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Mar 17

Optimizing NOTEARS Objectives via Topological Swaps

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.

A hybrid deep-learning-metaheuristic framework for bi-level network design problems

This study proposes a hybrid deep-learning-metaheuristic framework with a bi-level architecture for road network design problems (NDPs). We train a graph neural network (GNN) to approximate the solution of the user equilibrium (UE) traffic assignment problem and use inferences made by the trained model to calculate fitness function evaluations of a genetic algorithm (GA) to approximate solutions for NDPs. Using three test networks, two NDP variants and an exact solver as benchmark, we show that on average, our proposed framework can provide solutions within 1.5% gap of the best results in less than 0.5% of the time used by the exact solution procedure. Our framework can be utilized within an expert system for infrastructure planning to determine the best infrastructure planning and management decisions under different scenarios. Given the flexibility of the framework, it can easily be adapted to many other decision problems that can be modeled as bi-level problems on graphs. Moreover, we foreseen interesting future research directions, thus we also put forward a brief research agenda for this topic. The key observation from our research that can shape future research is that the fitness function evaluation time using the inferences made by the GNN model was in the order of milliseconds, which points to an opportunity and a need for novel heuristics that 1) can cope well with noisy fitness function values provided by deep learning models, and 2) can use the significantly enlarged efficiency of the evaluation step to explore the search space effectively (rather than efficiently). This opens a new avenue for a modern class of metaheuristics that are crafted for use with AI-powered predictors.

Multiobjective Optimization of Non-Smooth PDE-Constrained Problems

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".

Generating Private Synthetic Data with Genetic Algorithms

We study the problem of efficiently generating differentially private synthetic data that approximate the statistical properties of an underlying sensitive dataset. In recent years, there has been a growing line of work that approaches this problem using first-order optimization techniques. However, such techniques are restricted to optimizing differentiable objectives only, severely limiting the types of analyses that can be conducted. For example, first-order mechanisms have been primarily successful in approximating statistical queries only in the form of marginals for discrete data domains. In some cases, one can circumvent such issues by relaxing the task's objective to maintain differentiability. However, even when possible, these approaches impose a fundamental limitation in which modifications to the minimization problem become additional sources of error. Therefore, we propose Private-GSD, a private genetic algorithm based on zeroth-order optimization heuristics that do not require modifying the original objective. As a result, it avoids the aforementioned limitations of first-order optimization. We empirically evaluate Private-GSD against baseline algorithms on data derived from the American Community Survey across a variety of statistics--otherwise known as statistical queries--both for discrete and real-valued attributes. We show that Private-GSD outperforms the state-of-the-art methods on non-differential queries while matching accuracy in approximating differentiable ones.

BQ-NCO: Bisimulation Quotienting for Efficient Neural Combinatorial Optimization

Despite the success of neural-based combinatorial optimization methods for end-to-end heuristic learning, out-of-distribution generalization remains a challenge. In this paper, we present a novel formulation of Combinatorial Optimization Problems (COPs) as Markov Decision Processes (MDPs) that effectively leverages common symmetries of COPs to improve out-of-distribution robustness. Starting from a direct MDP formulation of a constructive method, we introduce a generic way to reduce the state space, based on Bisimulation Quotienting (BQ) in MDPs. Then, for COPs with a recursive nature, we specialize the bisimulation and show how the reduced state exploits the symmetries of these problems and facilitates MDP solving. Our approach is principled and we prove that an optimal policy for the proposed BQ-MDP actually solves the associated COPs. We illustrate our approach on five classical problems: the Euclidean and Asymmetric Traveling Salesman, Capacitated Vehicle Routing, Orienteering and Knapsack Problems. Furthermore, for each problem, we introduce a simple attention-based policy network for the BQ-MDPs, which we train by imitation of (near) optimal solutions of small instances from a single distribution. We obtain new state-of-the-art results for the five COPs on both synthetic and realistic benchmarks. Notably, in contrast to most existing neural approaches, our learned policies show excellent generalization performance to much larger instances than seen during training, without any additional search procedure.

Graph Learning-based Fleet Scheduling for Urban Air Mobility under Operational Constraints, Varying Demand & Uncertainties

This paper develops a graph reinforcement learning approach to online planning of the schedule and destinations of electric aircraft that comprise an urban air mobility (UAM) fleet operating across multiple vertiports. This fleet scheduling problem is formulated to consider time-varying demand, constraints related to vertiport capacity, aircraft capacity and airspace safety guidelines, uncertainties related to take-off delay, weather-induced route closures, and unanticipated aircraft downtime. Collectively, such a formulation presents greater complexity, and potentially increased realism, than in existing UAM fleet planning implementations. To address these complexities, a new policy architecture is constructed, primary components of which include: graph capsule conv-nets for encoding vertiport and aircraft-fleet states both abstracted as graphs; transformer layers encoding time series information on demand and passenger fare; and a Multi-head Attention-based decoder that uses the encoded information to compute the probability of selecting each available destination for an aircraft. Trained with Proximal Policy Optimization, this policy architecture shows significantly better performance in terms of daily averaged profits on unseen test scenarios involving 8 vertiports and 40 aircraft, when compared to a random baseline and genetic algorithm-derived optimal solutions, while being nearly 1000 times faster in execution than the latter.

