We investigate group fairness regularizers in federated learning, aiming to train a globally fair model in a distributed setting. Ensuring global fairness in distributed training presents unique challenges, as fairness regularizers typically involve probability metrics between distributions across all clients and are not naturally separable by client. To address this, we introduce a function-tracking scheme for the global fairness regularizer based on a Maximum Mean Discrepancy (MMD), which incurs a small communication overhead. This scheme seamlessly integrates into most federated learning algorithms while preserving rigorous convergence guarantees, as demonstrated in the context of FedAvg. Additionally, when enforcing differential privacy, the kernel-based MMD regularization enables straightforward analysis through a change of kernel, leveraging an intuitive interpretation of kernel convolution. Numerical experiments confirm our theoretical insights.

We study a robust auction design problem with a minimax regret objective, in which a seller seeks a mechanism for selling multiple items to multiple bidders with additive values. The seller knows that the bidders’ values range over a box uncertainty set but has no information on their probability distribution. The robust auction design model we study requires no distributional information except for upper bounds on the bidders’ values for each item. This model is relevant if there is no trustworthy distributional information or if any distributional information is costly or time-consuming to acquire. We propose a mechanism that sells each item separately via a second price auction with a random reserve price and prove that this mechanism is optimal using duality techniques from robust optimization. We then interpret the auction design problem as a zero-sum game between the seller, who chooses a mechanism, and a fictitious adversary or “nature,” who chooses the bidders’ values from within the uncertainty set with the aim to maximize the seller’s regret. We characterize the Nash equilibrium of this game analytically when the bidders are symmetric. The Nash strategy of the seller coincides with the optimal separable second price auction, whereas the Nash strategy of nature is mixed and constitutes a probability distribution on the uncertainty set under which each bidder’s values for the items are comonotonic. We also study a restricted auction design problem over deterministic mechanisms. In this setting, we characterize the suboptimality of a separable second price auction with deterministic reserve prices and show that this auction becomes optimal if the bidders are symmetric. The optimal mechanism is derived in closed form and can easily be implemented by practitioners. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was funded by the Swiss National Science Foundation [Grant BSCGI0_157733] and by the Ministry of Education, Singapore, under its Academic Research Fund Tier 2 [Grant MOE-T2EP20222-0003]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0275 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0275 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We develop a principled approach to end-to-end learning in stochastic optimization. First, we show that the standard end-to-end learning algorithm admits a Bayesian interpretation and trains a posterior Bayes action map. Building on the insights of this analysis, we then propose new end-to-end learning algorithms for training decision maps that output solutions of empirical risk minimization and distributionally robust optimization problems, two dominant modeling paradigms in optimization under uncertainty. Numerical results for a synthetic newsvendor problem illustrate the key differences between alternative training schemes. We also investigate an economic dispatch problem based on real data to showcase the impact of the neural network architecture of the decision maps on their test performance.

We introduce contextual stochastic bilevel optimization (CSBO) -- a stochastic bilevel optimization framework with the lower-level problem minimizing an expectation conditioned on some contextual information and the upper-level decision variable. This framework extends classical stochastic bilevel optimization when the lower-level decision maker responds optimally not only to the decision of the upper-level decision maker but also to some side information and when there are multiple or even infinite many followers. It captures important applications such as meta-learning, personalized federated learning, end-to-end learning, and Wasserstein distributionally robust optimization with side information (WDRO-SI). Due to the presence of contextual information, existing single-loop methods for classical stochastic bilevel optimization are unable to converge. To overcome this challenge, we introduce an efficient double-loop gradient method based on the Multilevel Monte-Carlo (MLMC) technique and establish its sample and computational complexities. When specialized to stochastic nonconvex optimization, our method matches existing lower bounds. For meta-learning, the complexity of our method does not depend on the number of tasks. Numerical experiments further validate our theoretical results.

Linear-Quadratic-Gaussian (LQG) control is a fundamental control paradigm that is studied in various fields such as engineering, computer science, economics, and neuroscience. It involves controlling a system with linear dynamics and imperfect observations, subject to additive noise, with the goal of minimizing a quadratic cost function for the state and control variables. In this work, we consider a generalization of the discrete-time, finite-horizon LQG problem, where the noise distributions are unknown and belong to Wasserstein ambiguity sets centered at nominal (Gaussian) distributions. The objective is to minimize a worst-case cost across all distributions in the ambiguity set, including non-Gaussian distributions. Despite the added complexity, we prove that a control policy that is linear in the observations is optimal for this problem, as in the classic LQG problem. We propose a numerical solution method that efficiently characterizes this optimal control policy. Our method uses the Frank-Wolfe algorithm to identify the least-favorable distributions within the Wasserstein ambiguity sets and computes the controller's optimal policy using Kalman filter estimation under these distributions.

