studied problem of assortment optimization. While the objectives in assortment optimization are known to be non-submodular (and non-monotone) even for simple choice models, we show that they are compatible with the notion of submodular order. Consequently, we obtain new and in some cases the first constant factor guarantee for constrained assortment optimization in fundamental choice models. As a second application of submodular order functions, we show an intriguing connection to the maximization of monotone submodular functions in the streaming model. We recover some best known guarantees for this problem as a corollary of our results.

Domain adaptation (DA) attempts to transfer the knowledge from a labeled source domain to an unlabeled target domain that follows different distribution from the source. To achieve this, DA methods include a source classification objective to extract the source knowledge and a domain alignment objective to diminish the domain shift, ensuring knowledge transfer. Typically, former DA methods adopt some weight hyper-parameters to linearly combine the training objectives to form an overall objective. However, the gradient directions of these objectives may conflict with each other due to domain shift. Under such circumstances, the linear optimization scheme might decrease the overall objective value at the expense of damaging one of the training objectives, leading to restricted solutions. In this paper, we rethink the optimization scheme for DA from a gradient-based perspective. We propose a Pareto Domain Adaptation (ParetoDA) approach to control the overall optimization direction, aiming to cooperatively optimize all training objectives. Specifically, to reach a desirable solution on the target domain, we design a surrogate loss mimicking target classification. To improve target-prediction accuracy to support the mimicking, we propose a target-prediction refining mechanism which exploits domain labels via Bayes' theorem. On the other hand, since prior knowledge of weighting schemes for objectives is often unavailable to guide optimization to approach the optimal solution on the target domain, we propose a dynamic preference mechanism to dynamically guide our cooperative optimization by the gradient of the surrogate loss on a held-out unlabeled target dataset. Extensive experiments on image classification and semantic segmentation benchmarks demonstrate the effectiveness of ParetoDA

We propose a reinforcement learning (RL) framework under a broad class of risk objectives, characterized by convex scoring functions. This class covers many common risk measures, such as variance, Expected Shortfall, entropic Value-at-Risk, and mean-risk utility. To resolve the time-inconsistency issue, we consider an augmented state space and an auxiliary variable and recast the problem as a two-state optimization problem. We propose a customized Actor-Critic algorithm and establish some theoretical approximation guarantees. A key theoretical contribution is that our results do not require the Markov decision process to be continuous. Additionally, we propose an auxiliary variable sampling method inspired by the alternating minimization algorithm, which is convergent under certain conditions. We validate our approach in simulation experiments with a financial application in statistical arbitrage trading, demonstrating the effectiveness of the algorithm.

Bayesian optimization (BO) is a powerful framework to optimize black-box expensive-to-evaluate functions via sequential interactions. In several important problems (e.g. drug discovery, circuit design, neural architecture search, etc.), though, such functions are defined over large combinatorial and unstructured spaces. This makes existing BO algorithms not feasible due to the intractable maximization of the acquisition function over these domains. To address this issue, we propose GameOpt, a novel game-theoretical approach to combinatorial BO. GameOpt establishes a cooperative game between the different optimization variables, and selects points that are game equilibria of an upper confidence bound acquisition function. These are stable configurations from which no variable has an incentive to deviate- analog to local optima in continuous domains. Crucially, this allows us to efficiently break down the complexity of the combinatorial domain into individual decision sets, making GameOpt scalable to large combinatorial spaces. We demonstrate the application of GameOpt to the challenging protein design problem and validate its performance on four real-world protein datasets. Each protein can take up to 20^{X} possible configurations, where X is the length of a protein, making standard BO methods infeasible. Instead, our approach iteratively selects informative protein configurations and very quickly discovers highly active protein variants compared to other baselines.

This paper studies distributional model risk in marginal problems, where each marginal measure is assumed to lie in a Wasserstein ball centered at a fixed reference measure with a given radius. Theoretically, we establish several fundamental results including strong duality, finiteness of the proposed Wasserstein distributional model risk, and the existence of an optimizer at each radius. In addition, we show continuity of the Wasserstein distributional model risk as a function of the radius. Using strong duality, we extend the well-known Makarov bounds for the distribution function of the sum of two random variables with given marginals to Wasserstein distributionally robust Markarov bounds. Practically, we illustrate our results on four distinct applications when the sample information comes from multiple data sources and only some marginal reference measures are identified. They are: partial identification of treatment effects; externally valid treatment choice via robust welfare functions; Wasserstein distributionally robust estimation under data combination; and evaluation of the worst aggregate risk measures.

We present metrics for measuring state similarity in Markov decision processes (MDPs) with infinitely many states, including MDPs with continuous state spaces. Such metrics provide a stable quantitative analogue of the notion of bisimulation for MDPs, and are suitable for use in MDP approximation. We show that the optimal value function associated with a discounted infinite horizon planning task varies continuously with respect to our metric distances.

The online knapsack problem is a classic problem in the field of online algorithms. Its canonical version asks how to pack items of different values and weights arriving online into a capacity-limited knapsack so as to maximize the total value of the admitted items. Although optimal competitive algorithms are known for this problem, they may be fundamentally unfair, i.e., individual items may be treated inequitably in different ways. We formalize a practically-relevant notion of time fairness which effectively models a trade off between static and dynamic pricing in a motivating application such as cloud resource allocation, and show that existing algorithms perform poorly under this metric. We propose a parameterized deterministic algorithm where the parameter precisely captures the Pareto-optimal trade-off between fairness (static pricing) and competitiveness (dynamic pricing). We show that randomization is theoretically powerful enough to be simultaneously competitive and fair; however, it does not work well in experiments. To further improve the trade-off between fairness and competitiveness, we develop a nearly-optimal learning-augmented algorithm which is fair, consistent, and robust (competitive), showing substantial performance improvements in numerical experiments.

The bandits with knapsack (BwK) framework models online decision-making problems in which an agent makes a sequence of decisions subject to resource consumption constraints. The traditional model assumes that each action consumes a non-negative amount of resources and the process ends when the initial budgets are fully depleted. We study a natural generalization of the BwK framework which allows non-monotonic resource utilization, i.e., resources can be replenished by a positive amount. We propose a best-of-both-worlds primal-dual template that can handle any online learning problem with replenishment for which a suitable primal regret minimizer exists. In particular, we provide the first positive results for the case of adversarial inputs by showing that our framework guarantees a constant competitive ratio alpha when B=Omega(T) or when the possible per-round replenishment is a positive constant. Moreover, under a stochastic input model, our algorithm yields an instance-independent O(T^{1/2}) regret bound which complements existing instance-dependent bounds for the same setting. Finally, we provide applications of our framework to some economic problems of practical relevance.

This work introduces MultiTRON, an approach that adapts Pareto front approximation techniques to multi-objective session-based recommender systems using a transformer neural network. Our approach optimizes trade-offs between key metrics such as click-through and conversion rates by training on sampled preference vectors. A significant advantage is that after training, a single model can access the entire Pareto front, allowing it to be tailored to meet the specific requirements of different stakeholders by adjusting an additional input vector that weights the objectives. We validate the model's performance through extensive offline and online evaluation. For broader application and research, the source code is made available at https://github.com/otto-de/MultiTRON. The results confirm the model's ability to manage multiple recommendation objectives effectively, offering a flexible tool for diverse business needs.

In the stochastic multi-armed bandit problem, a randomized probability matching policy called Thompson sampling (TS) has shown excellent performance in various reward models. In addition to the empirical performance, TS has been shown to achieve asymptotic problem-dependent lower bounds in several models. However, its optimality has been mainly addressed under light-tailed or one-parameter models that belong to exponential families. In this paper, we consider the optimality of TS for the Pareto model that has a heavy tail and is parameterized by two unknown parameters. Specifically, we discuss the optimality of TS with probability matching priors that include the Jeffreys prior and the reference priors. We first prove that TS with certain probability matching priors can achieve the optimal regret bound. Then, we show the suboptimality of TS with other priors, including the Jeffreys and the reference priors. Nevertheless, we find that TS with the Jeffreys and reference priors can achieve the asymptotic lower bound if one uses a truncation procedure. These results suggest carefully choosing noninformative priors to avoid suboptimality and show the effectiveness of truncation procedures in TS-based policies.

The prevalence and importance of algorithmic two-sided marketplaces has drawn attention to the issue of fairness in such settings. Algorithmic decisions are used in assigning students to schools, users to advertisers, and applicants to job interviews. These decisions should heed the preferences of individuals, and simultaneously be fair with respect to their merits (synonymous with fit, future performance, or need). Merits conditioned on observable features are always uncertain, a fact that is exacerbated by the widespread use of machine learning algorithms to infer merit from the observables. As our key contribution, we carefully axiomatize a notion of individual fairness in the two-sided marketplace setting which respects the uncertainty in the merits; indeed, it simultaneously recognizes uncertainty as the primary potential cause of unfairness and an approach to address it. We design a linear programming framework to find fair utility-maximizing distributions over allocations, and we show that the linear program is robust to perturbations in the estimated parameters of the uncertain merit distributions, a key property in combining the approach with machine learning techniques.

We theoretically study the impact of differential privacy on fairness in classification. We prove that, given a class of models, popular group fairness measures are pointwise Lipschitz-continuous with respect to the parameters of the model. This result is a consequence of a more general statement on accuracy conditioned on an arbitrary event (such as membership to a sensitive group), which may be of independent interest. We use the aforementioned Lipschitz property to prove a high probability bound showing that, given enough examples, the fairness level of private models is close to the one of their non-private counterparts.

We present a general convergent class of reinforcement learning algorithms that is founded on two distinct principles: (1) mapping value estimates to a different space using arbitrary functions from a broad class, and (2) linearly decomposing the reward signal into multiple channels. The first principle enables incorporating specific properties into the value estimator that can enhance learning. The second principle, on the other hand, allows for the value function to be represented as a composition of multiple utility functions. This can be leveraged for various purposes, e.g. dealing with highly varying reward scales, incorporating a priori knowledge about the sources of reward, and ensemble learning. Combining the two principles yields a general blueprint for instantiating convergent algorithms by orchestrating diverse mapping functions over multiple reward channels. This blueprint generalizes and subsumes algorithms such as Q-Learning, Log Q-Learning, and Q-Decomposition. In addition, our convergence proof for this general class relaxes certain required assumptions in some of these algorithms. Based on our theory, we discuss several interesting configurations as special cases. Finally, to illustrate the potential of the design space that our theory opens up, we instantiate a particular algorithm and evaluate its performance on the Atari suite.

Multi-objective reinforcement learning (MORL) excels at handling rapidly changing preferences in tasks that involve multiple criteria, even for unseen preferences. However, previous dominating MORL methods typically generate a fixed policy set or preference-conditioned policy through multiple training iterations exclusively for sampled preference vectors, and cannot ensure the efficient discovery of the Pareto front. Furthermore, integrating preferences into the input of policy or value functions presents scalability challenges, in particular as the dimension of the state and preference space grow, which can complicate the learning process and hinder the algorithm's performance on more complex tasks. To address these issues, we propose a two-stage Pareto front discovery algorithm called Constrained MORL (C-MORL), which serves as a seamless bridge between constrained policy optimization and MORL. Concretely, a set of policies is trained in parallel in the initialization stage, with each optimized towards its individual preference over the multiple objectives. Then, to fill the remaining vacancies in the Pareto front, the constrained optimization steps are employed to maximize one objective while constraining the other objectives to exceed a predefined threshold. Empirically, compared to recent advancements in MORL methods, our algorithm achieves more consistent and superior performances in terms of hypervolume, expected utility, and sparsity on both discrete and continuous control tasks, especially with numerous objectives (up to nine objectives in our experiments).