Hardness of Independent Learning and Sparse Equilibrium Computation in Markov Games

We consider the problem of decentralized multi-agent reinforcement learning in Markov games. A fundamental question is whether there exist algorithms that, when adopted by all agents and run independently in a decentralized fashion, lead to no-regret for each player, analogous to celebrated convergence results in normal-form games. While recent work has shown that such algorithms exist for restricted settings (notably, when regret is defined with respect to deviations to Markovian policies), the question of whether independent no-regret learning can be achieved in the standard Markov game framework was open. We provide a decisive negative resolution this problem, both from a computational and statistical perspective. We show that: - Under the widely-believed assumption that PPAD-hard problems cannot be solved in polynomial time, there is no polynomial-time algorithm that attains no-regret in general-sum Markov games when executed independently by all players, even when the game is known to the algorithm designer and the number of players is a small constant. - When the game is unknown, no algorithm, regardless of computational efficiency, can achieve no-regret without observing a number of episodes that is exponential in the number of players. Perhaps surprisingly, our lower bounds hold even for seemingly easier setting in which all agents are controlled by a a centralized algorithm. They are proven via lower bounds for a simpler problem we refer to as SparseCCE, in which the goal is to compute a coarse correlated equilibrium that is sparse in the sense that it can be represented as a mixture of a small number of product policies. The crux of our approach is a novel application of aggregation techniques from online learning, whereby we show that any algorithm for the SparseCCE problem can be used to compute approximate Nash equilibria for non-zero sum normal-form games.

Sparsity-Constrained Optimal Transport

Regularized optimal transport (OT) is now increasingly used as a loss or as a matching layer in neural networks. Entropy-regularized OT can be computed using the Sinkhorn algorithm but it leads to fully-dense transportation plans, meaning that all sources are (fractionally) matched with all targets. To address this issue, several works have investigated quadratic regularization instead. This regularization preserves sparsity and leads to unconstrained and smooth (semi) dual objectives, that can be solved with off-the-shelf gradient methods. Unfortunately, quadratic regularization does not give direct control over the cardinality (number of nonzeros) of the transportation plan. We propose in this paper a new approach for OT with explicit cardinality constraints on the transportation plan. Our work is motivated by an application to sparse mixture of experts, where OT can be used to match input tokens such as image patches with expert models such as neural networks. Cardinality constraints ensure that at most k tokens are matched with an expert, which is crucial for computational performance reasons. Despite the nonconvexity of cardinality constraints, we show that the corresponding (semi) dual problems are tractable and can be solved with first-order gradient methods. Our method can be thought as a middle ground between unregularized OT (recovered in the limit case k=1) and quadratically-regularized OT (recovered when k is large enough). The smoothness of the objectives increases as k increases, giving rise to a trade-off between convergence speed and sparsity of the optimal plan.

Derivative-Free Guidance in Continuous and Discrete Diffusion Models with Soft Value-Based Decoding

Diffusion models excel at capturing the natural design spaces of images, molecules, DNA, RNA, and protein sequences. However, rather than merely generating designs that are natural, we often aim to optimize downstream reward functions while preserving the naturalness of these design spaces. Existing methods for achieving this goal often require ``differentiable'' proxy models (e.g., classifier guidance or DPS) or involve computationally expensive fine-tuning of diffusion models (e.g., classifier-free guidance, RL-based fine-tuning). In our work, we propose a new method to address these challenges. Our algorithm is an iterative sampling method that integrates soft value functions, which looks ahead to how intermediate noisy states lead to high rewards in the future, into the standard inference procedure of pre-trained diffusion models. Notably, our approach avoids fine-tuning generative models and eliminates the need to construct differentiable models. This enables us to (1) directly utilize non-differentiable features/reward feedback, commonly used in many scientific domains, and (2) apply our method to recent discrete diffusion models in a principled way. Finally, we demonstrate the effectiveness of our algorithm across several domains, including image generation, molecule generation, and DNA/RNA sequence generation. The code is available at https://github.com/masa-ue/SVDD{https://github.com/masa-ue/SVDD}.

The Update-Equivalence Framework for Decision-Time Planning

The process of revising (or constructing) a policy at execution time -- known as decision-time planning -- has been key to achieving superhuman performance in perfect-information games like chess and Go. A recent line of work has extended decision-time planning to imperfect-information games, leading to superhuman performance in poker. However, these methods involve solving subgames whose sizes grow quickly in the amount of non-public information, making them unhelpful when the amount of non-public information is large. Motivated by this issue, we introduce an alternative framework for decision-time planning that is not based on solving subgames, but rather on update equivalence. In this update-equivalence framework, decision-time planning algorithms replicate the updates of last-iterate algorithms, which need not rely on public information. This facilitates scalability to games with large amounts of non-public information. Using this framework, we derive a provably sound search algorithm for fully cooperative games based on mirror descent and a search algorithm for adversarial games based on magnetic mirror descent. We validate the performance of these algorithms in cooperative and adversarial domains, notably in Hanabi, the standard benchmark for search in fully cooperative imperfect-information games. Here, our mirror descent approach exceeds or matches the performance of public information-based search while using two orders of magnitude less search time. This is the first instance of a non-public-information-based algorithm outperforming public-information-based approaches in a domain they have historically dominated.