We study a stochastic program where the probability distribution of the uncertain problem parameters is unknown and only indirectly observed via finitely many correlated samples generated by an unknown Markov chain with $d$ states. We propose a data-driven distributionally robust optimization model to estimate the problem’s objective function and optimal solution. By leveraging results from large deviations theory, we derive statistical guarantees on the quality of these estimators. The underlying worst-case expectation problem is nonconvex and involves $\mathcal O(d^2)$ decision variables. Thus, it cannot be solved efficiently for large $d$. By exploiting the structure of this problem, we devise a customized Frank-Wolfe algorithm with convex direction-finding subproblems of size $\mathcal O(d)$. We prove that this algorithm finds a stationary point efficiently under mild conditions. The efficiency of the method is predicated on a dimensionality reduction enabled by a dual reformulation. Numerical experiments indicate that our approach has better computational and statistical properties than the state-of-the-art methods.

Least squares estimators, when trained on few target domain samples, may predict poorly. Supervised domain adaptation aims to improve the predictive accuracy by exploiting additional labeled training samples from a source distribution that is close to the target distribution. Given available data, we investigate novel strategies to synthesize a family of least squares estimator experts that are robust with regard to moment conditions. When these moment conditions are specified using Kullback-Leibler or Wasserstein-type divergences, we can find the robust estimators efficiently using convex optimization. We use the Bernstein online aggregation algorithm on the proposed family of robust experts to generate predictions for the sequential stream of target test samples. Numerical experiments on real data show that the robust strategies systematically outperform non-robust interpolations of the empirical least squares estimators.

Training models that perform well under distribution shifts is a central challenge in machine learning. In this paper, we introduce a modeling framework where, in addition to training data, we have partial structural knowledge of the shifted test distribution. We employ the principle of minimum discriminating information to embed the available prior knowledge, and use distributionally robust optimization to account for uncertainty due to the limited samples. By leveraging large deviation results, we obtain explicit generalization bounds with respect to the unknown shifted distribution. Lastly, we demonstrate the versatility of our framework by demonstrating it on two rather distinct applications: (1) training classifiers on systematically biased data and (2) off-policy evaluation in Markov Decision Processes.

A fundamental problem arising in many areas of machine learning is the evaluation of the likelihood of a given observation under different nominal distributions. Frequently, these nominal distributions are themselves estimated from data, which makes them susceptible to estimation errors. We thus propose to replace each nominal distribution with an ambiguity set containing all distributions in its vicinity and to evaluate an optimistic likelihood, that is, the maximum of the likelihood over all distributions in the ambiguity set. When the proximity of distributions is quantified by the Fisher-Rao distance or the Kullback-Leibler divergence, the emerging optimistic likelihoods can be computed efficiently using either geodesic or standard convex optimization techniques. We showcase the advantages of working with optimistic likelihoods on a classification problem using synthetic as well as empirical data.

The likelihood function is a fundamental component in Bayesian statistics. However, evaluating the likelihood of an observation is computationally intractable in many applications. In this paper, we propose a non-parametric approximation of the likelihood that identifies a probability measure which lies in the neighborhood of the nominal measure and that maximizes the probability of observing the given sample point. We show that when the neighborhood is constructed by the Kullback-Leibler divergence, by moment conditions or by the Wasserstein distance, then our optimistic likelihood can be determined through the solution of a convex optimization problem, and it admits an analytical expression in particular cases. We also show that the posterior inference problem with our optimistic likelihood approximation enjoys strong theoretical performance guarantees, and it performs competitively in a probabilistic classification task.

The goal of regression and classification methods in supervised learning is to minimize the empirical risk, that is, the expectation of some loss function quantifying the prediction error under the empirical distribution. When facing scarce training data, overfitting is typically mitigated by adding regularization terms to the objective that penalize hypothesis complexity. In this paper we introduce new regularization techniques using ideas from distributionally robust optimization, and we give new probabilistic interpretations to existing techniques. Specifically, we propose to minimize the worst-case expected loss, where the worst case is taken over the ball of all (continuous or discrete) distributions that have a bounded transportation distance from the (discrete) empirical distribution. By choosing the radius of this ball judiciously, we can guarantee that the worst-case expected loss provides an upper confidence bound on the loss on test data, thus offering new generalization bounds. We prove that the resulting regularized learning problems are tractable and can be tractably kernelized for many popular loss functions. The proposed approach to regluarization is also extended to neural networks. We validate our theoretical out-of-sample guarantees through simulated and empirical experiments.

