Generalisation bounds are crucial for providing data-driven models with performance and safety guarantees. In this respect, bounds that do not require a held-out test set are particularly valuable as they allow the use of all data for training. While many such bounds do not improve upon the train-test approach, which remains the gold standard, the P2L algorithm (Paccagnan et al., 2023) has shown great potential. However, P2L comes with limitations, including computational overhead, reliance on consistent data, and restriction to non-Bayesian settings. In this work, we overcome these challenges in general settings and employ the corresponding results to show that classical Gaussian process (GP) training procedures can be interpreted as instantiations of P2L, thus inheriting tight, self-certified bounds. Three contributions underpin these conclusions. First, we introduce early stopping in P2L, equipping it with a tight generalisation bound to reduce training costs and address the non-consistent case. Second, we adapt P2L to the Bayesian setting and demonstrate its equivalence to posterior updating in a hierarchical model. Third, we show that greedy subset-of-data GPs are special P2L instantiations. Numerical evidence shows that the resulting P2L bounds we obtain compare favourably with the train-test and PAC-Bayes approaches on various real-world datasets.

When optimizing transportation systems, anticipating traffic flows is a central element. Yet, computing such traffic equilibria remains computationally expensive. Against this background, we introduce a novel combinatorial optimization augmented neural network pipeline that allows for fast and accurate traffic flow predictions. We propose WardropNet, a neural network that combines classical layers with a subsequent equilibrium layer: the first ones inform the latter by predicting the parameterization of the equilibrium problem's latency functions. Using supervised learning we minimize the difference between the actual traffic flow and the predicted output. We show how to leverage a Bregman divergence fitting the geometry of the equilibria, which allows for end-to-end learning. WardropNet outperforms pure learning-based approaches in predicting traffic equilibria for realistic and stylized traffic scenarios. On realistic scenarios, WardropNet improves on average for time-invariant predictions by up to 72\% and for time-variant predictions by up to 23\% over pure learning-based approaches.

Estimating player utilities from observed equilibria is crucial for many applications. Existing approaches to tackle this problem are either limited to specific games or do not scale well with the number of players. Our work addresses these issues by proposing a novel utility estimation method for general multi-player non-cooperative games. Our main idea consists in reformulating the inverse game problem as an inverse variational inequality problem and in selecting among all utility parameters consistent with the data, the so-called incenter. We show that the choice of the incenter can produce parameters that are most robust to the observed equilibrium behaviors. However, its computation is challenging, as the number of constraints in the corresponding optimization problem increases with the number of players and the behavior space size. To tackle this challenge, we propose a loss function-based algorithm, making our method scalable to games with many players or a continuous action space. Furthermore, we show that our method can be extended to incorporate prior knowledge of player utilities, and that it can handle inconsistent data, i.e., data where players do not play exact equilibria. Numerical experiments on three game applications demonstrate that our methods outperform the state of the art. The code, datasets, and supplementary material are available at https://github.com/cuilvye/Incenter-Project.

Product-form closed queueing networks are a useful formalism for analyzing the availability of vehicles within a vehicle sharing system with stochastic user behavior. However, existing models assume that the provider has full control over the system. In reality, competition between providers within a single geographic area is quite common. In this paper, we introduce a non-cooperative game that extends a closed queueing model into a competitive environment, in which players decide the number of jobs that they wish to submit. This can be used to model vehicle sharing systems with multiple providers, in which they receive fares from trips but are responsible for the costs of their own fleet. The core technical results of this paper include conditions that guarantee the existence of a pure Nash equilibrium, and an efficient equilibrium-finding algorithm. We then present a case study, using our model, of a vehicle sharing system in Oslo with multiple providers. In this case study, we find that adding an additional competitor can increase the number of trips by up to 18.9%, and a highly competitive market can increase this by up to 30%.

Understanding the impact of exploration on the behaviour of multi-agent learning has, so far, benefited from the restriction to potential, or network zero-sum games in which convergence to an equilibrium can be shown. Outside of these classes, learning dynamics rarely converge and little is known about the effect of exploration in the face of non-convergence. To progress this front, we study the smooth Q- Learning dynamics. We show that, in any network game, exploration by agents results in the convergence of Q-Learning to a neighbourhood of an equilibrium. This holds independently of whether the dynamics reach the equilibrium or display complex behaviours. We show that increasing the exploration rate decreases the size of this neighbourhood and also decreases the ability of all agents to improve their payoffs. Furthermore, in a broad class of games, the payoff performance of Q-Learning dynamics, measured by Social Welfare, decreases when the exploration rate increases. Our experiments show this to be a general phenomenon, namely that exploration leads to improved convergence of Q-Learning, at the cost of payoff performance.

Generalization bounds are valuable both for theory and applications. On the one hand, they shed light on the mechanisms that underpin the learning processes; on the other, they certify how well a learned model performs against unseen inputs. In this work we build upon a recent breakthrough in compression theory to develop a new framework yielding tight generalization bounds of wide practical applicability. The core idea is to embed any given learning algorithm into a suitably-constructed meta-algorithm (here called Pick-to-Learn, P2L) in order to instill desirable compression properties. When applied to the MNIST classification dataset and to a synthetic regression problem, P2L not only attains generalization bounds that compare favorably with the state of the art (test-set and PAC-Bayes bounds), but it also learns models with better post-training performance.

Over the past decade, GPS-enabled traffic applications such as Google Maps and Waze have become ubiquitous and have had a significant influence on billions of daily commuters’ travel patterns. A consequence of the online route suggestions of such applications, for example, via greedy routing, has often been an increase in traffic congestion since the induced travel patterns may be far from the system optimum. Spurred by the widespread impact of traffic applications on travel patterns, this work studies online traffic routing in the context of capacity-constrained parallel road networks and analyzes this problem from two perspectives. First, we perform a worst-case analysis to identify the limits of deterministic online routing. Although we find that deterministic online algorithms achieve finite, problem/instance-dependent competitive ratios in special cases, we show that for a general setting the competitive ratio is unbounded. This result motivates us to move beyond worst-case analysis. Here, we consider algorithms that exploit knowledge of past problem instances and show how to design data-driven algorithms whose performance can be quantified and formally generalized to unseen future instances. We then present numerical experiments based on an application case for the San Francisco Bay Area to evaluate the performance of the proposed data-driven algorithms compared with the greedy algorithm and two look-ahead heuristics with access to additional information on the values of time and arrival time parameters of users. Our results show that the developed data-driven algorithms outperform commonly used greedy online-routing algorithms. Furthermore, our work sheds light on the interplay between data availability and achievable solution quality. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Funding: This work was supported by National Science Foundation (NSF) Award 1830554 and by the German Research Foundation (DFG) under [Grant 449261765]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2023.1275 .

We consider the problem of minimizing social cost in atomic congestion games and show, perhaps surprisingly, that efficiently computed taxation mechanisms yield the same performance achievable by the best polynomial time algorithm, even when the latter has full control over the players' actions. It follows that no other tractable approach geared at incentivizing desirable system behavior can improve upon this result, regardless of whether it is based on taxations, coordination mechanisms, information provision, or any other principle. Three technical contributions underpin this conclusion. First, we show that computing the minimum social cost is NP-hard to approximate within a given factor depending solely on the admissible resource costs. Second, we design a tractable taxation mechanism whose efficiency (price of anarchy) matches this hardness factor, and thus is optimal. As these results extend to coarse correlated equilibria, any no-regret algorithm inherits the same performances, allowing us to devise polynomial time algorithms with optimal approximation.