Contextual Bandits in Payment Processing: Non-uniform Exploration and Supervised Learning at Adyen

Uniform random exploration in decision-making systems supports off-policy learning via supervision but incurs high regret, making it impractical for many applications. Conversely, non-uniform exploration offers better immediate performance but lacks support for off-policy learning. Recent research suggests that regression oracles can bridge this gap by combining non-uniform exploration with supervised learning. In this paper, we analyze these approaches within a real-world industrial context at Adyen, a large global payments processor characterized by batch logged delayed feedback, short-term memory, and dynamic action spaces under the Empirical Risk Minimization (ERM) framework. Our analysis reveals that while regression oracles significantly improve performance, they introduce challenges due to rigid algorithmic assumptions. Specifically, we observe that as a policy improves, subsequent generations may perform worse due to shifts in the reward distribution and increased class imbalance in the training data. This degradation occurs de spite improvements in other aspects of the training data, leading to decreased performance in successive policy iterations. We further explore the long-term impact of regression oracles, identifying a potential "oscillation effect." This effect arises when regression oracles influence probability estimates and the realizability of subsequent policy models, leading to fluctuations in performance across iterations. Our findings highlight the need for more adaptable algorithms that can leverage the benefits of regression oracles without introducing instability in policy performance over time.

Submodular Reinforcement Learning

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.

Fine-Tuning Discrete Diffusion Models via Reward Optimization with Applications to DNA and Protein Design

Recent studies have demonstrated the strong empirical performance of diffusion models on discrete sequences across domains from natural language to biological sequence generation. For example, in the protein inverse folding task, conditional diffusion models have achieved impressive results in generating natural-like sequences that fold back into the original structure. However, practical design tasks often require not only modeling a conditional distribution but also optimizing specific task objectives. For instance, we may prefer protein sequences with high stability. To address this, we consider the scenario where we have pre-trained discrete diffusion models that can generate natural-like sequences, as well as reward models that map sequences to task objectives. We then formulate the reward maximization problem within discrete diffusion models, analogous to reinforcement learning (RL), while minimizing the KL divergence against pretrained diffusion models to preserve naturalness. To solve this RL problem, we propose a novel algorithm, DRAKES, that enables direct backpropagation of rewards through entire trajectories generated by diffusion models, by making the originally non-differentiable trajectories differentiable using the Gumbel-Softmax trick. Our theoretical analysis indicates that our approach can generate sequences that are both natural-like and yield high rewards. While similar tasks have been recently explored in diffusion models for continuous domains, our work addresses unique algorithmic and theoretical challenges specific to discrete diffusion models, which arise from their foundation in continuous-time Markov chains rather than Brownian motion. Finally, we demonstrate the effectiveness of DRAKES in generating DNA and protein sequences that optimize enhancer activity and protein stability, respectively, important tasks for gene therapies and protein-based therapeutics.

Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve mgg 1 lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling I blocks and sampling B samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of O(mepsilon^{-3I(I<m)}{II} + mepsilon^{-3}{IB}) for finding an epsilon-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

Multi-Agent Reinforcement Learning for Offloading Cellular Communications with Cooperating UAVs

Effective solutions for intelligent data collection in terrestrial cellular networks are crucial, especially in the context of Internet of Things applications. The limited spectrum and coverage area of terrestrial base stations pose challenges in meeting the escalating data rate demands of network users. Unmanned aerial vehicles, known for their high agility, mobility, and flexibility, present an alternative means to offload data traffic from terrestrial BSs, serving as additional access points. This paper introduces a novel approach to efficiently maximize the utilization of multiple UAVs for data traffic offloading from terrestrial BSs. Specifically, the focus is on maximizing user association with UAVs by jointly optimizing UAV trajectories and users association indicators under quality of service constraints. Since, the formulated UAVs control problem is nonconvex and combinatorial, this study leverages the multi agent reinforcement learning framework. In this framework, each UAV acts as an independent agent, aiming to maintain inter UAV cooperative behavior. The proposed approach utilizes the finite state Markov decision process to account for UAVs velocity constraints and the relationship between their trajectories and state space. A low complexity distributed state action reward state action algorithm is presented to determine UAVs optimal sequential decision making policies over training episodes. The extensive simulation results validate the proposed analysis and offer valuable insights into the optimal UAV trajectories. The derived trajectories demonstrate superior average UAV association performance compared to benchmark techniques such as Q learning and particle swarm optimization.

Random Network Distillation Based Deep Reinforcement Learning for AGV Path Planning

With the flourishing development of intelligent warehousing systems, the technology of Automated Guided Vehicle (AGV) has experienced rapid growth. Within intelligent warehousing environments, AGV is required to safely and rapidly plan an optimal path in complex and dynamic environments. Most research has studied deep reinforcement learning to address this challenge. However, in the environments with sparse extrinsic rewards, these algorithms often converge slowly, learn inefficiently or fail to reach the target. Random Network Distillation (RND), as an exploration enhancement, can effectively improve the performance of proximal policy optimization, especially enhancing the additional intrinsic rewards of the AGV agent which is in sparse reward environments. Moreover, most of the current research continues to use 2D grid mazes as experimental environments. These environments have insufficient complexity and limited action sets. To solve this limitation, we present simulation environments of AGV path planning with continuous actions and positions for AGVs, so that it can be close to realistic physical scenarios. Based on our experiments and comprehensive analysis of the proposed method, the results demonstrate that our proposed method enables AGV to more rapidly complete path planning tasks with continuous actions in our environments. A video of part of our experiments can be found at https://youtu.be/lwrY9YesGmw.