Recently, remarkable progress has been made by approximating Nash equilibrium (NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE) through function approximation that trains a neural network to predict equilibria from game representations. Furthermore, equivariant architectures are widely adopted in designing such equilibrium approximators in normal-form games. In this paper, we theoretically characterize benefits and limitations of equivariant equilibrium approximators. For the benefits, we show that they enjoy better generalizability than general ones and can achieve better approximations when the payoff distribution is permutation-invariant. For the limitations, we discuss their drawbacks in terms of equilibrium selection and social welfare. Together, our results help to understand the role of equivariance in equilibrium approximators.

Reinforcement learning from human feedback (RLHF) is a standard approach for fine-tuning large language models to follow instructions. As part of this process, learned reward models are used to approximately model human preferences. However, as imperfect representations of the "true" reward, these learned reward models are susceptible to overoptimization. Gao et al. (2023) studied this phenomenon in a synthetic human feedback setup with a significantly larger "gold" reward model acting as the true reward (instead of humans) and showed that overoptimization remains a persistent problem regardless of the size of the proxy reward model and training data used. Using a similar setup, we conduct a systematic study to evaluate the efficacy of using ensemble-based conservative optimization objectives, specifically worst-case optimization (WCO) and uncertainty-weighted optimization (UWO), for mitigating reward model overoptimization when using two optimization methods: (a) best-of-n sampling (BoN) (b) proximal policy optimization (PPO). We additionally extend the setup of Gao et al. (2023) to include 25% label noise to better mirror real-world conditions. Both with and without label noise, we find that conservative optimization practically eliminates overoptimization and improves performance by up to 70% for BoN sampling. For PPO, ensemble-based conservative optimization always reduces overoptimization and outperforms single reward model optimization. Moreover, combining it with a small KL penalty successfully prevents overoptimization at no performance cost. Overall, our results demonstrate that ensemble-based conservative optimization can effectively counter overoptimization.

We introduce a boosting algorithm to pre-process data for fairness. Starting from an initial fair but inaccurate distribution, our approach shifts towards better data fitting while still ensuring a minimal fairness guarantee. To do so, it learns the sufficient statistics of an exponential family with boosting-compliant convergence. Importantly, we are able to theoretically prove that the learned distribution will have a representation rate and statistical rate data fairness guarantee. Unlike recent optimization based pre-processing methods, our approach can be easily adapted for continuous domain features. Furthermore, when the weak learners are specified to be decision trees, the sufficient statistics of the learned distribution can be examined to provide clues on sources of (un)fairness. Empirical results are present to display the quality of result on real-world data.

Model merging has emerged as an effective approach to combine multiple single-task models into a multitask model. This process typically involves computing a weighted average of the model parameters without any additional training. Existing model-merging methods focus on enhancing average task accuracy. However, interference and conflicts between the objectives of different tasks can lead to trade-offs during the merging process. In real-world applications, a set of solutions with various trade-offs can be more informative, helping practitioners make decisions based on diverse preferences. In this paper, we introduce a novel and low-compute algorithm, Model Merging with Amortized Pareto Front (MAP). MAP efficiently identifies a Pareto set of scaling coefficients for merging multiple models, reflecting the trade-offs involved. It amortizes the substantial computational cost of evaluations needed to estimate the Pareto front by using quadratic approximation surrogate models derived from a pre-selected set of scaling coefficients. Experimental results on vision and natural language processing tasks demonstrate that MAP can accurately identify the Pareto front, providing practitioners with flexible solutions to balance competing task objectives. We also introduce Bayesian MAP for scenarios with a relatively low number of tasks and Nested MAP for situations with a high number of tasks, further reducing the computational cost of evaluation.

In the context of reinforcement learning from human feedback (RLHF), the reward function is generally derived from maximum likelihood estimation of a random utility model based on pairwise comparisons made by humans. The problem of learning a reward function is one of preference aggregation that, we argue, largely falls within the scope of social choice theory. From this perspective, we can evaluate different aggregation methods via established axioms, examining whether these methods meet or fail well-known standards. We demonstrate that both the Bradley-Terry-Luce Model and its broad generalizations fail to meet basic axioms. In response, we develop novel rules for learning reward functions with strong axiomatic guarantees. A key innovation from the standpoint of social choice is that our problem has a linear structure, which greatly restricts the space of feasible rules and leads to a new paradigm that we call linear social choice.

Mehta and Panigrahi (2012) proposed Online Matching with Stochastic Rewards, which generalizes the Online Bipartite Matching problem of Karp, Vazirani, and Vazirani (1990) by associating the edges with success probabilities. This new feature captures the pay-per-click model in online advertising. Recently, Huang and Zhang (2020) studied this problem under the online primal dual framework using the Configuration Linear Program (LP), and got the best known competitive ratios of the Stochastic Balance algorithm. Their work suggests that the more expressive Configuration LP is more suitable for this problem than the Matching LP. This paper advances the theory of Configuration LP in two directions. Our technical contribution includes a characterization of the joint matching outcome of an offline vertex and all its neighbors. This characterization may be of independent interest, and is aligned with the spirit of Configuration LP. By contrast, previous analyses of Ranking generally focus on only one neighbor. Second, we designed a Stochastic Configuration LP that captures a stochastic benchmark proposed by Goyal and Udwani (2020), who used a Path-based LP. The Stochastic Configuration LP is smaller and simpler than the Path-based LP. Moreover, using the new LP we improved the competitive ratio of Stochastic Balance from 0.596 to 0.611 when the success probabilities are infinitesimal, and to 0.613 when the success probabilities are further equal.

Scaling data and compute is critical to the success of machine learning. However, scaling demands predictability: we want methods to not only perform well with more compute or data, but also have their performance be predictable from small-scale runs, without running the large-scale experiment. In this paper, we show that value-based off-policy RL methods are predictable despite community lore regarding their pathological behavior. First, we show that data and compute requirements to attain a given performance level lie on a Pareto frontier, controlled by the updates-to-data (UTD) ratio. By estimating this frontier, we can predict this data requirement when given more compute, and this compute requirement when given more data. Second, we determine the optimal allocation of a total resource budget across data and compute for a given performance and use it to determine hyperparameters that maximize performance for a given budget. Third, this scaling behavior is enabled by first estimating predictable relationships between hyperparameters, which is used to manage effects of overfitting and plasticity loss unique to RL. We validate our approach using three algorithms: SAC, BRO, and PQL on DeepMind Control, OpenAI gym, and IsaacGym, when extrapolating to higher levels of data, compute, budget, or performance.

Learning neural subset selection tasks, such as compound selection in AI-aided drug discovery, have become increasingly pivotal across diverse applications. The existing methodologies in the field primarily concentrate on constructing models that capture the relationship between utility function values and subsets within their respective supersets. However, these approaches tend to overlook the valuable information contained within the superset when utilizing neural networks to model set functions. In this work, we address this oversight by adopting a probabilistic perspective. Our theoretical findings demonstrate that when the target value is conditioned on both the input set and subset, it is essential to incorporate an invariant sufficient statistic of the superset into the subset of interest for effective learning. This ensures that the output value remains invariant to permutations of the subset and its corresponding superset, enabling identification of the specific superset from which the subset originated. Motivated by these insights, we propose a simple yet effective information aggregation module designed to merge the representations of subsets and supersets from a permutation invariance perspective. Comprehensive empirical evaluations across diverse tasks and datasets validate the enhanced efficacy of our approach over conventional methods, underscoring the practicality and potency of our proposed strategies in real-world contexts.

The performance of acquisition functions for Bayesian optimisation to locate the global optimum of continuous functions is investigated in terms of the Pareto front between exploration and exploitation. We show that Expected Improvement (EI) and the Upper Confidence Bound (UCB) always select solutions to be expensively evaluated on the Pareto front, but Probability of Improvement is not guaranteed to do so and Weighted Expected Improvement does so only for a restricted range of weights. We introduce two novel epsilon-greedy acquisition functions. Extensive empirical evaluation of these together with random search, purely exploratory, and purely exploitative search on 10 benchmark problems in 1 to 10 dimensions shows that epsilon-greedy algorithms are generally at least as effective as conventional acquisition functions (e.g., EI and UCB), particularly with a limited budget. In higher dimensions epsilon-greedy approaches are shown to have improved performance over conventional approaches. These results are borne out on a real world computational fluid dynamics optimisation problem and a robotics active learning problem. Our analysis and experiments suggest that the most effective strategy, particularly in higher dimensions, is to be mostly greedy, occasionally selecting a random exploratory solution.

We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.

We make two contributions in the field of AI fairness over continuous protected attributes. First, we show that the Hirschfeld-Gebelein-Renyi (HGR) indicator (the only one currently available for such a case) is valuable but subject to a few crucial limitations regarding semantics, interpretability, and robustness. Second, we introduce a family of indicators that are: 1) complementary to HGR in terms of semantics; 2) fully interpretable and transparent; 3) robust over finite samples; 4) configurable to suit specific applications. Our approach also allows us to define fine-grained constraints to permit certain types of dependence and forbid others selectively. By expanding the available options for continuous protected attributes, our approach represents a significant contribution to the area of fair artificial intelligence.

We study the impact on mechanisms for facility location of moving from one dimension to two (or more) dimensions and Euclidean or Manhattan distances. We consider three fundamental axiomatic properties: anonymity which is a basic fairness property, Pareto optimality which is one of the most important efficiency properties, and strategy proofness which ensures agents do not have an incentive to mis-report. We also consider how well such mechanisms can approximate the optimal welfare. Our results are somewhat negative. Moving from one dimension to two (or more) dimensions often makes these axiomatic properties more difficult to achieve. For example, with two facilities in Euclidean space or with just a single facility in Manhattan space, no mechanism is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the line exist with all three properties.We also show that approximation ratios may increase when moving to two (or more) dimensions. All our impossibility results are minimal. If we drop one of the three axioms (anonymity, Pareto optimality or strategy proofness) multiple mechanisms satisfy the other two axioms.

Computing stable partitions in hedonic games is a challenging task because there exist games in which stable outcomes do not exist. Even more, these No-instances can often be leveraged to prove computational hardness results. We make this impression rigorous in a dynamic model of cardinal hedonic games by providing meta theorems. These imply hardness of deciding about the possible or necessary convergence of deviation dynamics based on the mere existence of No-instances. Our results hold for additively separable, fractional, and modified fractional hedonic games (ASHGs, FHGs, and MFHGs). Moreover, they encompass essentially all reasonable stability notions based on single-agent deviations. In addition, we propose dynamics as a method to find individually rational and contractually individual stable (CIS) partitions in ASHGs. In particular, we find that CIS dynamics from the singleton partition possibly converge after a linear number of deviations but may require an exponential number of deviations in the worst case.