We study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the non-convex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filter that hedges against model risk.

This paper proposes a distributionally robust approach to logistic regression. We use the Wasserstein distance to construct a ball in the space of probability distributions centered at the uniform distribution on the training samples. If the radius of this Wasserstein ball is chosen judiciously, we can guarantee that it contains the unknown data-generating distribution with high confidence. We then formulate a distributionally robust logistic regression model that minimizes a worst-case expected logloss function, where the worst case is taken over all distributions in the Wasserstein ball. We prove that this optimization problem admits a tractable reformulation and encapsulates the classical as well as the popular regularized logistic regression problems as special cases. We further propose a distributionally robust approach based on Wasserstein balls to compute upper and lower confidence bounds on the misclassification probability of the resulting classifier. These bounds are given by the optimal values of two highly tractable linear programs. We validate our theoretical out-of-sample guarantees through simulated and empirical experiments.

In stochastic optimal control the distribution of the exogenous noise is typically unknown and must be inferred from limited data before dynamic programming (DP)-based solution schemes can be applied. If the conditional expectations in the DP recursions are estimated via kernel regression, however, the historical sample paths enter the solution procedure directly as they determine the evaluation points of the cost-to-go functions. The resulting data-driven DP scheme is asymptotically consistent and admits efficient computational solution when combined with parametric value function approximations. If training data is sparse, however, the estimated cost-to-go functions display a high variability and an optimistic bias, while the corresponding control policies perform poorly in out-of-sample tests. To mitigate these small sample effects, we propose a robust data-driven DP scheme, which replaces the expectations in the DP recursions with worst-case expectations over a set of distributions close to the best estimate. We show that the arising min-max problems in the DP recursions reduce to tractable conic programs. We also demonstrate that this robust algorithm dominates state-of-the-art benchmark algorithms in out-of-sample tests across several application domains.

The purpose of software partitioning is to assign code segments of a given computer program to a range of execution locations such as general-purpose processors or specialist hardware components. These execution locations differ in speed, communication characteristics, and size. In particular, hardware components offering high speed tend to accommodate only few code segments. The goal of software partitioning is to find an assignment of code segments to execution locations that minimizes the overall program run time and respects the size constraints. In this paper we demonstrate that an additional speedup is obtained if we allow code segments to be instantiated on more than one location. We further show that the program run time not only depends on the execution frequency of the code segments but also on their execution order if there are multiply instantiated code segments. Unlike frequency information, however, sequence information is not available at the time when the software partition is selected. This motivates us to formulate the software-partitioning problem as a robust optimization problem with decision-dependent uncertainty. We transform this problem to a mixed-integer linear program of moderate size and report on promising numerical results.

When users submit new queries to a distributed stream processing system (DSPS), a query planner must allocate physical resources, such as CPU cores, memory and network bandwidth, from a set of hosts to queries. Allocation decisions must provide the correct mix of resources required by queries, while achieving an efficient overall allocation to scale in the number of admitted queries. By exploiting overlap between queries and reusing partial results, a query planner can conserve resources but has to carry out more complex planning decisions. In this paper, we describe SQPR, a query planner that targets DSPSs in data centre environments with heterogeneous resources. SQPR models query admission, allocation and reuse as a single constrained optimisation problem and solves an approximate version to achieve scalability. It prevents individual resources from becoming bottlenecks by re-planning past allocation decisions and supports different allocation objectives. As our experimental evaluation in comparison with a state-of-the-art planner shows SQPR makes efficient resource allocation decisions, even with a high utilisation of resources, with acceptable overheads.

We formulate the service composition problem as a multi-objective stochastic program which simultaneously optimizes the following quality of service (QoS) parameters: workflow duration, service invocation costs, availability, and reliability. All of these quality measures are modelled as decision-dependent random variables. Our model minimizes the average value-at- risk (AVaR) of the workflow duration and costs while imposing constraints on the workflow availability and reliability. AVaR is a popular risk measure in decision theory which quantifies the expected shortfall below some percentile of a loss distribution. By replacing the random durations and costs with their expected values, our risk-aware model reduces to the nominal problem formulation prevalent in literature. We argue that this nominal model can lead to overly risky decisions. Finally, we report on the scalability properties of our model.