This paper provides a systematic study of the robust Stackelberg equilibrium (RSE), which naturally generalizes the widely adopted solution concept of the strong Stackelberg equilibrium (SSE). The RSE accounts for any possible up-to-$\delta$ suboptimal follower responses in Stackelberg games and is adopted to improve the robustness of the leader's strategy. While a few variants of robust Stackelberg equilibrium have been considered in previous literature, the RSE solution concept we consider is importantly different -- in some sense, it relaxes previously studied robust Stackelberg strategies and is applicable to much broader sources of uncertainties. We provide a thorough investigation of several fundamental properties of RSE, including its utility guarantees, algorithmics, and learnability. We first show that the RSE we defined always exists and thus is well-defined. Then we characterize how the leader's utility in RSE changes with the robustness level considered. On the algorithmic side, we show that, in sharp contrast to the tractability of computing an SSE, it is NP-hard to obtain a fully polynomial approximation scheme (FPTAS) for any constant robustness level. Nevertheless, we develop a quasi-polynomial approximation scheme (QPTAS) for RSE. Finally, we examine the learnability of the RSE in a natural learning scenario, where both players' utilities are not known in advance, and provide almost tight sample complexity results on learning the RSE. As a corollary of this result, we also obtain an algorithm for learning SSE, which strictly improves a key result of Bai et al. in terms of both utility guarantee and computational efficiency.
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$l^q$-regularization has been demonstrated to be an attractive technique in machine learning and statistical modeling. It attempts to improve the generalization (prediction) capability of a machine (model) through appropriately shrinking its coefficients. The shape of a $l^q$ estimator differs in varying choices of the regularization order $q$. In particular, $l^1$ leads to the LASSO estimate, while $l^{2}$ corresponds to the smooth ridge regression. This makes the order $q$ a potential tuning parameter in applications. To facilitate the use of $l^{q}$-regularization, we intend to seek for a modeling strategy where an elaborative selection on $q$ is avoidable. In this spirit, we place our investigation within a general framework of $l^{q}$-regularized kernel learning under a sample dependent hypothesis space (SDHS). For a designated class of kernel functions, we show that all $l^{q}$ estimators for $0< q < \infty$ attain similar generalization error bounds. These estimated bounds are almost optimal in the sense that up to a logarithmic factor, the upper and lower bounds are asymptotically identical. This finding tentatively reveals that, in some modeling contexts, the choice of $q$ might not have a strong impact in terms of the generalization capability. From this perspective, $q$ can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..
Despite extraordinary progress, current machine learning systems have been shown to be brittle against adversarial examples: seemingly innocuous but carefully crafted perturbations of test examples that cause machine learning predictors to misclassify. Can we learn predictors robust to adversarial examples? and how? There has been much empirical interest in this contemporary challenge in machine learning, and in this thesis, we address it from a theoretical perspective. In this thesis, we explore what robustness properties can we hope to guarantee against adversarial examples and develop an understanding of how to algorithmically guarantee them. We illustrate the need to go beyond traditional approaches and principles such as empirical risk minimization and uniform convergence, and make contributions that can be categorized as follows: (1) introducing problem formulations capturing aspects of emerging practical challenges in robust learning, (2) designing new learning algorithms with provable robustness guarantees, and (3) characterizing the complexity of robust learning and fundamental limitations on the performance of any algorithm.
We model a system of n asymmetric firms selling a homogeneous good in a common market through a pay-as-bid auction. Every producer chooses as its strategy a supply function returning the quantity S(p) that it is willing to sell at a minimum unit price p. The market clears at the price at which the aggregate demand intersects the total supply and firms are paid the bid prices. We study a game theoretic model of competition among such firms and focus on its equilibria (Supply function equilibrium). The game we consider is a generalization of both models where firms can either set a fixed quantity (Cournot model) or set a fixed price (Bertrand model). Our main result is to prove existence and provide a characterization of (pure strategy) Nash equilibria in the space of K-Lipschitz supply functions.
Although reinforcement learning (RL) is considered the gold standard for policy design, it may not always provide a robust solution in various scenarios. This can result in severe performance degradation when the environment is exposed to potential disturbances. Adversarial training using a two-player max-min game has been proven effective in enhancing the robustness of RL agents. In this work, we extend the two-player game by introducing an adversarial herd, which involves a group of adversaries, in order to address ($\textit{i}$) the difficulty of the inner optimization problem, and ($\textit{ii}$) the potential over pessimism caused by the selection of a candidate adversary set that may include unlikely scenarios. We first prove that adversarial herds can efficiently approximate the inner optimization problem. Then we address the second issue by replacing the worst-case performance in the inner optimization with the average performance over the worst-$k$ adversaries. We evaluate the proposed method on multiple MuJoCo environments. Experimental results demonstrate that our approach consistently generates more robust policies.