We study Pareto optimality in multi-objective multi-armed bandit by providing a formulation of adversarial multi-objective multi-armed bandit and defining its Pareto regrets that can be applied to both stochastic and adversarial settings. The regrets do not rely on any scalarization functions and reflect Pareto optimality compared to scalarized regrets. We also present new algorithms assuming both with and without prior information of the multi-objective multi-armed bandit setting. The algorithms are shown optimal in adversarial settings and nearly optimal up to a logarithmic factor in stochastic settings simultaneously by our established upper bounds and lower bounds on Pareto regrets. Moreover, the lower bound analyses show that the new regrets are consistent with the existing Pareto regret for stochastic settings and extend an adversarial attack mechanism from bandit to the multi-objective one.

Dealing with multi-task trade-offs during inference can be addressed via Pareto Front Learning (PFL) methods that parameterize the Pareto Front with a single model, contrary to traditional Multi-Task Learning (MTL) approaches that optimize for a single trade-off which has to be decided prior to training. However, recent PFL methodologies suffer from limited scalability, slow convergence and excessive memory requirements compared to MTL approaches while exhibiting inconsistent mappings from preference space to objective space. In this paper, we introduce PaLoRA, a novel parameter-efficient method that augments the original model with task-specific low-rank adapters and continuously parameterizes the Pareto Front in their convex hull. Our approach dedicates the original model and the adapters towards learning general and task-specific features, respectively. Additionally, we propose a deterministic sampling schedule of preference vectors that reinforces this division of labor, enabling faster convergence and scalability to real world networks. Our experimental results show that PaLoRA outperforms MTL and PFL baselines across various datasets, scales to large networks and provides a continuous parameterization of the Pareto Front, reducing the memory overhead 23.8-31.7 times compared with competing PFL baselines in scene understanding benchmarks.

Efficient LLM pre-training requires well-tuned hyperparameters (HPs), including learning rate {\eta} and weight decay {\lambda}. We study scaling laws for HPs: formulas for how to scale HPs as we scale model size N, dataset size D, and batch size B. Recent work suggests the AdamW timescale, B/({\eta}{\lambda}D), should remain constant across training settings, and we verify the implication that optimal {\lambda} scales linearly with B, for a fixed N,D. However, as N,D scale, we show the optimal timescale obeys a precise power law in the tokens-per-parameter ratio, D/N. This law thus provides a method to accurately predict {\lambda}opt in advance of large-scale training. We also study scaling laws for optimal batch size Bopt (the B enabling lowest loss at a given N,D) and critical batch size Bcrit (the B beyond which further data parallelism becomes ineffective). In contrast with prior work, we find both Bopt and Bcrit scale as power laws in D, independent of model size, N. Finally, we analyze how these findings inform the real-world selection of Pareto-optimal N and D under dual training time and compute objectives.

In this study, we propose a multicriteria group decision making (MCGDM) algorithm under uncertainty where data is collected as intervals. The proposed MCGDM algorithm aggregates the data, determines the optimal weights for criteria and ranks alternatives with no further input. The intervals give flexibility to experts in assessing alternatives against criteria and provide an opportunity to gain maximum information. We also propose a novel method to aggregate expert judgements using cloud models. We introduce an experimental approach to check the validity of the aggregation method. After that, we use the aggregation method for an MCGDM problem. Here, we find the optimal weights for each criterion by proposing a bilevel optimisation model. Then, we extend the technique for order of preference by similarity to ideal solution (TOPSIS) for data based on cloud models to prioritise alternatives. As a result, the algorithm can gain information from decision makers with different levels of uncertainty and examine alternatives with no more information from decision-makers. The proposed MCGDM algorithm is implemented on a case study of a cybersecurity problem to illustrate its feasibility and effectiveness. The results verify the robustness and validity of the proposed MCGDM using sensitivity analysis and comparison with other existing algorithms.

The prevalent deployment of learning from human preferences through reinforcement learning (RLHF) relies on two important approximations: the first assumes that pairwise preferences can be substituted with pointwise rewards. The second assumes that a reward model trained on these pointwise rewards can generalize from collected data to out-of-distribution data sampled by the policy. Recently, Direct Preference Optimisation (DPO) has been proposed as an approach that bypasses the second approximation and learn directly a policy from collected data without the reward modelling stage. However, this method still heavily relies on the first approximation. In this paper we try to gain a deeper theoretical understanding of these practical algorithms. In particular we derive a new general objective called PsiPO for learning from human preferences that is expressed in terms of pairwise preferences and therefore bypasses both approximations. This new general objective allows us to perform an in-depth analysis of the behavior of RLHF and DPO (as special cases of PsiPO) and to identify their potential pitfalls. We then consider another special case for PsiPO by setting Psi simply to Identity, for which we can derive an efficient optimisation procedure, prove performance guarantees and demonstrate its empirical superiority to DPO on some illustrative examples.

We consider strategy proof mechanisms for facility location which maximize equitability between agents. As is common in the literature, we measure equitability with the Gini index. We first prove a simple but fundamental impossibility result that no strategy proof mechanism can bound the approximation ratio of the optimal Gini index of utilities for one or more facilities. We propose instead computing approximation ratios of the complemented Gini index of utilities, and consider how well both deterministic and randomized mechanisms approximate this. In addition, as Nash welfare is often put forwards as an equitable compromise between egalitarian and utilitarian outcomes, we consider how well mechanisms approximate the Nash welfare.

We consider a class of assortment optimization problems in an offline data-driven setting. A firm does not know the underlying customer choice model but has access to an offline dataset consisting of the historically offered assortment set, customer choice, and revenue. The objective is to use the offline dataset to find an optimal assortment. Due to the combinatorial nature of assortment optimization, the problem of insufficient data coverage is likely to occur in the offline dataset. Therefore, designing a provably efficient offline learning algorithm becomes a significant challenge. To this end, we propose an algorithm referred to as Pessimistic ASsortment opTimizAtion (PASTA for short) designed based on the principle of pessimism, that can correctly identify the optimal assortment by only requiring the offline data to cover the optimal assortment under general settings. In particular, we establish a regret bound for the offline assortment optimization problem under the celebrated multinomial logit model. We also propose an efficient computational procedure to solve our pessimistic assortment optimization problem. Numerical studies demonstrate the superiority of the proposed method over the existing baseline method.

We consider the problem of subset selection where one is given multiple rankings of items and the goal is to select the highest ``quality'' subset. Score functions from the multiwinner voting literature have been used to aggregate rankings into quality scores for subsets. We study this setting of subset selection problems when, in addition, rankings may contain systemic or unconscious biases toward a group of items. For a general model of input rankings and biases, we show that requiring the selected subset to satisfy group fairness constraints can improve the quality of the selection with respect to unbiased rankings. Importantly, we show that for fairness constraints to be effective, different multiwinner score functions may require a drastically different number of rankings: While for some functions, fairness constraints need an exponential number of rankings to recover a close-to-optimal solution, for others, this dependency is only polynomial. This result relies on a novel notion of ``smoothness'' of submodular functions in this setting that quantifies how well a function can ``correctly'' assess the quality of items in the presence of bias. The results in this paper can be used to guide the choice of multiwinner score functions for the subset selection setting considered here; we additionally provide a tool to empirically enable this.

We consider stochastic sequential learning problems where the learner can observe the average reward of several actions. Such a setting is interesting in many applications involving monitoring and surveillance, where the set of the actions to observe represent some (geographical) area. The importance of this setting is that in these applications, it is actually cheaper to observe average reward of a group of actions rather than the reward of a single action. We show that when the reward is smooth over a given graph representing the neighboring actions, we can maximize the cumulative reward of learning while minimizing the sensing cost. In this paper we propose CheapUCB, an algorithm that matches the regret guarantees of the known algorithms for this setting and at the same time guarantees a linear cost again over them. As a by-product of our analysis, we establish a Ω(dT) lower bound on the cumulative regret of spectral bandits for a class of graphs with effective dimension d.

Self-supervised learning (SSL) for graph neural networks (GNNs) has attracted increasing attention from the graph machine learning community in recent years, owing to its capability to learn performant node embeddings without costly label information. One weakness of conventional SSL frameworks for GNNs is that they learn through a single philosophy, such as mutual information maximization or generative reconstruction. When applied to various downstream tasks, these frameworks rarely perform equally well for every task, because one philosophy may not span the extensive knowledge required for all tasks. To enhance the task generalization across tasks, as an important first step forward in exploring fundamental graph models, we introduce PARETOGNN, a multi-task SSL framework for node representation learning over graphs. Specifically, PARETOGNN is self-supervised by manifold pretext tasks observing multiple philosophies. To reconcile different philosophies, we explore a multiple-gradient descent algorithm, such that PARETOGNN actively learns from every pretext task while minimizing potential conflicts. We conduct comprehensive experiments over four downstream tasks (i.e., node classification, node clustering, link prediction, and partition prediction), and our proposal achieves the best overall performance across tasks on 11 widely adopted benchmark datasets. Besides, we observe that learning from multiple philosophies enhances not only the task generalization but also the single task performances, demonstrating that PARETOGNN achieves better task generalization via the disjoint yet complementary knowledge learned from different philosophies. Our code is publicly available at https://github.com/jumxglhf/ParetoGNN.

We consider the problem of locating a facility to serve a set of agents located along a line. The Nash welfare objective function, defined as the product of the agents' utilities, is known to provide a compromise between fairness and efficiency in resource allocation problems. We apply this welfare notion to the facility location problem, converting individual costs to utilities and analyzing the facility placement that maximizes the Nash welfare. We give a polynomial-time approximation algorithm to compute this facility location, and prove results suggesting that it achieves a good balance of fairness and efficiency. Finally, we take a mechanism design perspective and propose a strategy-proof mechanism with a bounded approximation ratio for Nash welfare.

For continuous action spaces, actor-critic methods are widely used in online reinforcement learning (RL). However, unlike RL algorithms for discrete actions, which generally model the optimal value function using the Bellman optimality operator, RL algorithms for continuous actions typically model Q-values for the current policy using the Bellman operator. These algorithms for continuous actions rely exclusively on policy updates for improvement, which often results in low sample efficiency. This study examines the effectiveness of incorporating the Bellman optimality operator into actor-critic frameworks. Experiments in a simple environment show that modeling optimal values accelerates learning but leads to overestimation bias. To address this, we propose an annealing approach that gradually transitions from the Bellman optimality operator to the Bellman operator, thereby accelerating learning while mitigating bias. Our method, combined with TD3 and SAC, significantly outperforms existing approaches across various locomotion and manipulation tasks, demonstrating improved performance and robustness to hyperparameters related to optimality. The code for this study is available at https://github.com/motokiomura/annealed-q-learning.

We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a (delta,epsilon)-stationary point from O(epsilon^{-4}delta^{-1}) stochastic gradient queries to O(epsilon^{-3}delta^{-1}), which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to online learning, after which our results follow from standard regret bounds in online learning. For deterministic and second-order smooth objectives, applying more advanced optimistic online learning techniques enables a new complexity of O(epsilon^{-1.5}delta^{-0.5}). Our techniques also recover all optimal or best-known results for finding epsilon stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

We seek to understand fundamental tradeoffs between the accuracy of prior information that a learner has on a given problem and its learning performance. We introduce the notion of prioritized risk, which differs from traditional notions of minimax and Bayes risk by allowing us to study such fundamental tradeoffs in settings where reality does not necessarily conform to the learner's prior. We present a general reduction-based approach for extending classical minimax lower-bound techniques in order to lower bound the prioritized risk for statistical estimation problems. We also introduce a novel generalization of Fano's inequality (which may be of independent interest) for lower bounding the prioritized risk in more general settings involving unbounded losses. We illustrate the ability of our framework to provide insights into tradeoffs between prior information and learning performance for problems in estimation, regression, and reinforcement learning.