RARTS: An Efficient First-Order Relaxed Architecture Search Method

Differentiable architecture search (DARTS) is an effective method for data-driven neural network design based on solving a bilevel optimization problem. Despite its success in many architecture search tasks, there are still some concerns about the accuracy of first-order DARTS and the efficiency of the second-order DARTS. In this paper, we formulate a single level alternative and a relaxed architecture search (RARTS) method that utilizes the whole dataset in architecture learning via both data and network splitting, without involving mixed second derivatives of the corresponding loss functions like DARTS. In our formulation of network splitting, two networks with different but related weights cooperate in search of a shared architecture. The advantage of RARTS over DARTS is justified by a convergence theorem and an analytically solvable model. Moreover, RARTS outperforms DARTS and its variants in accuracy and search efficiency, as shown in adequate experimental results. For the task of searching topological architecture, i.e., the edges and the operations, RARTS obtains a higher accuracy and 60\% reduction of computational cost than second-order DARTS on CIFAR-10. RARTS continues to out-perform DARTS upon transfer to ImageNet and is on par with recent variants of DARTS even though our innovation is purely on the training algorithm without modifying search space. For the task of searching width, i.e., the number of channels in convolutional layers, RARTS also outperforms the traditional network pruning benchmarks. Further experiments on the public architecture search benchmark like NATS-Bench also support the preeminence of RARTS.

Generating Dispatching Rules for the Interrupting Swap-Allowed Blocking Job Shop Problem Using Graph Neural Network and Reinforcement Learning

The interrupting swap-allowed blocking job shop problem (ISBJSSP) is a complex scheduling problem that is able to model many manufacturing planning and logistics applications realistically by addressing both the lack of storage capacity and unforeseen production interruptions. Subjected to random disruptions due to machine malfunction or maintenance, industry production settings often choose to adopt dispatching rules to enable adaptive, real-time re-scheduling, rather than traditional methods that require costly re-computation on the new configuration every time the problem condition changes dynamically. To generate dispatching rules for the ISBJSSP problem, a method that uses graph neural networks and reinforcement learning is proposed. ISBJSSP is formulated as a Markov decision process. Using proximal policy optimization, an optimal scheduling policy is learnt from randomly generated instances. Employing a set of reported benchmark instances, we conduct a detailed experimental study on ISBJSSP instances with a range of machine shutdown probabilities to show that the scheduling policies generated can outperform or are at least as competitive as existing dispatching rules with predetermined priority. This study shows that the ISBJSSP, which requires real-time adaptive solutions, can be scheduled efficiently with the proposed machine learning method when production interruptions occur with random machine shutdowns.

Revisiting Design Choices in Offline Model-Based Reinforcement Learning

Offline reinforcement learning enables agents to leverage large pre-collected datasets of environment transitions to learn control policies, circumventing the need for potentially expensive or unsafe online data collection. Significant progress has been made recently in offline model-based reinforcement learning, approaches which leverage a learned dynamics model. This typically involves constructing a probabilistic model, and using the model uncertainty to penalize rewards where there is insufficient data, solving for a pessimistic MDP that lower bounds the true MDP. Existing methods, however, exhibit a breakdown between theory and practice, whereby pessimistic return ought to be bounded by the total variation distance of the model from the true dynamics, but is instead implemented through a penalty based on estimated model uncertainty. This has spawned a variety of uncertainty heuristics, with little to no comparison between differing approaches. In this paper, we compare these heuristics, and design novel protocols to investigate their interaction with other hyperparameters, such as the number of models, or imaginary rollout horizon. Using these insights, we show that selecting these key hyperparameters using Bayesian Optimization produces superior configurations that are vastly different to those currently used in existing hand-tuned state-of-the-art methods, and result in drastically stronger performance.

Challenging the Need for Packet Spraying in Large-Scale Distributed Training

Large-scale distributed training in production datacenters constitutes a challenging workload bottlenecked by network communication. In response, both major industry players (e.g., Ultra Ethernet Consortium) and parts of academia have surprisingly, and almost unanimously, agreed that packet spraying is necessary to improve the performance of large-scale distributed training workloads. In this paper, we challenge this prevailing belief and pose the question: How close can a singlepath transport approach an optimal multipath transport? We demonstrate that singlepath transport (from a NIC's perspective) is sufficient and can perform nearly as well as an ideal multipath transport with packet spraying, particularly in the context of distributed training in leaf-spine topologies. Our assertion is based on four key observations about workloads driven by collective communication patterns: (i) flows within a collective start almost simultaneously, (ii) flow sizes are nearly equal, (iii) the completion time of a collective is more crucial than individual flow completion times, and (iv) flows can be split upon arrival. We analytically prove that singlepath transport, using minimal flow splitting (at the application layer), is equivalent to an ideal multipath transport with packet spraying in terms of maximum congestion. Our preliminary evaluations support our claims. This paper suggests an alternative agenda for developing next-generation transport protocols tailored for large-scale distributed training.

