Although stochastic gradient descent (SGD) method and its variants (e.g., stochastic momentum methods, AdaGrad) are the choice of algorithms for solving non-convex problems (especially deep learning), there still remain big gaps between the theory and the practice with many questions unresolved. For example, there is still a lack of theories of convergence for SGD and its variants that use stagewise step size and return an averaged solution in practice. In addition, theoretical insights of why adaptive step size of AdaGrad could improve non-adaptive step size of {\sgd} is still missing for non-convex optimization. This paper aims to address these questions and fill the gap between theory and practice. We propose a universal stagewise optimization framework for a broad family of {\bf non-smooth non-convex} (namely weakly convex) problems with the following key features: (i) at each stage any suitable stochastic convex optimization algorithms (e.g., SGD or AdaGrad) that return an averaged solution can be employed for minimizing a regularized convex problem; (ii) the step size is decreased in a stagewise manner; (iii) an averaged solution is returned as the final solution that is selected from all stagewise averaged solutions with sampling probabilities {\it increasing} as the stage number. Our theoretical results of stagewise AdaGrad exhibit its adaptive convergence, therefore shed insights on its faster convergence for problems with sparse stochastic gradients than stagewise SGD. To the best of our knowledge, these new results are the first of their kind for addressing the unresolved issues of existing theories mentioned earlier.

In order to achieve a good level of autonomy in unmanned helicopters, an accurate replication of vehicle dynamics is required, which is achievable through precise mathematical modeling. This paper aims to identify a parametric state-space system for an unmanned helicopter to a good level of accuracy using Invasive Weed Optimization (IWO) algorithm. The flight data of Align TREX 550 flybarless helicopter is used in the identification process. The rigid-body dynamics of the helicopter is modeled in a state-space form that has 40 parameters, which serve as control variables for the IWO algorithm. The results after 1000 iterations were compared with the traditionally used Prediction Error Minimization (PEM) method and also with Genetic Algorithm (GA), which serve as references. Results show a better level of correlation between the actual and estimated responses of the system identified using IWO to that of PEM and GA.

Many optimization techniques evaluate solutions consecutively, where the next candidate for evaluation is determined by the results of previous evaluations. For example, these include iterative methods, "black box" optimization algorithms, simulated annealing, evolutionary algorithms and tabu search, to name a few. When solving an optimization problem, these algorithms evaluate a large number of solutions, which raises the following question: Is it possible to learn something about the optimum using these solutions? In this paper, we define this "learning" question in terms of a logistic regression model and explore its predictive accuracy computationally. The proposed model uses a collection of solutions to predict the components of the optimal solutions. To illustrate the utility of such predictions, we embed the logistic regression model into the tabu search algorithm for job shop scheduling problem. The resulting framework is simple to implement, yet provides a significant boost to the performance of the standard tabu search.

We introduce and analyze an extension to the matching problem on a weighted bipartite graph: Assignment with Type Constraints. The two parts of the graph are partitioned into subsets called types and blocks; we seek a matching with the largest sum of weights under the constraint that there is a pre-specified cap on the number of vertices matched in every type-block pair. Our primary motivation stems from the public housing program of Singapore, accounting for over 70\% of its residential real estate. To promote ethnic diversity within its housing projects, Singapore imposes ethnicity quotas: each new housing development comprises blocks of flats and each ethnicity-based group in the population must not own more than a certain percentage of flats in a block. Other domains using similar hard capacity constraints include matching prospective students to schools or medical residents to hospitals. Limiting agents' choices for ensuring diversity in this manner naturally entails some welfare loss. One of our goals is to study the trade-off between diversity and social welfare in such settings. We first show that, while the classic assignment program is polynomial-time computable, adding diversity constraints makes it computationally intractable; however, we identify a $\tfrac{1}{2}$-approximation algorithm, as well as reasonable assumptions on the weights that permit poly-time algorithms. Next, we provide two upper bounds on the {\em price of diversity} -- a measure of the loss in welfare incurred by imposing diversity constraints -- as functions of natural problem parameters. We conclude the paper with simulations based on publicly available data from two diversity-constrained allocation problems -- Singapore Public Housing and Chicago School Choice -- which shed light on how the constrained maximization as well as lottery-based variants perform in practice.