We propose a new definition of instance optimality for differentially private estimation algorithms. Our definition requires an optimal algorithm to compete, simultaneously for every dataset D, with the best private benchmark algorithm that (a) knows D in advance and (b) is evaluated by its worst-case performance on large subsets of D. That is, the benchmark algorithm need not perform well when potentially extreme points are added to D; it only has to handle the removal of a small number of real data points that already exist. This makes our benchmark significantly stronger than those proposed in prior work. We nevertheless show, for real-valued datasets, how to construct private algorithms that achieve our notion of instance optimality when estimating a broad class of dataset properties, including means, quantiles, and ell_p-norm minimizers. For means in particular, we provide a detailed analysis and show that our algorithm simultaneously matches or exceeds the asymptotic performance of existing algorithms under a range of distributional assumptions.

In this paper, we study risk-sensitive Reinforcement Learning (RL), focusing on the objective of Conditional Value at Risk (CVaR) with risk tolerance tau. Starting with multi-arm bandits (MABs), we show the minimax CVaR regret rate is Omega(tau^{-1AK}), where A is the number of actions and K is the number of episodes, and that it is achieved by an Upper Confidence Bound algorithm with a novel Bernstein bonus. For online RL in tabular Markov Decision Processes (MDPs), we show a minimax regret lower bound of Omega(tau^{-1SAK}) (with normalized cumulative rewards), where S is the number of states, and we propose a novel bonus-driven Value Iteration procedure. We show that our algorithm achieves the optimal regret of widetilde O(tau^{-1SAK}) under a continuity assumption and in general attains a near-optimal regret of widetilde O(tau^{-1}SAK), which is minimax-optimal for constant tau. This improves on the best available bounds. By discretizing rewards appropriately, our algorithms are computationally efficient.

In this paper, we introduce a conformal prediction method to construct prediction sets in a oneshot federated learning setting. More specifically, we define a quantile-of-quantiles estimator and prove that for any distribution, it is possible to output prediction sets with desired coverage in only one round of communication. To mitigate privacy issues, we also describe a locally differentially private version of our estimator. Finally, over a wide range of experiments, we show that our method returns prediction sets with coverage and length very similar to those obtained in a centralized setting. Overall, these results demonstrate that our method is particularly well-suited to perform conformal predictions in a one-shot federated learning setting.

Offline preference optimization is a key method for enhancing and controlling the quality of Large Language Model (LLM) outputs. Typically, preference optimization is approached as an offline supervised learning task using manually-crafted convex loss functions. While these methods are based on theoretical insights, they are inherently constrained by human creativity, so the large search space of possible loss functions remains under explored. We address this by performing LLM-driven objective discovery to automatically discover new state-of-the-art preference optimization algorithms without (expert) human intervention. Specifically, we iteratively prompt an LLM to propose and implement new preference optimization loss functions based on previously-evaluated performance metrics. This process leads to the discovery of previously-unknown and performant preference optimization algorithms. The best performing of these we call Discovered Preference Optimization (DiscoPOP), a novel algorithm that adaptively blends logistic and exponential losses. Experiments demonstrate the state-of-the-art performance of DiscoPOP and its successful transfer to held-out tasks.

Seven years ago, researchers proposed a postprocessing method to equalize the error rates of a model across different demographic groups. The work launched hundreds of papers purporting to improve over the postprocessing baseline. We empirically evaluate these claims through thousands of model evaluations on several tabular datasets. We find that the fairness-accuracy Pareto frontier achieved by postprocessing contains all other methods we were feasibly able to evaluate. In doing so, we address two common methodological errors that have confounded previous observations. One relates to the comparison of methods with different unconstrained base models. The other concerns methods achieving different levels of constraint relaxation. At the heart of our study is a simple idea we call unprocessing that roughly corresponds to the inverse of postprocessing. Unprocessing allows for a direct comparison of methods using different underlying models and levels of relaxation.

A major barrier to deploying current machine learning models lies in their non-reliability to dataset shifts. To resolve this problem, most existing studies attempted to transfer stable information to unseen environments. Particularly, independent causal mechanisms-based methods proposed to remove mutable causal mechanisms via the do-operator. Compared to previous methods, the obtained stable predictors are more effective in identifying stable information. However, a key question remains: which subset of this whole stable information should the model transfer, in order to achieve optimal generalization ability? To answer this question, we present a comprehensive minimax analysis from a causal perspective. Specifically, we first provide a graphical condition for the whole stable set to be optimal. When this condition fails, we surprisingly find with an example that this whole stable set, although can fully exploit stable information, is not the optimal one to transfer. To identify the optimal subset under this case, we propose to estimate the worst-case risk with a novel optimization scheme over the intervention functions on mutable causal mechanisms. We then propose an efficient algorithm to search for the subset with minimal worst-case risk, based on a newly defined equivalence relation between stable subsets. Compared to the exponential cost of exhaustively searching over all subsets, our searching strategy enjoys a polynomial complexity. The effectiveness and efficiency of our methods are demonstrated on synthetic data and the diagnosis of Alzheimer's disease.

Combinatorial optimization (CO) is naturally discrete, making machine learning based on differentiable optimization inapplicable. Karalias & Loukas (2020) adapted the probabilistic method to incorporate CO into differentiable optimization. Their work ignited the research on unsupervised learning for CO, composed of two main components: probabilistic objectives and derandomization. However, each component confronts unique challenges. First, deriving objectives under various conditions (e.g., cardinality constraints and minimum) is nontrivial. Second, the derandomization process is underexplored, and the existing derandomization methods are either random sampling or naive rounding. In this work, we aim to tackle prevalent (i.e., commonly involved) conditions in unsupervised CO. First, we concretize the targets for objective construction and derandomization with theoretical justification. Then, for various conditions commonly involved in different CO problems, we derive nontrivial objectives and derandomization to meet the targets. Finally, we apply the derivations to various CO problems. Via extensive experiments on synthetic and real-world graphs, we validate the correctness of our derivations and show our empirical superiority w.r.t. both optimization quality and speed.

In this paper, we investigate the problem of offline Preference-based Reinforcement Learning (PbRL) with human feedback where feedback is available in the form of preference between trajectory pairs rather than explicit rewards. Our proposed algorithm consists of two main steps: (1) estimate the implicit reward using Maximum Likelihood Estimation (MLE) with general function approximation from offline data and (2) solve a distributionally robust planning problem over a confidence set around the MLE. We consider the general reward setting where the reward can be defined over the whole trajectory and provide a novel guarantee that allows us to learn any target policy with a polynomial number of samples, as long as the target policy is covered by the offline data. This guarantee is the first of its kind with general function approximation. To measure the coverage of the target policy, we introduce a new single-policy concentrability coefficient, which can be upper bounded by the per-trajectory concentrability coefficient. We also establish lower bounds that highlight the necessity of such concentrability and the difference from standard RL, where state-action-wise rewards are directly observed. We further extend and analyze our algorithm when the feedback is given over action pairs.

We demonstrate that from an algorithm guaranteeing an approximation factor for the ratio of submodular (RS) optimization problem, we can build another algorithm having a different kind of approximation guarantee -- weaker than the classical one -- for the difference of submodular (DS) optimization problem, and vice versa. We also illustrate the link between these two problems by analyzing a Greedy algorithm which approximately maximizes objective functions of the form Ψ(f,g), where f,g are two non-negative, monotone, submodular functions and Ψ is a {quasiconvex} 2-variables function, which is non decreasing with respect to the first variable. For the choice Ψ(f,g)triangleq f/g, we recover RS, and for the choice Ψ(f,g)triangleq f-g, we recover DS. To the best of our knowledge, this greedy approach is new for DS optimization. For RS optimization, it reduces to the standard GreedRatio algorithm that has already been analyzed previously. However, our analysis is novel for this case.

Model merging has shown great promise at combining expert models, but the benefit of merging is unclear when merging ``generalist'' models trained on many tasks. We explore merging in the context of large (sim100B) models, by recycling checkpoints that exhibit tradeoffs among different tasks. Such checkpoints are often created in the process of developing a frontier model, and many suboptimal ones are usually discarded. Given a pool of model checkpoints obtained from different training runs (e.g., different stages, objectives, hyperparameters, and data mixtures), which naturally show tradeoffs across different language capabilities (e.g., instruction following vs. code generation), we investigate whether merging can recycle such suboptimal models into a Pareto-optimal one. Our optimization algorithm tunes the weight of each checkpoint in a linear combination, resulting in a Pareto-optimal models that outperforms both individual models and merge-based baselines. Further analysis shows that good merges tend to include almost all checkpoints with with non-zero weights, indicating that even seemingly bad initial checkpoints can contribute to good final merges.

We consider the problem of constructing small coresets for k-Median in Euclidean spaces. Given a large set of data points Psubset R^d, a coreset is a much smaller set Ssubset R^d, so that the k-Median costs of any k centers w.r.t. P and S are close. Existing literature mainly focuses on the high-dimension case and there has been great success in obtaining dimension-independent bounds, whereas the case for small d is largely unexplored. Considering many applications of Euclidean clustering algorithms are in small dimensions and the lack of systematic studies in the current literature, this paper investigates coresets for k-Median in small dimensions. For small d, a natural question is whether existing near-optimal dimension-independent bounds can be significantly improved. We provide affirmative answers to this question for a range of parameters. Moreover, new lower bound results are also proved, which are the highest for small d. In particular, we completely settle the coreset size bound for 1-d k-Median (up to log factors). Interestingly, our results imply a strong separation between 1-d 1-Median and 1-d 2-Median. As far as we know, this is the first such separation between k=1 and k=2 in any dimension.

We consider the obnoxious facility location problem (in which agents prefer the facility location to be far from them) and propose a hierarchy of distance-based proportional fairness concepts for the problem. These fairness axioms ensure that groups of agents at the same location are guaranteed to be a distance from the facility proportional to their group size. We consider deterministic and randomized mechanisms, and compute tight bounds on the price of proportional fairness. In the deterministic setting, we show that our proportional fairness axioms are incompatible with strategyproofness, and prove asymptotically tight epsilon-price of anarchy and stability bounds for proportionally fair welfare-optimal mechanisms. In the randomized setting, we identify proportionally fair and strategyproof mechanisms that give an expected welfare within a constant factor of the optimal welfare. Finally, we prove existence results for two extensions to our model.

Many continuous control tasks have easily formulated objectives, yet using them directly as a reward in reinforcement learning (RL) leads to suboptimal policies. Therefore, many classical control tasks guide RL training using complex rewards, which require tedious hand-tuning. We automate the reward search with AutoRL, an evolutionary layer over standard RL that treats reward tuning as hyperparameter optimization and trains a population of RL agents to find a reward that maximizes the task objective. AutoRL, evaluated on four Mujoco continuous control tasks over two RL algorithms, shows improvements over baselines, with the the biggest uplift for more complex tasks. The video can be found at: https://youtu.be/svdaOFfQyC8.