Last Switch Dependent Bandits with Monotone Payoff Functions

In a recent work, Laforgue et al. introduce the model of last switch dependent (LSD) bandits, in an attempt to capture nonstationary phenomena induced by the interaction between the player and the environment. Examples include satiation, where consecutive plays of the same action lead to decreased performance, or deprivation, where the payoff of an action increases after an interval of inactivity. In this work, we take a step towards understanding the approximability of planning LSD bandits, namely, the (NP-hard) problem of computing an optimal arm-pulling strategy under complete knowledge of the model. In particular, we design the first efficient constant approximation algorithm for the problem and show that, under a natural monotonicity assumption on the payoffs, its approximation guarantee (almost) matches the state-of-the-art for the special and well-studied class of recharging bandits (also known as delay-dependent). In this attempt, we develop new tools and insights for this class of problems, including a novel higher-dimensional relaxation and the technique of mirroring the evolution of virtual states. We believe that these novel elements could potentially be used for approaching richer classes of action-induced nonstationary bandits (e.g., special instances of restless bandits). In the case where the model parameters are initially unknown, we develop an online learning adaptation of our algorithm for which we provide sublinear regret guarantees against its full-information counterpart.

Online Information Acquisition: Hiring Multiple Agents

We investigate the mechanism design problem faced by a principal who hires multiple agents to gather and report costly information. Then, the principal exploits the information to make an informed decision. We model this problem as a game, where the principal announces a mechanism consisting in action recommendations and a payment function, a.k.a. scoring rule. Then, each agent chooses an effort level and receives partial information about an underlying state of nature based on the effort. Finally, the agents report the information (possibly non-truthfully), the principal takes a decision based on this information, and the agents are paid according to the scoring rule. While previous work focuses on single-agent problems, we consider multi-agents settings. This poses the challenge of coordinating the agents' efforts and aggregating correlated information. Indeed, we show that optimal mechanisms must correlate agents' efforts, which introduces externalities among the agents, and hence complex incentive compatibility constraints and equilibrium selection problems. First, we design a polynomial-time algorithm to find an optimal incentive compatible mechanism. Then, we study an online problem, where the principal repeatedly interacts with a group of unknown agents. We design a no-regret algorithm that provides mathcal{O}(T^{2/3}) regret with respect to an optimal mechanism, matching the state-of-the-art bound for single-agent settings.

Asymmetric Graph Error Control with Low Complexity in Causal Bandits

In this paper, the causal bandit problem is investigated, in which the objective is to select an optimal sequence of interventions on nodes in a causal graph. It is assumed that the graph is governed by linear structural equations; it is further assumed that both the causal topology and the distribution of interventions are unknown. By exploiting the causal relationships between the nodes whose signals contribute to the reward, interventions are optimized. First, based on the difference between the two types of graph identification errors (false positives and negatives), a causal graph learning method is proposed, which strongly reduces sample complexity relative to the prior art by learning sub-graphs. Under the assumption of Gaussian exogenous inputs and minimum-mean squared error weight estimation, a new uncertainty bound tailored to the causal bandit problem is derived. This uncertainty bound drives an upper confidence bound based intervention selection to optimize the reward. To cope with non-stationary bandits, a sub-graph change detection mechanism is proposed, with high sample efficiency. Numerical results compare the new methodology to existing schemes and show a substantial performance improvement in both stationary and non-stationary settings. Compared to existing approaches, the proposed scheme takes 67% fewer samples to learn the causal structure and achieves an average reward gain of 85%.

Rethinking the "Heatmap + Monte Carlo Tree Search" Paradigm for Solving Large Scale TSP

The Travelling Salesman Problem (TSP) remains a fundamental challenge in combinatorial optimization, inspiring diverse algorithmic strategies. This paper revisits the "heatmap + Monte Carlo Tree Search (MCTS)" paradigm that has recently gained traction for learning-based TSP solutions. Within this framework, heatmaps encode the likelihood of edges forming part of the optimal tour, and MCTS refines this probabilistic guidance to discover optimal solutions. Contemporary approaches have predominantly emphasized the refinement of heatmap generation through sophisticated learning models, inadvertently sidelining the critical role of MCTS. Our extensive empirical analysis reveals two pivotal insights: 1) The configuration of MCTS strategies profoundly influences the solution quality, demanding meticulous tuning to leverage their full potential; 2) Our findings demonstrate that a rudimentary and parameter-free heatmap, derived from the intrinsic k-nearest nature of TSP, can rival or even surpass the performance of complicated heatmaps, with strong generalizability across various scales. Empirical evaluations across various TSP scales underscore the efficacy of our approach, achieving competitive results. These observations challenge the prevailing focus on heatmap sophistication, advocating a reevaluation of the paradigm to harness both components synergistically. Our code is available at: https://github.com/LOGO-CUHKSZ/rethink_mcts_tsp.