In the context of reducing carbon emissions in the automotive supply chain, collaboration between vehicle manufacturers and retailers has proven to be an effective measure for enhancing carbon emission reduction within the enterprise. This study aims to evaluate the effectiveness of such collaboration by constructing a differential game model that incorporates carbon trading and consumer preferences for low-carbon products. The model examines the decision-making process of an automotive supply chain comprising a vehicle manufacturer and multiple retailers. By utilizing the Hamilton-Jacobi-Bellman equation, we analyze the equilibrium strategies of the participants under both a decentralized model and a Stackelberg leader-follower game model. In the decentralized model, the vehicle manufacturer optimizes its carbon emission reduction efforts, while each retailer independently determines its low-carbon promotion efforts and vehicle retail price. In the Stackelberg leader-follower game model, the vehicle manufacturer cooperates with the retailers by offering them a subsidy. Consequently, the manufacturer plays as the leader, making decisions on carbon emission reduction efforts and the subsidy rate, while the retailers, as followers, compute their promotion efforts and retail prices accordingly. Through theoretical analysis and numerical experiments considering the manufacturer's and retailers' efforts, the low-carbon reputation of vehicles, and the overall system profits under both models, we conclude that compared to the decentralized model, where each party pursues individual profits, the collaboration in the Stackelberg game yields greater benefits for both parties. Furthermore, this collaborative approach promotes the long-term development of the automotive supply chain.
In this era of exoplanet characterisation with JWST, the need for a fast implementation of classical forward models to understand the chemical and physical processes in exoplanet atmospheres is more important than ever. Notably, the time-dependent ordinary differential equations to be solved by chemical kinetics codes are very time-consuming to compute. In this study, we focus on the implementation of neural networks to replace mathematical frameworks in one-dimensional chemical kinetics codes. Using the gravity profile, temperature-pressure profiles, initial mixing ratios, and stellar flux of a sample of hot-Jupiters atmospheres as free parameters, the neural network is built to predict the mixing ratio outputs in steady state. The architecture of the network is composed of individual autoencoders for each input variable to reduce the input dimensionality, which is then used as the input training data for an LSTM-like neural network. Results show that the autoencoders for the mixing ratios, stellar spectra, and pressure profiles are exceedingly successful in encoding and decoding the data. Our results show that in 90% of the cases, the fully trained model is able to predict the evolved mixing ratios of the species in the hot-Jupiter atmosphere simulations. The fully trained model is ~1000 times faster than the simulations done with the forward, chemical kinetics model while making accurate predictions.
A drawback of the classic approach for complexity analysis of distributed graph problems is that it mostly informs about the complexity of notorious classes of ``worst case'' graphs. Algorithms that are used to prove a tight (existential) bound are essentially optimized to perform well on such worst case graphs. However, such graphs are often either unlikely or actively avoided in practice, where benign graph instances usually admit much faster solutions. To circumnavigate these drawbacks, the concept of universal complexity analysis in the distributed setting was suggested by [Kutten and Peleg, PODC'95] and actively pursued by [Haeupler et al., STOC'21]. Here, the aim is to gauge the complexity of a distributed graph problem depending on the given graph instance. The challenge is to identify and understand the graph property that allows to accurately quantify the complexity of a distributed problem on a given graph. In the present work, we consider distributed shortest paths problems in the HYBRID model of distributed computing, where nodes have simultaneous access to two different modes of communication: one is restricted by locality and the other is restricted by congestion. We identify the graph parameter of neighborhood quality and show that it accurately describes a universal bound for the complexity of certain class of shortest paths problems in the HYBRID model.
The $1-N$ generalized Stackelberg game (single-leader multi-follower game) is intricately intertwined with the interaction between a leader and followers (hierarchical interaction) and the interaction among followers (simultaneous interaction). However, obtaining the optimal strategy of the leader is generally challenging due to the complex interactions among the leader and followers. Here, we propose a general methodology to find a generalized Stackelberg equilibrium of a $1-N$ generalized Stackelberg game. Specifically, we first provide the conditions where a generalized Stackelberg equilibrium always exists using the variational equilibrium concept. Next, to find an equilibrium in polynomial time, we transformed the $1-N$ generalized Stackelberg game into a $1-1$ Stackelberg game whose Stackelberg equilibrium is identical to that of the original. Finally, we propose an effective computation procedure based on the projected implicit gradient descent algorithm to find a Stackelberg equilibrium of the transformed $1-1$ Stackelberg game. We validate the proposed approaches using the two problems of deriving operating strategies for EV charging stations: (1) the first problem is optimizing the one-time charging price for EV users, in which a platform operator determines the price of electricity and EV users determine the optimal amount of charging for their satisfaction; and (2) the second problem is to determine the spatially varying charging price to optimally balance the demand and supply over every charging station.
Multi-agent influence diagrams (MAIDs) are a popular form of graphical model that, for certain classes of games, have been shown to offer key complexity and explainability advantages over traditional extensive form game (EFG) representations. In this paper, we extend previous work on MAIDs by introducing the concept of a MAID subgame, as well as subgame perfect and trembling hand perfect equilibrium refinements. We then prove several equivalence results between MAIDs and EFGs. Finally, we describe an open source implementation for reasoning about MAIDs and computing their equilibria.
As data are increasingly being stored in different silos and societies becoming more aware of data privacy issues, the traditional centralized training of artificial intelligence (AI) models is facing efficiency and privacy challenges. Recently, federated learning (FL) has emerged as an alternative solution and continue to thrive in this new reality. Existing FL protocol design has been shown to be vulnerable to adversaries within or outside of the system, compromising data privacy and system robustness. Besides training powerful global models, it is of paramount importance to design FL systems that have privacy guarantees and are resistant to different types of adversaries. In this paper, we conduct the first comprehensive survey on this topic. Through a concise introduction to the concept of FL, and a unique taxonomy covering: 1) threat models; 2) poisoning attacks and defenses against robustness; 3) inference attacks and defenses against privacy, we provide an accessible review of this important topic. We highlight the intuitions, key techniques as well as fundamental assumptions adopted by various attacks and defenses. Finally, we discuss promising future research directions towards robust and privacy-preserving federated learning.
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