Robust Markov decision processes (MDPs) address the challenge of model uncertainty by optimizing the worst-case performance over an uncertainty set of MDPs. In this paper, we focus on the robust average-reward MDPs under the model-free setting. We first theoretically characterize the structure of solutions to the robust average-reward Bellman equation, which is essential for our later convergence analysis. We then design two model-free algorithms, robust relative value iteration (RVI) TD and robust RVI Q-learning, and theoretically prove their convergence to the optimal solution. We provide several widely used uncertainty sets as examples, including those defined by the contamination model, total variation, Chi-squared divergence, Kullback-Leibler (KL) divergence and Wasserstein distance.

We study offline multi-agent reinforcement learning (RL) in Markov games, where the goal is to learn an approximate equilibrium -- such as Nash equilibrium and (Coarse) Correlated Equilibrium -- from an offline dataset pre-collected from the game. Existing works consider relatively restricted tabular or linear models and handle each equilibria separately. In this work, we provide the first framework for sample-efficient offline learning in Markov games under general function approximation, handling all 3 equilibria in a unified manner. By using Bellman-consistent pessimism, we obtain interval estimation for policies' returns, and use both the upper and the lower bounds to obtain a relaxation on the gap of a candidate policy, which becomes our optimization objective. Our results generalize prior works and provide several additional insights. Importantly, we require a data coverage condition that improves over the recently proposed "unilateral concentrability". Our condition allows selective coverage of deviation policies that optimally trade-off between their greediness (as approximate best responses) and coverage, and we show scenarios where this leads to significantly better guarantees. As a new connection, we also show how our algorithmic framework can subsume seemingly different solution concepts designed for the special case of two-player zero-sum games.

Crafting adversarial examples is crucial for evaluating and enhancing the robustness of Deep Neural Networks (DNNs), presenting a challenge equivalent to maximizing a non-differentiable 0-1 loss function. However, existing single objective methods, namely adversarial attacks focus on a surrogate loss function, do not fully harness the benefits of engaging multiple loss functions, as a result of insufficient understanding of their synergistic and conflicting nature. To overcome these limitations, we propose the Multi-Objective Set-based Attack (MOS Attack), a novel adversarial attack framework leveraging multiple loss functions and automatically uncovering their interrelations. The MOS Attack adopts a set-based multi-objective optimization strategy, enabling the incorporation of numerous loss functions without additional parameters. It also automatically mines synergistic patterns among various losses, facilitating the generation of potent adversarial attacks with fewer objectives. Extensive experiments have shown that our MOS Attack outperforms single-objective attacks. Furthermore, by harnessing the identified synergistic patterns, MOS Attack continues to show superior results with a reduced number of loss functions. Our code is available at https://github.com/pgg3/MOS-Attack.

Saliency post-hoc explainability methods are important tools for understanding increasingly complex NLP models. While these methods can reflect the model's reasoning, they may not align with human intuition, making the explanations not plausible. In this work, we present a methodology for incorporating rationales, which are text annotations explaining human decisions, into text classification models. This incorporation enhances the plausibility of post-hoc explanations while preserving their faithfulness. Our approach is agnostic to model architectures and explainability methods. We introduce the rationales during model training by augmenting the standard cross-entropy loss with a novel loss function inspired by contrastive learning. By leveraging a multi-objective optimization algorithm, we explore the trade-off between the two loss functions and generate a Pareto-optimal frontier of models that balance performance and plausibility. Through extensive experiments involving diverse models, datasets, and explainability methods, we demonstrate that our approach significantly enhances the quality of model explanations without causing substantial (sometimes negligible) degradation in the original model's performance.

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games under a restricted set of policies. The latter case can be further reduce to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.

Foundation models are first pre-trained on vast unsupervised datasets and then fine-tuned on labeled data. Reinforcement learning, notably from human feedback (RLHF), can further align the network with the intended usage. Yet the imperfections in the proxy reward may hinder the training and lead to suboptimal results; the diversity of objectives in real-world tasks and human opinions exacerbate the issue. This paper proposes embracing the heterogeneity of diverse rewards by following a multi-policy strategy. Rather than focusing on a single a priori reward, we aim for Pareto-optimal generalization across the entire space of preferences. To this end, we propose rewarded soup, first specializing multiple networks independently (one for each proxy reward) and then interpolating their weights linearly. This succeeds empirically because we show that the weights remain linearly connected when fine-tuned on diverse rewards from a shared pre-trained initialization. We demonstrate the effectiveness of our approach for text-to-text (summarization, Q&A, helpful assistant, review), text-image (image captioning, text-to-image generation, visual grounding, VQA), and control (locomotion) tasks. We hope to enhance the alignment of deep models, and how they interact with the world in all its diversity.

We study infinite-horizon average-reward Markov decision processes (AMDPs) in the context of general function approximation. Specifically, we propose a novel algorithmic framework named Local-fitted Optimization with OPtimism (LOOP), which incorporates both model-based and value-based incarnations. In particular, LOOP features a novel construction of confidence sets and a low-switching policy updating scheme, which are tailored to the average-reward and function approximation setting. Moreover, for AMDPs, we propose a novel complexity measure -- average-reward generalized eluder coefficient (AGEC) -- which captures the challenge of exploration in AMDPs with general function approximation. Such a complexity measure encompasses almost all previously known tractable AMDP models, such as linear AMDPs and linear mixture AMDPs, and also includes newly identified cases such as kernel AMDPs and AMDPs with Bellman eluder dimensions. Using AGEC, we prove that LOOP achieves a sublinear mathcal{O}(poly(d, sp(V^*)) Tbeta ) regret, where d and beta correspond to AGEC and log-covering number of the hypothesis class respectively, sp(V^*) is the span of the optimal state bias function, T denotes the number of steps, and mathcal{O} (cdot) omits logarithmic factors. When specialized to concrete AMDP models, our regret bounds are comparable to those established by the existing algorithms designed specifically for these special cases. To the best of our knowledge, this paper presents the first comprehensive theoretical framework capable of handling nearly all AMDPs.

We study the autonomous exploration (AX) problem proposed by Lim & Auer (2012). In this setting, the objective is to discover a set of epsilon-optimal policies reaching a set S_L^{rightarrow} of incrementally L-controllable states. We introduce a novel layered decomposition of the set of incrementally L-controllable states that is based on the iterative application of a state-expansion operator. We leverage these results to design Layered Autonomous Exploration (LAE), a novel algorithm for AX that attains a sample complexity of mathcal{O}(LS^{rightarrow}_{L(1+epsilon)}Gamma_{L(1+epsilon)} A ln^{12}(S^{rightarrow}_{L(1+epsilon)})/epsilon^2), where S^{rightarrow}_{L(1+epsilon)} is the number of states that are incrementally L(1+epsilon)-controllable, A is the number of actions, and Gamma_{L(1+epsilon)} is the branching factor of the transitions over such states. LAE improves over the algorithm of Tarbouriech et al. (2020a) by a factor of L^2 and it is the first algorithm for AX that works in a countably-infinite state space. Moreover, we show that, under a certain identifiability assumption, LAE achieves minimax-optimal sample complexity of mathcal{O}(LS^{rightarrow}_{L}Aln^{12}(S^{rightarrow}_{L})/epsilon^2), outperforming existing algorithms and matching for the first time the lower bound proved by Cai et al. (2022) up to logarithmic factors.

In the field of algorithmic fairness, significant attention has been put on group fairness criteria, such as Demographic Parity and Equalized Odds. Nevertheless, these objectives, measured as global averages, have raised concerns about persistent local disparities between sensitive groups. In this work, we address the problem of local fairness, which ensures that the predictor is unbiased not only in terms of expectations over the whole population, but also within any subregion of the feature space, unknown at training time. To enforce this objective, we introduce ROAD, a novel approach that leverages the Distributionally Robust Optimization (DRO) framework within a fair adversarial learning objective, where an adversary tries to infer the sensitive attribute from the predictions. Using an instance-level re-weighting strategy, ROAD is designed to prioritize inputs that are likely to be locally unfair, i.e. where the adversary faces the least difficulty in reconstructing the sensitive attribute. Numerical experiments demonstrate the effectiveness of our method: it achieves Pareto dominance with respect to local fairness and accuracy for a given global fairness level across three standard datasets, and also enhances fairness generalization under distribution shift.

A coreset is a tiny weighted subset of an input set, that closely resembles the loss function, with respect to a certain set of queries. Coresets became prevalent in machine learning as they have shown to be advantageous for many applications. While coreset research is an active research area, unfortunately, coresets are constructed in a problem-dependent manner, where for each problem, a new coreset construction algorithm is usually suggested, a process that may take time or may be hard for new researchers in the field. Even the generic frameworks require additional (problem-dependent) computations or proofs to be done by the user. Besides, many problems do not have (provable) small coresets, limiting their applicability. To this end, we suggest an automatic practical framework for constructing coresets, which requires (only) the input data and the desired cost function from the user, without the need for any other task-related computation to be done by the user. To do so, we reduce the problem of approximating a loss function to an instance of vector summation approximation, where the vectors we aim to sum are loss vectors of a specific subset of the queries, such that we aim to approximate the image of the function on this subset. We show that while this set is limited, the coreset is quite general. An extensive experimental study on various machine learning applications is also conducted. Finally, we provide a ``plug and play" style implementation, proposing a user-friendly system that can be easily used to apply coresets for many problems. Full open source code can be found at https://github.com/alaamaalouf/AutoCoreset{https://github.com/alaamaalouf/AutoCoreset}. We believe that these contributions enable future research and easier use and applications of coresets.

In order to solve a task using reinforcement learning, it is necessary to first formalise the goal of that task as a reward function. However, for many real-world tasks, it is very difficult to manually specify a reward function that never incentivises undesirable behaviour. As a result, it is increasingly popular to use reward learning algorithms, which attempt to learn a reward function from data. However, the theoretical foundations of reward learning are not yet well-developed. In particular, it is typically not known when a given reward learning algorithm with high probability will learn a reward function that is safe to optimise. This means that reward learning algorithms generally must be evaluated empirically, which is expensive, and that their failure modes are difficult to anticipate in advance. One of the roadblocks to deriving better theoretical guarantees is the lack of good methods for quantifying the difference between reward functions. In this paper we provide a solution to this problem, in the form of a class of pseudometrics on the space of all reward functions that we call STARC (STAndardised Reward Comparison) metrics. We show that STARC metrics induce both an upper and a lower bound on worst-case regret, which implies that our metrics are tight, and that any metric with the same properties must be bilipschitz equivalent to ours. Moreover, we also identify a number of issues with reward metrics proposed by earlier works. Finally, we evaluate our metrics empirically, to demonstrate their practical efficacy. STARC metrics can be used to make both theoretical and empirical analysis of reward learning algorithms both easier and more principled.