Offline Planning and Online Learning under Recovering Rewards

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.

DIFF2: Differential Private Optimization via Gradient Differences for Nonconvex Distributed Learning

Differential private optimization for nonconvex smooth objective is considered. In the previous work, the best known utility bound is widetilde O(d/(nvarepsilon_DP)) in terms of the squared full gradient norm, which is achieved by Differential Private Gradient Descent (DP-GD) as an instance, where n is the sample size, d is the problem dimensionality and varepsilon_DP is the differential privacy parameter. To improve the best known utility bound, we propose a new differential private optimization framework called DIFF2 (DIFFerential private optimization via gradient DIFFerences) that constructs a differential private global gradient estimator with possibly quite small variance based on communicated gradient differences rather than gradients themselves. It is shown that DIFF2 with a gradient descent subroutine achieves the utility of widetilde O(d^{2/3}/(nvarepsilon_DP)^{4/3}), which can be significantly better than the previous one in terms of the dependence on the sample size n. To the best of our knowledge, this is the first fundamental result to improve the standard utility widetilde O(d/(nvarepsilon_DP)) for nonconvex objectives. Additionally, a more computational and communication efficient subroutine is combined with DIFF2 and its theoretical analysis is also given. Numerical experiments are conducted to validate the superiority of DIFF2 framework.

Optimal Horizon-Free Reward-Free Exploration for Linear Mixture MDPs

We study reward-free reinforcement learning (RL) with linear function approximation, where the agent works in two phases: (1) in the exploration phase, the agent interacts with the environment but cannot access the reward; and (2) in the planning phase, the agent is given a reward function and is expected to find a near-optimal policy based on samples collected in the exploration phase. The sample complexities of existing reward-free algorithms have a polynomial dependence on the planning horizon, which makes them intractable for long planning horizon RL problems. In this paper, we propose a new reward-free algorithm for learning linear mixture Markov decision processes (MDPs), where the transition probability can be parameterized as a linear combination of known feature mappings. At the core of our algorithm is uncertainty-weighted value-targeted regression with exploration-driven pseudo-reward and a high-order moment estimator for the aleatoric and epistemic uncertainties. When the total reward is bounded by 1, we show that our algorithm only needs to explore tilde O( d^2varepsilon^{-2}) episodes to find an varepsilon-optimal policy, where d is the dimension of the feature mapping. The sample complexity of our algorithm only has a polylogarithmic dependence on the planning horizon and therefore is ``horizon-free''. In addition, we provide an Omega(d^2varepsilon^{-2}) sample complexity lower bound, which matches the sample complexity of our algorithm up to logarithmic factors, suggesting that our algorithm is optimal.

Rethinking Architecture Selection in Differentiable NAS

Differentiable Neural Architecture Search is one of the most popular Neural Architecture Search (NAS) methods for its search efficiency and simplicity, accomplished by jointly optimizing the model weight and architecture parameters in a weight-sharing supernet via gradient-based algorithms. At the end of the search phase, the operations with the largest architecture parameters will be selected to form the final architecture, with the implicit assumption that the values of architecture parameters reflect the operation strength. While much has been discussed about the supernet's optimization, the architecture selection process has received little attention. We provide empirical and theoretical analysis to show that the magnitude of architecture parameters does not necessarily indicate how much the operation contributes to the supernet's performance. We propose an alternative perturbation-based architecture selection that directly measures each operation's influence on the supernet. We re-evaluate several differentiable NAS methods with the proposed architecture selection and find that it is able to extract significantly improved architectures from the underlying supernets consistently. Furthermore, we find that several failure modes of DARTS can be greatly alleviated with the proposed selection method, indicating that much of the poor generalization observed in DARTS can be attributed to the failure of magnitude-based architecture selection rather than entirely the optimization of its supernet.

Near-optimal Conservative Exploration in Reinforcement Learning under Episode-wise Constraints

This paper investigates conservative exploration in reinforcement learning where the performance of the learning agent is guaranteed to be above a certain threshold throughout the learning process. It focuses on the tabular episodic Markov Decision Process (MDP) setting that has finite states and actions. With the knowledge of an existing safe baseline policy, an algorithm termed as StepMix is proposed to balance the exploitation and exploration while ensuring that the conservative constraint is never violated in each episode with high probability. StepMix features a unique design of a mixture policy that adaptively and smoothly interpolates between the baseline policy and the optimistic policy. Theoretical analysis shows that StepMix achieves near-optimal regret order as in the constraint-free setting, indicating that obeying the stringent episode-wise conservative constraint does not compromise the learning performance. Besides, a randomization-based EpsMix algorithm is also proposed and shown to achieve the same performance as StepMix. The algorithm design and theoretical analysis are further extended to the setting where the baseline policy is not given a priori but must be learned from an offline dataset, and it is proved that similar conservative guarantee and regret can be achieved if the offline dataset is sufficiently large. Experiment results corroborate the theoretical analysis and demonstrate the effectiveness of the proposed conservative exploration strategies.

