High-dimensional nonconvex problems are popular in today's machine learning and statistical genetics research. Recently, Ghadimi and Lan [1] proposed an algorithm to optimize nonconvex high-dimensional problems. There are several parameters in their algorithm that are to be set before running the algorithm. It is not trivial how to choose these parameters nor there is, to the best of our knowledge, an explicit rule on how to select the parameters to make the algorithm converges faster. We analyze Ghadimi and Lan's algorithm to gain an interpretation based on the inequality constraints for convergence and the upper bound for the norm of the gradient analogue. Our interpretation of their algorithm suggests this to be a damped Nesterov's acceleration scheme. Based on this, we propose an approach on how to select the parameters to improve convergence of the algorithm. Our numerical studies using high-dimensional nonconvex sparse learning problems, motivated by image denoising and statistical genetics applications, show that convergence can be made, on average, considerably faster than that of the conventional ISTA algorithm for such optimization problems with over $10000$ variables should the parameters be chosen using our proposed approach.

Optimal transport (OT) distances are increasingly used as loss functions for statistical inference, notably in the learning of generative models or supervised learning. Yet, the behavior of minimum Wasserstein estimators is poorly understood, notably in high-dimensional regimes or under model misspecification. In this work we adopt the viewpoint of projection robust (PR) OT, which seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wasserstein distances, complementing and improving previous literature that has been restricted to one-dimensional and well-specified cases. Next, we propose the integral PR Wasserstein (IPRW) distance as an alternative to the PRW distance, by averaging rather than optimizing on subspaces. Our complexity bounds can help explain why both PRW and IPRW distances outperform Wasserstein distances empirically in high-dimensional inference tasks. Finally, we consider parametric inference using the PRW distance. We provide an asymptotic guarantee of two types of minimum PRW estimators and formulate a central limit theorem for max-sliced Wasserstein estimator under model misspecification. To enable our analysis on PRW with projection dimension larger than one, we devise a novel combination of variational analysis and statistical theory.

In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. The pheromone-based communication of biological ants is often the predominant paradigm used.[2] Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima. As an example, Ant colony optimization[3] is a class of optimization algorithms modeled on the actions of an ant colony. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.[4]

Federated learning is typically approached as an optimization problem, where the goal is to minimize a global loss function by distributing computation across client devices that possess local data and specify different parts of the global objective. We present an alternative perspective and formulate federated learning as a posterior inference problem, where the goal is to infer a global posterior distribution by having client devices each infer the posterior of their local data. While exact inference is often intractable, this perspective provides a principled way to search for global optima in federated settings. Further, starting with the analysis of federated quadratic objectives, we develop a computation- and communication-efficient approximate posterior inference algorithm -- federated posterior averaging (FedPA). Our algorithm uses MCMC for approximate inference of local posteriors on the clients and efficiently communicates their statistics to the server, where the latter uses them to refine a global estimate of the posterior mode. Finally, we show that FedPA generalizes federated averaging (FedAvg), can similarly benefit from adaptive optimizers, and yields state-of-the-art results on four realistic and challenging benchmarks, converging faster, to better optima.

Counterfactual explanations (CFE) are being widely used to explain algorithmic decisions, especially in consequential decision-making contexts (e.g., loan approval or pretrial bail). In this context, CFEs aim to provide individuals affected by an algorithmic decision with the most similar individual (i.e., nearest individual) with a different outcome. However, while an increasing number of works propose algorithms to compute CFEs, such approaches either lack in optimality of distance (i.e., they do not return the nearest individual) and perfect coverage (i.e., they do not provide a CFE for all individuals); or they cannot handle complex models, such as neural networks. In this work, we provide a framework based on Mixed-Integer Programming (MIP) to compute nearest counterfactual explanations with provable guarantees and with runtimes comparable to gradient-based approaches. Our experiments on the Adult, COMPAS, and Credit datasets show that, in contrast with previous methods, our approach allows for efficiently computing diverse CFEs with both distance guarantees and perfect coverage.