We consider the problem of fairly allocating a sequence of indivisible items that arrive online in an arbitrary order to a group of n agents with additive normalized valuation functions. We consider both the allocation of goods and chores and propose algorithms for approximating maximin share (MMS) allocations. When agents have identical valuation functions the problem coincides with the semi-online machine covering problem (when items are goods) and load balancing problem (when items are chores), for both of which optimal competitive ratios have been achieved. In this paper, we consider the case when agents have general additive valuation functions. For the allocation of goods, we show that no competitive algorithm exists even when there are only three agents and propose an optimal 0.5-competitive algorithm for the case of two agents. For the allocation of chores, we propose a (2-1/n)-competitive algorithm for n>=3 agents and a square root of 2 (approximately 1.414)-competitive algorithm for two agents. Additionally, we show that no algorithm can do better than 15/11 (approximately 1.364)-competitive for two agents.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps R to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between R^{C-1} and the projected probability simplex Delta^{C-1} by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.

Prior works in multi-objective reinforcement learning typically use linear reward scalarization with fixed weights, which provably fail to capture non-convex Pareto fronts and thus yield suboptimal results. This limitation becomes especially critical in online preference alignment for large language models. Here, stochastic trajectories generated by parameterized policies create highly non-linear and non-convex mappings from parameters to objectives that no single static weighting scheme can find optimal trade-offs. We address this limitation by introducing dynamic reward weighting, which adaptively adjusts reward weights during the online reinforcement learning process. Unlike existing approaches that rely on fixed-weight interpolation, our dynamic weighting continuously balances and prioritizes objectives in training, facilitating effective exploration of Pareto fronts in objective space. We introduce two approaches of increasing sophistication and generalizability: (1) hypervolume-guided weight adaptation and (2) gradient-based weight optimization, offering a versatile toolkit for online multi-objective alignment. Our extensive experiments demonstrate their compatibility with commonly used online reinforcement learning algorithms (including GRPO, REINFORCE, and RLOO), effectiveness across multiple mathematical reasoning datasets, and applicability to different model families, consistently achieving Pareto dominant solutions with fewer training steps than fixed-weight linear scalarization baselines.

We resolve an open problem of Hanneke on the subject of universally consistent online learning with non-i.i.d. processes and unbounded losses. The notion of an optimistically universal learning rule was defined by Hanneke in an effort to study learning theory under minimal assumptions. A given learning rule is said to be optimistically universal if it achieves a low long-run average loss whenever the data generating process makes this goal achievable by some learning rule. Hanneke posed as an open problem whether, for every unbounded loss, the family of processes admitting universal learning are precisely those having a finite number of distinct values almost surely. In this paper, we completely resolve this problem, showing that this is indeed the case. As a consequence, this also offers a dramatically simpler formulation of an optimistically universal learning rule for any unbounded loss: namely, the simple memorization rule already suffices. Our proof relies on constructing random measurable partitions of the instance space and could be of independent interest for solving other open questions. We extend the results to the non-realizable setting thereby providing an optimistically universal Bayes consistent learning rule.

Probabilistic dynamics model ensemble is widely used in existing model-based reinforcement learning methods as it outperforms a single dynamics model in both asymptotic performance and sample efficiency. In this paper, we provide both practical and theoretical insights on the empirical success of the probabilistic dynamics model ensemble through the lens of Lipschitz continuity. We find that, for a value function, the stronger the Lipschitz condition is, the smaller the gap between the true dynamics- and learned dynamics-induced Bellman operators is, thus enabling the converged value function to be closer to the optimal value function. Hence, we hypothesize that the key functionality of the probabilistic dynamics model ensemble is to regularize the Lipschitz condition of the value function using generated samples. To test this hypothesis, we devise two practical robust training mechanisms through computing the adversarial noise and regularizing the value network's spectral norm to directly regularize the Lipschitz condition of the value functions. Empirical results show that combined with our mechanisms, model-based RL algorithms with a single dynamics model outperform those with an ensemble of probabilistic dynamics models. These findings not only support the theoretical insight, but also provide a practical solution for developing computationally efficient model-based RL algorithms.

Congestion is a common failure mode of markets, where consumers compete inefficiently on the same subset of goods (e.g., chasing the same small set of properties on a vacation rental platform). The typical economic story is that prices decongest by balancing supply and demand. But in modern online marketplaces, prices are typically set in a decentralized way by sellers, and the information about items is inevitably partial. The power of a platform is limited to controlling representations -- the subset of information about items presented by default to users. This motivates the present study of decongestion by representation, where a platform seeks to learn representations that reduce congestion and thus improve social welfare. The technical challenge is twofold: relying only on revealed preferences from the choices of consumers, rather than true preferences; and the combinatorial problem associated with representations that determine the features to reveal in the default view. We tackle both challenges by proposing a differentiable proxy of welfare that can be trained end-to-end on consumer choice data. We develop sufficient conditions for when decongestion promotes welfare, and present the results of extensive experiments on both synthetic and real data that demonstrate the utility of our approach.

We consider the exploration-exploitation dilemma in finite-horizon reinforcement learning problems whose state-action space is endowed with a metric. We introduce Kernel-UCBVI, a model-based optimistic algorithm that leverages the smoothness of the MDP and a non-parametric kernel estimator of the rewards and transitions to efficiently balance exploration and exploitation. For problems with K episodes and horizon H, we provide a regret bound of Oleft( H^3 K^{2d{2d+1}}right), where d is the covering dimension of the joint state-action space. This is the first regret bound for kernel-based RL using smoothing kernels, which requires very weak assumptions on the MDP and has been previously applied to a wide range of tasks. We empirically validate our approach in continuous MDPs with sparse rewards.

This paper investigates the problem of ensembling multiple strategies for sequential portfolios to outperform individual strategies in terms of long-term wealth. Due to the uncertainty of strategies' performances in the future market, which are often based on specific models and statistical assumptions, investors often mitigate risk and enhance robustness by combining multiple strategies, akin to common approaches in collective learning prediction. However, the absence of a distribution-free and consistent preference framework complicates decisions of combination due to the ambiguous objective. To address this gap, we introduce a novel framework for decision-making in combining strategies, irrespective of market conditions, by establishing the investor's preference between decisions and then forming a clear objective. Through this framework, we propose a combinatorial strategy construction, free from statistical assumptions, for any scale of component strategies, even infinite, such that it meets the determined criterion. Finally, we test the proposed strategy along with its accelerated variant and some other multi-strategies. The numerical experiments show results in favor of the proposed strategies, albeit with small tradeoffs in their Sharpe ratios, in which their cumulative wealths eventually exceed those of the best component strategies while the accelerated strategy significantly improves performance.

Quantum optimization holds promise for addressing classically intractable combinatorial problems, yet a standardized framework for benchmarking its performance, particularly in terms of solution quality, computational speed, and scalability is still lacking. In this work, we introduce a comprehensive benchmarking framework designed to systematically evaluate a range of quantum optimization techniques against well-established NP-hard combinatorial problems. Our framework focuses on key problem classes, including the Multi-Dimensional Knapsack Problem (MDKP), Maximum Independent Set (MIS), Quadratic Assignment Problem (QAP), and Market Share Problem (MSP). Our study evaluates gate-based quantum approaches, including the Variational Quantum Eigensolver (VQE) and its CVaR-enhanced variant, alongside advanced quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) and its extensions. To address resource constraints, we incorporate qubit compression techniques like Pauli Correlation Encoding (PCE) and Quantum Random Access Optimization (QRAO). Experimental results, obtained from simulated quantum environments and classical solvers, provide key insights into feasibility, optimality gaps, and scalability. Our findings highlight both the promise and current limitations of quantum optimization, offering a structured pathway for future research and practical applications in quantum-enhanced decision-making.

Unfair treatment and discrimination are critical ethical concerns in AI systems, particularly as their adoption expands across diverse domains. Addressing these challenges, the recent introduction of the EU AI Act establishes a unified legal framework to ensure legal certainty for AI innovation and investment while safeguarding public interests, such as health, safety, fundamental rights, democracy, and the rule of law (Recital 8). The Act encourages stakeholders to initiate dialogue on existing AI fairness notions to address discriminatory outcomes of AI systems. However, these notions often overlook the critical role of Socio-Economic Status (SES), inadvertently perpetuating biases that favour the economically advantaged. This is concerning, given that principles of equalization advocate for equalizing resources or opportunities to mitigate disadvantages beyond an individual's control. While provisions for discrimination are laid down in the AI Act, specialized directions should be broadened, particularly in addressing economic disparities perpetuated by AI systems. In this work, we explore the limitations of popular AI fairness notions using a real-world dataset (Adult), highlighting their inability to address SES-driven disparities. To fill this gap, we propose a novel fairness notion, Socio-Economic Parity (SEP), which incorporates SES and promotes positive actions for underprivileged groups while accounting for factors within an individual's control, such as working hours, which can serve as a proxy for effort. We define a corresponding fairness measure and optimize a model constrained by SEP to demonstrate practical utility. Our results show the effectiveness of SEP in mitigating SES-driven biases. By analyzing the AI Act alongside our method, we lay a foundation for aligning AI fairness with SES factors while ensuring legal compliance.

Reinforcement learning (RL) for continuous control typically employs distributions whose support covers the entire action space. In this work, we investigate the colloquially known phenomenon that trained agents often prefer actions at the boundaries of that space. We draw theoretical connections to the emergence of bang-bang behavior in optimal control, and provide extensive empirical evaluation across a variety of recent RL algorithms. We replace the normal Gaussian by a Bernoulli distribution that solely considers the extremes along each action dimension - a bang-bang controller. Surprisingly, this achieves state-of-the-art performance on several continuous control benchmarks - in contrast to robotic hardware, where energy and maintenance cost affect controller choices. Since exploration, learning,and the final solution are entangled in RL, we provide additional imitation learning experiments to reduce the impact of exploration on our analysis. Finally, we show that our observations generalize to environments that aim to model real-world challenges and evaluate factors to mitigate the emergence of bang-bang solutions. Our findings emphasize challenges for benchmarking continuous control algorithms, particularly in light of potential real-world applications.

Model-free algorithms for reinforcement learning typically require a condition called Bellman completeness in order to successfully operate off-policy with function approximation, unless additional conditions are met. However, Bellman completeness is a requirement that is much stronger than realizability and that is deemed to be too strong to hold in practice. In this work, we relax this structural assumption and analyze the statistical complexity of off-policy reinforcement learning when only realizability holds for the prescribed function class. We establish finite-sample guarantees for off-policy reinforcement learning that are free of the approximation error term known as inherent Bellman error, and that depend on the interplay of three factors. The first two are well known: they are the metric entropy of the function class and the concentrability coefficient that represents the cost of learning off-policy. The third factor is new, and it measures the violation of Bellman completeness, namely the mis-alignment between the chosen function class and its image through the Bellman operator. In essence, these error bounds establish that off-policy reinforcement learning remains statistically viable even in absence of Bellman completeness, and characterize the intermediate situation between the favorable Bellman complete setting and the worst-case scenario where exponential lower bounds are in force. Our analysis directly applies to the solution found by temporal difference algorithms when they converge.