Differentially Private Sequential Learning

In a differentially private sequential learning setting, agents introduce endogenous noise into their actions to maintain privacy. Applying this to a standard sequential learning model leads to different outcomes for continuous vs. binary signals. For continuous signals with a nonzero privacy budget, we introduce a novel smoothed randomized response mechanism that adapts noise based on distance to a threshold, unlike traditional randomized response, which applies uniform noise. This enables agents' actions to better reflect both private signals and observed history, accelerating asymptotic learning speed to Theta_{epsilon}(log(n)), compared to Theta(log(n)) in the non-private regime where privacy budget is infinite. Moreover, in the non-private setting, the expected stopping time for the first correct decision and the number of incorrect actions diverge, meaning early agents may make mistakes for an unreasonably long period. In contrast, under a finite privacy budget epsilon in (0,1), both remain finite, highlighting a stark contrast between private and non-private learning. Learning with continuous signals in the private regime is more efficient, as smooth randomized response enhances the log-likelihood ratio over time, improving information aggregation. Conversely, for binary signals, differential privacy noise hinders learning, as agents tend to use a constant randomized response strategy before an information cascade forms, reducing action informativeness and hampering the overall process.

Gradient is All You Need?

In this paper we provide a novel analytical perspective on the theoretical understanding of gradient-based learning algorithms by interpreting consensus-based optimization (CBO), a recently proposed multi-particle derivative-free optimization method, as a stochastic relaxation of gradient descent. Remarkably, we observe that through communication of the particles, CBO exhibits a stochastic gradient descent (SGD)-like behavior despite solely relying on evaluations of the objective function. The fundamental value of such link between CBO and SGD lies in the fact that CBO is provably globally convergent to global minimizers for ample classes of nonsmooth and nonconvex objective functions, hence, on the one side, offering a novel explanation for the success of stochastic relaxations of gradient descent. On the other side, contrary to the conventional wisdom for which zero-order methods ought to be inefficient or not to possess generalization abilities, our results unveil an intrinsic gradient descent nature of such heuristics. This viewpoint furthermore complements previous insights into the working principles of CBO, which describe the dynamics in the mean-field limit through a nonlinear nonlocal partial differential equation that allows to alleviate complexities of the nonconvex function landscape. Our proofs leverage a completely nonsmooth analysis, which combines a novel quantitative version of the Laplace principle (log-sum-exp trick) and the minimizing movement scheme (proximal iteration). In doing so, we furnish useful and precise insights that explain how stochastic perturbations of gradient descent overcome energy barriers and reach deep levels of nonconvex functions. Instructive numerical illustrations support the provided theoretical insights.

Ensembling Portfolio Strategies for Long-Term Investments: A Distribution-Free Preference Framework for Decision-Making and Algorithms

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.

Probabilistic Mixture-of-Experts for Efficient Deep Reinforcement Learning

Deep reinforcement learning (DRL) has successfully solved various problems recently, typically with a unimodal policy representation. However, grasping distinguishable skills for some tasks with non-unique optima can be essential for further improving its learning efficiency and performance, which may lead to a multimodal policy represented as a mixture-of-experts (MOE). To our best knowledge, present DRL algorithms for general utility do not deploy this method as policy function approximators due to the potential challenge in its differentiability for policy learning. In this work, we propose a probabilistic mixture-of-experts (PMOE) implemented with a Gaussian mixture model (GMM) for multimodal policy, together with a novel gradient estimator for the indifferentiability problem, which can be applied in generic off-policy and on-policy DRL algorithms using stochastic policies, e.g., Soft Actor-Critic (SAC) and Proximal Policy Optimisation (PPO). Experimental results testify the advantage of our method over unimodal polices and two different MOE methods, as well as a method of option frameworks, based on the above two types of DRL algorithms, on six MuJoCo tasks. Different gradient estimations for GMM like the reparameterisation trick (Gumbel-Softmax) and the score-ratio trick are also compared with our method. We further empirically demonstrate the distinguishable primitives learned with PMOE and show the benefits of our method in terms of exploration.

Efficiently Training Deep-Learning Parametric Policies using Lagrangian Duality

Constrained Markov Decision Processes (CMDPs) are critical in many high-stakes applications, where decisions must optimize cumulative rewards while strictly adhering to complex nonlinear constraints. In domains such as power systems, finance, supply chains, and precision robotics, violating these constraints can result in significant financial or societal costs. Existing Reinforcement Learning (RL) methods often struggle with sample efficiency and effectiveness in finding feasible policies for highly and strictly constrained CMDPs, limiting their applicability in these environments. Stochastic dual dynamic programming is often used in practice on convex relaxations of the original problem, but they also encounter computational challenges and loss of optimality. This paper introduces a novel approach, Two-Stage Deep Decision Rules (TS-DDR), to efficiently train parametric actor policies using Lagrangian Duality. TS-DDR is a self-supervised learning algorithm that trains general decision rules (parametric policies) using stochastic gradient descent (SGD); its forward passes solve {\em deterministic} optimization problems to find feasible policies, and its backward passes leverage duality theory to train the parametric policy with closed-form gradients. TS-DDR inherits the flexibility and computational performance of deep learning methodologies to solve CMDP problems. Applied to the Long-Term Hydrothermal Dispatch (LTHD) problem using actual power system data from Bolivia, TS-DDR is shown to enhance solution quality and to reduce computation times by several orders of magnitude when compared to current state-of-the-art methods.