This paper contributes to the treatment of extensive constraints in evolutionary many-constraint optimization through consideration of the relationships between pair-wise constraints. In a conflicting relationship, the functional value of one constraint increases as the value in another constraint decreases. In a harmonious relationship, the improvement in one constraint is rewarded with simultaneous improvement in the other constraint. In an independent relationship, the adjustment to one constraint never affects the adjustment to the other. Based on the different features, methods for identifying constraint relationships are discussed, helping to simplify many-constraint optimization problems (MCOPs). Additionally, the transitivity of the relationships is further discussed at the aim of determining the relationship in a new pair of constraints.

Epidemiologists model the dynamics of epidemics in order to propose control strategies based on pharmaceutical and non-pharmaceutical interventions (contact limitation, lock down, vaccination, etc). Hand-designing such strategies is not trivial because of the number of possible interventions and the difficulty to predict long-term effects. This task can be cast as an optimization problem where state-of-the-art machine learning algorithms such as deep reinforcement learning, might bring significant value. However, the specificity of each domain -- epidemic modelling or solving optimization problem -- requires strong collaborations between researchers from different fields of expertise. This is why we introduce EpidemiOptim, a Python toolbox that facilitates collaborations between researchers in epidemiology and optimization. EpidemiOptim turns epidemiological models and cost functions into optimization problems via a standard interface commonly used by optimization practitioners (OpenAI Gym). Reinforcement learning algorithms based on Q-Learning with deep neural networks (DQN) and evolutionary algorithms (NSGA-II) are already implemented. We illustrate the use of EpidemiOptim to find optimal policies for dynamical on-off lock-down control under the optimization of death toll and economic recess using a Susceptible-Exposed-Infectious-Removed (SEIR) model for COVID-19. Using EpidemiOptim and its interactive visualization platform in Jupyter notebooks, epidemiologists, optimization practitioners and others (e.g. economists) can easily compare epidemiological models, costs functions and optimization algorithms to address important choices to be made by health decision-makers.

Meta continual learning algorithms seek to train a model when faced with similar tasks observed in a sequential manner. Despite promising methodological advancements, there is a lack of theoretical frameworks that enable analysis of learning challenges such as generalization and catastrophic forgetting. To that end, we develop a new theoretical approach for meta continual learning~(MCL) where we mathematically model the learning dynamics using dynamic programming, and we establish conditions of optimality for the MCL problem. Moreover, using the theoretical framework, we derive a new dynamic-programming-based MCL method that adopts stochastic-gradient-driven alternating optimization to balance generalization and catastrophic forgetting. We show that, on MCL benchmark data sets, our theoretically grounded method achieves accuracy better than or comparable to that of existing state-of-the-art methods.

Algorithmic fairness has received lots of interests in machine learning recently. In this paper, we focus on the bipartite ranking scenario, where the instances come from either the positive or negative class and the goal is to learn a ranking function that ranks positive instances higher than negative ones. In an unfair setting, the probabilities of ranking the positives higher than negatives are different across different protected groups. We propose a general post-processing framework, xOrder, for achieving fairness in bipartite ranking while maintaining the algorithm classification performance. In particular, we optimize a weighted sum of the utility and fairness by directly adjusting the relative ordering across groups. We formulate this problem as identifying an optimal warping path across {different} protected groups and solve it through a dynamic programming process. xOrder is compatible with various classification models and applicable to a variety of ranking fairness metrics. We evaluate our proposed algorithm on four benchmark data sets and two real world patient electronic health record repository. The experimental results show that our approach can achieve great balance between the algorithm utility and ranking fairness. Our algorithm can also achieve robust performance when training and testing ranking score distributions are significantly different.

Quasi-Maximum Likelihood (QML) procedures are theoretically appealing and widely used for statistical inference. While there are extensive references on QML estimation in batch settings, the QML estimation in streaming settings has attracted little attention until recently. An investigation of the convergence properties of the QML procedure in a general conditionally heteroscedastic time series model is conducted, and the classical batch optimization routines extended to the framework of streaming and large-scale problems. An adaptive recursive estimation routine for GARCH models named AdaVol is presented. The AdaVol procedure relies on stochastic approximations combined with the technique of Variance Targeting Estimation (VTE). This recursive method has computationally efficient properties, while VTE alleviates some convergence difficulties encountered by the usual QML estimation due to a lack of convexity. Empirical results demonstrate a favorable trade-off between AdaVol's stability and the ability to adapt to time-varying estimates for real-life data.