Quantile regression is a fundamental problem in statistical learning motivated by a need to quantify uncertainty in predictions, or to model a diverse population without being overly reductive. For instance, epidemiological forecasts, cost estimates, and revenue predictions all benefit from being able to quantify the range of possible values accurately. As such, many models have been developed for this problem over many years of research in statistics, machine learning, and related fields. Rather than proposing yet another (new) algorithm for quantile regression we adopt a meta viewpoint: we investigate methods for aggregating any number of conditional quantile models, in order to improve accuracy and robustness. We consider weighted ensembles where weights may vary over not only individual models, but also over quantile levels, and feature values. All of the models we consider in this paper can be fit using modern deep learning toolkits, and hence are widely accessible (from an implementation point of view) and scalable. To improve the accuracy of the predicted quantiles (or equivalently, prediction intervals), we develop tools for ensuring that quantiles remain monotonically ordered, and apply conformal calibration methods. These can be used without any modification of the original library of base models. We also review some basic theory surrounding quantile aggregation and related scoring rules, and contribute a few new results to this literature (for example, the fact that post sorting or post isotonic regression can only improve the weighted interval score). Finally, we provide an extensive suite of empirical comparisons across 34 data sets from two different benchmark repositories.

We investigate stochastic combinatorial multi-armed bandit with semi-bandit feedback (CMAB). In CMAB, the question of the existence of an efficient policy with an optimal asymptotic regret (up to a factor poly-logarithmic with the action size) is still open for many families of distributions, including mutually independent outcomes, and more generally the multivariate sub-Gaussian family. We propose to answer the above question for these two families by analyzing variants of the Combinatorial Thompson Sampling policy (CTS). For mutually independent outcomes in [0,1], we propose a tight analysis of CTS using Beta priors. We then look at the more general setting of multivariate sub-Gaussian outcomes and propose a tight analysis of CTS using Gaussian priors. This last result gives us an alternative to the Efficient Sampling for Combinatorial Bandit policy (ESCB), which, although optimal, is not computationally efficient.

Algorithms for min-max optimization and variational inequalities are often studied under monotonicity assumptions. Motivated by non-monotone machine learning applications, we follow the line of works [Diakonikolas et al., 2021, Lee and Kim, 2021, Pethick et al., 2022, B\"ohm, 2022] aiming at going beyond monotonicity by considering the weaker negative comonotonicity assumption. In particular, we provide tight complexity analyses for the Proximal Point, Extragradient, and Optimistic Gradient methods in this setup, closing some questions on their working guarantees beyond monotonicity.

Mobile edge computing (MEC) is essential for next-generation mobile network applications that prioritize various performance metrics, including delays and energy efficiency. However, conventional single-objective scheduling solutions cannot be directly applied to practical systems in which the preferences (i.e., the weights of different objectives) are often unknown or challenging to specify in advance. In this study, we formulate a multi-objective offloading problem for MEC with multiple edges to minimize the sum of expected long-term energy consumption and delay while considering unknown preferences. To address the challenge of unknown preferences and the potentially diverse MEC systems, we propose a generalizable multi-objective (deep) reinforcement learning (GMORL)-based tasks offloading framework, which employs the Discrete Soft Actor-Critic (Discrete-SAC) method. Our method uses a single policy model to efficiently schedule tasks based on varying preferences and adapt to heterogeneous MEC systems with different CPU frequencies and server quantities. Under the proposed framework, we introduce a histogram-based state encoding method for constructing features for multiple edges in MEC systems, a sophisticated reward function for accurately computing the utilities of delay and energy consumption, and a novel neural network architecture for improving generalization. Simulation results demonstrate that our proposed GMORL scheme enhances the hypervolume of the Pareto front by up to 121.0% compared to benchmarks. Our code are avavilable at https://github.com/gracefulning/Generalizable-Pareto-Optimal-Offloading-with-Reinforcement-Learning-in-Mobile-Edge-Computing

Inverse reinforcement learning (IRL) denotes a powerful family of algorithms for recovering a reward function justifying the behavior demonstrated by an expert agent. A well-known limitation of IRL is the ambiguity in the choice of the reward function, due to the existence of multiple rewards that explain the observed behavior. This limitation has been recently circumvented by formulating IRL as the problem of estimating the feasible reward set, i.e., the region of the rewards compatible with the expert's behavior. In this paper, we make a step towards closing the theory gap of IRL in the case of finite-horizon problems with a generative model. We start by formally introducing the problem of estimating the feasible reward set, the corresponding PAC requirement, and discussing the properties of particular classes of rewards. Then, we provide the first minimax lower bound on the sample complexity for the problem of estimating the feasible reward set of order {Omega}Bigl( H^3SA{epsilon^2} bigl( log bigl(1{delta}bigl) + S bigl)Bigl), being S and A the number of states and actions respectively, H the horizon, epsilon the desired accuracy, and delta the confidence. We analyze the sample complexity of a uniform sampling strategy (US-IRL), proving a matching upper bound up to logarithmic factors. Finally, we outline several open questions in IRL and propose future research directions.

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimization challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fr\'echet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

Recent AI algorithms are blackbox models whose decisions are difficult to interpret. eXplainable AI (XAI) seeks to address lack of AI interpretability and trust by explaining to customers their AI decision, e.g., decision to reject a loan application. The common wisdom is that regulating AI by mandating fully transparent XAI leads to greater social welfare. This paper challenges this notion through a game theoretic model for a policy-maker who maximizes social welfare, firms in a duopoly competition that maximize profits, and heterogenous consumers. The results show that XAI regulation may be redundant. In fact, mandating fully transparent XAI may make firms and customers worse off. This reveals a trade-off between maximizing welfare and receiving explainable AI outputs. We also discuss managerial implications for policy-maker and firms.

In this paper, we consider the minimum submodular cost allocation (MSCA) problem. The input of MSCA is k non-negative submodular functions f_1,ldots,f_k on the ground set N given by evaluation oracles, and the goal is to partition N into k (possibly empty) sets X_1,ldots,X_k so that sum_{i=1}^k f_i(X_i) is minimized. In this paper, we focus on the case when f_1,ldots,f_k are monotone (denoted by Mono-MSCA). We provide a natural LP-relaxation for Mono-MSCA, which is equivalent to the convex program relaxation introduced by Chekuri and Ene. We show that the integrality gap of the LP-relaxation is at most k/2, which yields a k/2-approximation algorithm for Mono-MSCA. We also show that the integrality gap of the LP-relaxation is at least k/2-epsilon for any constant epsilon>0 when k is fixed.

We introduce and analyze distance preservation games (DPGs). In DPGs, agents express ideal distances to other agents and need to choose locations in the unit interval while preserving their ideal distances as closely as possible. We analyze the existence and computation of location profiles that are jump stable (i.e., no agent can benefit by moving to another location) or welfare optimal for DPGs, respectively. Specifically, we prove that there are DPGs without jump stable location profiles and identify important cases where such outcomes always exist and can be computed efficiently. Similarly, we show that finding welfare optimal location profiles is NP-complete and present approximation algorithms for finding solutions with social welfare close to optimal. Finally, we prove that DPGs have a price of anarchy of at most 2.

Multi-objective evolutionary algorithms (MOEAs) are among the most widely and successfully applied optimizers for multi-objective problems. However, to store many optimal trade-offs (the Pareto optima) at once, MOEAs are typically run with a large, static population of solution candidates, which can slow down the algorithm. We propose the dynamic NSGA-II (dNSGA-II), which is based on the popular NSGA-II and features a non-static population size. The dNSGA-II starts with a small initial population size of four and doubles it after a user-specified number tau of function evaluations, up to a maximum size of mu. Via a mathematical runtime analysis, we prove that the dNSGA-II with parameters mu geq 4(n + 1) and tau geq 256{50} e n computes the full Pareto front of the OneMinMax benchmark of size n in O(log(mu) tau + mu log(n)) function evaluations, both in expectation and with high probability. For an optimal choice of mu and tau, the resulting O(n log(n)) runtime improves the optimal expected runtime of the classic NSGA-II by a factor of Theta(n). In addition, we show that the parameter tau can be removed when utilizing concurrent runs of the dNSGA-II. This approach leads to a mild slow-down by a factor of O(log(n)) compared to an optimal choice of tau for the dNSGA-II, which is still a speed-up of Theta(n / log(n)) over the classic NSGA-II.

We consider the problem of preference based reinforcement learning (PbRL), where, unlike traditional reinforcement learning, an agent receives feedback only in terms of a 1 bit (0/1) preference over a trajectory pair instead of absolute rewards for them. The success of the traditional RL framework crucially relies on the underlying agent-reward model, which, however, depends on how accurately a system designer can express an appropriate reward function and often a non-trivial task. The main novelty of our framework is the ability to learn from preference-based trajectory feedback that eliminates the need to hand-craft numeric reward models. This paper sets up a formal framework for the PbRL problem with non-markovian rewards, where the trajectory preferences are encoded by a generalized linear model of dimension d. Assuming the transition model is known, we then propose an algorithm with almost optimal regret guarantee of mathcal{O}left( SH d log (T / delta) T right). We further, extend the above algorithm to the case of unknown transition dynamics, and provide an algorithm with near optimal regret guarantee mathcal{O}((d + H^2 + |S|)dT +|mathcal{S||A|TH} ). To the best of our knowledge, our work is one of the first to give tight regret guarantees for preference based RL problems with trajectory preferences.

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

The goal of this article is to provide an introduction to the desirability function approach to multi-objective optimization (direct and surrogate model-based), and multi-objective hyperparameter tuning. This work is based on the paper by Kuhn (2016). It presents a `Python` implementation of Kuhn's `R` package `desirability`. The `Python` package `spotdesirability` is available as part of the `sequential parameter optimization` framework. After a brief introduction to the desirability function approach is presented, three examples are given that demonstrate how to use the desirability functions for classical optimization, surrogate-model based optimization, and hyperparameter tuning.

In this paper, we study the predict-then-optimize problem where the output of a machine learning prediction task is used as the input of some downstream optimization problem, say, the objective coefficient vector of a linear program. The problem is also known as predictive analytics or contextual linear programming. The existing approaches largely suffer from either (i) optimization intractability (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii) requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach to the problem called maximum optimality margin which designs the machine learning loss function by the optimality condition of the downstream optimization. The max-margin formulation enjoys both computational efficiency and good theoretical properties for the learning procedure. More importantly, our new approach only needs the observations of the optimal solution in the training data rather than the objective function, which makes it a new and natural approach to the inverse linear programming problem under both contextual and context-free settings; we also analyze the proposed method under both offline and online settings, and demonstrate its performance using numerical experiments.

Is it possible to understand or imitate a policy maker's rationale by looking at past decisions they made? We formalize this question as the problem of learning social welfare functions belonging to the well-studied family of power mean functions. We focus on two learning tasks; in the first, the input is vectors of utilities of an action (decision or policy) for individuals in a group and their associated social welfare as judged by a policy maker, whereas in the second, the input is pairwise comparisons between the welfares associated with a given pair of utility vectors. We show that power mean functions are learnable with polynomial sample complexity in both cases, even if the comparisons are social welfare information is noisy. Finally, we design practical algorithms for these tasks and evaluate their performance.

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.

Reward functions are difficult to design and often hard to align with human intent. Preference-based Reinforcement Learning (RL) algorithms address these problems by learning reward functions from human feedback. However, the majority of preference-based RL methods na\"ively combine supervised reward models with off-the-shelf RL algorithms. Contemporary approaches have sought to improve performance and query complexity by using larger and more complex reward architectures such as transformers. Instead of using highly complex architectures, we develop a new and parameter-efficient algorithm, Inverse Preference Learning (IPL), specifically designed for learning from offline preference data. Our key insight is that for a fixed policy, the Q-function encodes all information about the reward function, effectively making them interchangeable. Using this insight, we completely eliminate the need for a learned reward function. Our resulting algorithm is simpler and more parameter-efficient. Across a suite of continuous control and robotics benchmarks, IPL attains competitive performance compared to more complex approaches that leverage transformer-based and non-Markovian reward functions while having fewer algorithmic hyperparameters and learned network parameters. Our code is publicly released.