PARL: A Unified Framework for Policy Alignment in Reinforcement Learning

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.

Stochastic-Robust Planning of Networked Hydrogen-Electrical Microgrids: A Study on Induced Refueling Demand

Hydrogen-electrical microgrids are increasingly assuming an important role on the pathway toward decarbonization of energy and transportation systems. This paper studies networked hydrogen-electrical microgrids planning (NHEMP), considering a critical but often-overlooked issue, i.e., the demand-inducing effect (DIE) associated with infrastructure development decisions. Specifically, higher refueling capacities will attract more refueling demand of hydrogen-powered vehicles (HVs). To capture such interactions between investment decisions and induced refueling demand, we introduce a decision-dependent uncertainty (DDU) set and build a trilevel stochastic-robust formulation. The upper-level determines optimal investment strategies for hydrogen-electrical microgrids, the lower-level optimizes the risk-aware operation schedules across a series of stochastic scenarios, and, for each scenario, the middle-level identifies the "worst" situation of refueling demand within an individual DDU set to ensure economic feasibility. Then, an adaptive and exact decomposition algorithm, based on Parametric Column-and-Constraint Generation (PC&CG), is customized and developed to address the computational challenge and to quantitatively analyze the impact of DIE. Case studies on an IEEE exemplary system validate the effectiveness of the proposed NHEMP model and the PC&CG algorithm. It is worth highlighting that DIE can make an important contribution to the economic benefits of NHEMP, yet its significance will gradually decrease when the main bottleneck transits to other system restrictions.

Neur2RO: Neural Two-Stage Robust Optimization

Robust optimization provides a mathematical framework for modeling and solving decision-making problems under worst-case uncertainty. This work addresses two-stage robust optimization (2RO) problems (also called adjustable robust optimization), wherein first-stage and second-stage decisions are made before and after uncertainty is realized, respectively. This results in a nested min-max-min optimization problem which is extremely challenging computationally, especially when the decisions are discrete. We propose Neur2RO, an efficient machine learning-driven instantiation of column-and-constraint generation (CCG), a classical iterative algorithm for 2RO. Specifically, we learn to estimate the value function of the second-stage problem via a novel neural network architecture that is easy to optimize over by design. Embedding our neural network into CCG yields high-quality solutions quickly as evidenced by experiments on two 2RO benchmarks, knapsack and capital budgeting. For knapsack, Neur2RO finds solutions that are within roughly 2% of the best-known values in a few seconds compared to the three hours of the state-of-the-art exact branch-and-price algorithm; for larger and more complex instances, Neur2RO finds even better solutions. For capital budgeting, Neur2RO outperforms three variants of the k-adaptability algorithm, particularly on the largest instances, with a 10 to 100-fold reduction in solution time. Our code and data are available at https://github.com/khalil-research/Neur2RO.

Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approach

We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. We show that PE is equivalent to maintaining the martingale condition of a process. From this perspective, we find that the mean--square TD error approximates the quadratic variation of the martingale and thus is not a suitable objective for PE. We present two methods to use the martingale characterization for designing PE algorithms. The first one minimizes a "martingale loss function", whose solution is proved to be the best approximation of the true value function in the mean--square sense. This method interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the "martingale orthogonality conditions" with test functions. Solving these equations in different ways recovers various classical TD algorithms, such as TD(lambda), LSTD, and GTD. Different choices of test functions determine in what sense the resulting solutions approximate the true value function. Moreover, we prove that any convergent time-discretized algorithm converges to its continuous-time counterpart as the mesh size goes to zero, and we provide the convergence rate. We demonstrate the theoretical results and corresponding algorithms with numerical experiments and applications.

Variance Reduced Halpern Iteration for Finite-Sum Monotone Inclusions

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.

Free from Bellman Completeness: Trajectory Stitching via Model-based Return-conditioned Supervised Learning

Off-policy dynamic programming (DP) techniques such as Q-learning have proven to be important in sequential decision-making problems. In the presence of function approximation, however, these techniques often diverge due to the absence of Bellman completeness in the function classes considered, a crucial condition for the success of DP-based methods. In this paper, we show how off-policy learning techniques based on return-conditioned supervised learning (RCSL) are able to circumvent these challenges of Bellman completeness, converging under significantly more relaxed assumptions inherited from supervised learning. We prove there exists a natural environment in which if one uses two-layer multilayer perceptron as the function approximator, the layer width needs to grow linearly with the state space size to satisfy Bellman completeness while a constant layer width is enough for RCSL. These findings take a step towards explaining the superior empirical performance of RCSL methods compared to DP-based methods in environments with near-optimal datasets. Furthermore, in order to learn from sub-optimal datasets, we propose a simple framework called MBRCSL, granting RCSL methods the ability of dynamic programming to stitch together segments from distinct trajectories. MBRCSL leverages learned dynamics models and forward sampling to accomplish trajectory stitching while avoiding the need for Bellman completeness that plagues all dynamic programming algorithms. We propose both theoretical analysis and experimental evaluation to back these claims, outperforming state-of-the-art model-free and model-based offline RL algorithms across several simulated robotics problems.