In machine learning, the notion of multi-armed bandits refers to a class of online learning problems, in which an agent is supposed to simultaneously explore and exploit a given set of choice alternatives in the course of a sequential decision process. In the standard setting, the agent learns from stochastic feedback in the form of real-valued rewards. In many applications, however, numerical reward signals are not readily available -- instead, only weaker information is provided, in particular relative preferences in the form of qualitative comparisons between pairs of alternatives. This observation has motivated the study of variants of the multi-armed bandit problem, in which more general representations are used both for the type of feedback to learn from and the target of prediction. The aim of this paper is to provide a survey of the state of the art in this field, referred to as preference-based multi-armed bandits or dueling bandits. To this end, we provide an overview of problems that have been considered in the literature as well as methods for tackling them. Our taxonomy is mainly based on the assumptions made by these methods about the data-generating process and, related to this, the properties of the preference-based feedback.

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

Motivated by emerging applications such as live-streaming e-commerce, promotions and recommendations, we introduce and solve a general class of non-stationary multi-armed bandit problems that have the following two features: (i) the decision maker can pull and collect rewards from up to K,(ge 1) out of N different arms in each time period; (ii) the expected reward of an arm immediately drops after it is pulled, and then non-parametrically recovers as the arm's idle time increases. With the objective of maximizing the expected cumulative reward over T time periods, we design a class of ``Purely Periodic Policies'' that jointly set a period to pull each arm. For the proposed policies, we prove performance guarantees for both the offline problem and the online problems. For the offline problem when all model parameters are known, the proposed periodic policy obtains an approximation ratio that is at the order of 1-mathcal O(1/K), which is asymptotically optimal when K grows to infinity. For the online problem when the model parameters are unknown and need to be dynamically learned, we integrate the offline periodic policy with the upper confidence bound procedure to construct on online policy. The proposed online policy is proved to approximately have mathcal O(NT) regret against the offline benchmark. Our framework and policy design may shed light on broader offline planning and online learning applications with non-stationary and recovering rewards.

When trying to solve a computational problem, we are often faced with a choice between algorithms that are guaranteed to return the right answer but differ in their runtime distributions (e.g., SAT solvers, sorting algorithms). This paper aims to lay theoretical foundations for such choices by formalizing preferences over runtime distributions. It might seem that we should simply prefer the algorithm that minimizes expected runtime. However, such preferences would be driven by exactly how slow our algorithm is on bad inputs, whereas in practice we are typically willing to cut off occasional, sufficiently long runs before they finish. We propose a principled alternative, taking a utility-theoretic approach to characterize the scoring functions that describe preferences over algorithms. These functions depend on the way our value for solving our problem decreases with time and on the distribution from which captimes are drawn. We describe examples of realistic utility functions and show how to leverage a maximum-entropy approach for modeling underspecified captime distributions. Finally, we show how to efficiently estimate an algorithm's expected utility from runtime samples.

When a robot autonomously performs a complex task, it frequently must balance competing objectives while maintaining safety. This becomes more difficult in uncertain environments with stochastic outcomes. Enhancing transparency in the robot's behavior and aligning with user preferences are also crucial. This paper introduces a novel framework for multi-objective reinforcement learning that ensures safe task execution, optimizes trade-offs between objectives, and adheres to user preferences. The framework has two main layers: a multi-objective task planner and a high-level selector. The planning layer generates a set of optimal trade-off plans that guarantee satisfaction of a temporal logic task. The selector uses active inference to decide which generated plan best complies with user preferences and aids learning. Operating iteratively, the framework updates a parameterized learning model based on collected data. Case studies and benchmarks on both manipulation and mobile robots show that our framework outperforms other methods and (i) learns multiple optimal trade-offs, (ii) adheres to a user preference, and (iii) allows the user to adjust the balance between (i) and (ii).

We present a novel unified bilevel optimization-based framework, PARL, formulated to address the recently highlighted critical issue of policy alignment in reinforcement learning using utility or preference-based feedback. We identify a major gap within current algorithmic designs for solving policy alignment due to a lack of precise characterization of the dependence of the alignment objective on the data generated by policy trajectories. This shortfall contributes to the sub-optimal performance observed in contemporary algorithms. Our framework addressed these concerns by explicitly parameterizing the distribution of the upper alignment objective (reward design) by the lower optimal variable (optimal policy for the designed reward). Interestingly, from an optimization perspective, our formulation leads to a new class of stochastic bilevel problems where the stochasticity at the upper objective depends upon the lower-level variable. To demonstrate the efficacy of our formulation in resolving alignment issues in RL, we devised an algorithm named A-PARL to solve PARL problem, establishing sample complexity bounds of order O(1/T). Our empirical results substantiate that the proposed PARL can address the alignment concerns in RL by showing significant improvements (up to 63\% in terms of required samples) for policy alignment in large-scale environments of the Deepmind control suite and Meta world tasks.

Population Based Training (PBT) is an efficient hyperparameter optimization algorithm. PBT is a single-objective algorithm, but many real-world hyperparameter optimization problems involve two or more conflicting objectives. In this work, we therefore introduce a multi-objective version of PBT, MO-PBT. Our experiments on diverse multi-objective hyperparameter optimization problems (Precision/Recall, Accuracy/Fairness, Accuracy/Adversarial Robustness) show that MO-PBT outperforms random search, single-objective PBT, and the state-of-the-art multi-objective hyperparameter optimization algorithm MO-ASHA.

Asymmetrical distance structures (quasimetrics) are ubiquitous in our lives and are gaining more attention in machine learning applications. Imposing such quasimetric structures in model representations has been shown to improve many tasks, including reinforcement learning (RL) and causal relation learning. In this work, we present four desirable properties in such quasimetric models, and show how prior works fail at them. We propose Interval Quasimetric Embedding (IQE), which is designed to satisfy all four criteria. On three quasimetric learning experiments, IQEs show strong approximation and generalization abilities, leading to better performance and improved efficiency over prior methods. Project Page: https://www.tongzhouwang.info/interval_quasimetric_embedding Quasimetric Learning Code Package: https://www.github.com/quasimetric-learning/torch-quasimetric

In reinforcement learning (RL), rewards of states are typically considered additive, and following the Markov assumption, they are independent of states visited previously. In many important applications, such as coverage control, experiment design and informative path planning, rewards naturally have diminishing returns, i.e., their value decreases in light of similar states visited previously. To tackle this, we propose submodular RL (SubRL), a paradigm which seeks to optimize more general, non-additive (and history-dependent) rewards modelled via submodular set functions which capture diminishing returns. Unfortunately, in general, even in tabular settings, we show that the resulting optimization problem is hard to approximate. On the other hand, motivated by the success of greedy algorithms in classical submodular optimization, we propose SubPO, a simple policy gradient-based algorithm for SubRL that handles non-additive rewards by greedily maximizing marginal gains. Indeed, under some assumptions on the underlying Markov Decision Process (MDP), SubPO recovers optimal constant factor approximations of submodular bandits. Moreover, we derive a natural policy gradient approach for locally optimizing SubRL instances even in large state- and action- spaces. We showcase the versatility of our approach by applying SubPO to several applications, such as biodiversity monitoring, Bayesian experiment design, informative path planning, and coverage maximization. Our results demonstrate sample efficiency, as well as scalability to high-dimensional state-action spaces.

Self-attention in Transformers comes with a high computational cost because of their quadratic computational complexity, but their effectiveness in addressing problems in language and vision has sparked extensive research aimed at enhancing their efficiency. However, diverse experimental conditions, spanning multiple input domains, prevent a fair comparison based solely on reported results, posing challenges for model selection. To address this gap in comparability, we perform a large-scale benchmark of more than 45 models for image classification, evaluating key efficiency aspects, including accuracy, speed, and memory usage. Our benchmark provides a standardized baseline for efficiency-oriented transformers. We analyze the results based on the Pareto front -- the boundary of optimal models. Surprisingly, despite claims of other models being more efficient, ViT remains Pareto optimal across multiple metrics. We observe that hybrid attention-CNN models exhibit remarkable inference memory- and parameter-efficiency. Moreover, our benchmark shows that using a larger model in general is more efficient than using higher resolution images. Thanks to our holistic evaluation, we provide a centralized resource for practitioners and researchers, facilitating informed decisions when selecting or developing efficient transformers.

Wider adoption of neural networks in many critical domains such as finance and healthcare is being hindered by the need to explain their predictions and to impose additional constraints on them. Monotonicity constraint is one of the most requested properties in real-world scenarios and is the focus of this paper. One of the oldest ways to construct a monotonic fully connected neural network is to constrain signs on its weights. Unfortunately, this construction does not work with popular non-saturated activation functions as it can only approximate convex functions. We show this shortcoming can be fixed by constructing two additional activation functions from a typical unsaturated monotonic activation function and employing each of them on the part of neurons. Our experiments show this approach of building monotonic neural networks has better accuracy when compared to other state-of-the-art methods, while being the simplest one in the sense of having the least number of parameters, and not requiring any modifications to the learning procedure or post-learning steps. Finally, we prove it can approximate any continuous monotone function on a compact subset of R^n.

Multi-armed bandits a simple but very powerful framework for algorithms that make decisions over time under uncertainty. An enormous body of work has accumulated over the years, covered in several books and surveys. This book provides a more introductory, textbook-like treatment of the subject. Each chapter tackles a particular line of work, providing a self-contained, teachable technical introduction and a brief review of the further developments; many of the chapters conclude with exercises. The book is structured as follows. The first four chapters are on IID rewards, from the basic model to impossibility results to Bayesian priors to Lipschitz rewards. The next three chapters cover adversarial rewards, from the full-feedback version to adversarial bandits to extensions with linear rewards and combinatorially structured actions. Chapter 8 is on contextual bandits, a middle ground between IID and adversarial bandits in which the change in reward distributions is completely explained by observable contexts. The last three chapters cover connections to economics, from learning in repeated games to bandits with supply/budget constraints to exploration in the presence of incentives. The appendix provides sufficient background on concentration and KL-divergence. The chapters on "bandits with similarity information", "bandits with knapsacks" and "bandits and agents" can also be consumed as standalone surveys on the respective topics.

In problem-solving, we humans can come up with multiple novel solutions to the same problem. However, reinforcement learning algorithms can only produce a set of monotonous policies that maximize the cumulative reward but lack diversity and novelty. In this work, we address the problem of generating novel policies in reinforcement learning tasks. Instead of following the multi-objective framework used in existing methods, we propose to rethink the problem under a novel perspective of constrained optimization. We first introduce a new metric to evaluate the difference between policies and then design two practical novel policy generation methods following the new perspective. The two proposed methods, namely the Constrained Task Novel Bisector (CTNB) and the Interior Policy Differentiation (IPD), are derived from the feasible direction method and the interior point method commonly known in the constrained optimization literature. Experimental comparisons on the MuJoCo control suite show our methods can achieve substantial improvement over previous novelty-seeking methods in terms of both the novelty of policies and their performances in the primal task.