Settings such as lending and policing can be modeled by a centralized agent allocating a resource (loans or police officers) amongst several groups, in order to maximize some objective (loans given that are repaid or criminals that are apprehended). Often in such problems fairness is also a concern. A natural notion of fairness, based on general principles of equality of opportunity, asks that conditional on an individual being a candidate for the resource, the probability of actually receiving it is approximately independent of the individual's group. In lending this means that equally creditworthy individuals in different racial groups have roughly equal chances of receiving a loan. In policing it means that two individuals committing the same crime in different districts would have roughly equal chances of being arrested. We formalize this fairness notion for allocation problems and investigate its algorithmic consequences. Our main technical results include an efficient learning algorithm that converges to an optimal fair allocation even when the frequency of candidates (creditworthy individuals or criminals) in each group is unknown. The algorithm operates in a censored feedback model in which only the number of candidates who received the resource in a given allocation can be observed, rather than the true number of candidates. This models the fact that we do not learn the creditworthiness of individuals we do not give loans to nor learn about crimes committed if the police presence in a district is low. As an application of our framework, we consider the predictive policing problem. The learning algorithm is trained on arrest data gathered from its own deployments on previous days, resulting in a potential feedback loop that our algorithm provably overcomes. We empirically investigate the performance of our algorithm on the Philadelphia Crime Incidents dataset.

Gradient boosting is a prediction method that iteratively combines weak learners to produce a complex and accurate model. From an optimization point of view, the learning procedure of gradient boosting mimics a gradient descent on a functional variable. This paper proposes to build upon the proximal point algorithm when the empirical risk to minimize is not differentiable. In addition, the novel boosting approach, called accelerated proximal boosting, benefits from Nesterov's acceleration in the same way as gradient boosting [Biau et al., 2018]. Advantages of leveraging proximal methods for boosting are illustrated by numerical experiments on simulated and real-world data. In particular, we exhibit a favorable comparison over gradient boosting regarding convergence rate and prediction accuracy.

Solving statistical learning problems often involves nonconvex optimization. Despite the empirical success of nonconvex statistical optimization methods, their global dynamics, especially convergence to the desirable local minima, remain less well understood in theory. In this paper, we propose a new analytic paradigm based on diffusion processes to characterize the global dynamics of nonconvex statistical optimization. As a concrete example, we study stochastic gradient descent (SGD) for the tensor decomposition formulation of independent component analysis. In particular, we cast different phases of SGD into diffusion processes, i.e., solutions to stochastic differential equations. Initialized from an unstable equilibrium, the global dynamics of SGD transit over three consecutive phases: (i) an unstable Ornstein-Uhlenbeck process slowly departing from the initialization, (ii) the solution to an ordinary differential equation, which quickly evolves towards the desirable local minimum, and (iii) a stable Ornstein-Uhlenbeck process oscillating around the desirable local minimum. Our proof techniques are based upon Stroock and Varadhan's weak convergence of Markov chains to diffusion processes, which are of independent interest.

We propose a novel geometric approach for learning bilingual mappings given monolingual embeddings and a bilingual dictionary. Our approach decouples learning the transformation from the source language to the target language into (a) learning rotations for language-specific embeddings to align them to a common space, and (b) learning a similarity metric in the common space to model similarities between the embeddings. We model the bilingual mapping problem as an optimization problem on smooth Riemannian manifolds. We show that our approach outperforms previous approaches on the bilingual lexicon induction and cross-lingual word similarity tasks. We also generalize our framework to represent multiple languages in a common latent space. In particular, the latent space representations for several languages are learned jointly, given bilingual dictionaries for multiple language pairs. We illustrate the effectiveness of joint learning for multiple languages in zero-shot word translation setting.

We introduce a novel discriminative latent-variable model for the task of bilingual lexicon induction. Our model combines the bipartite matching dictionary prior of Haghighi et al. (2008) with a state-of-the-art embedding-based approach. To train the model, we derive an efficient Viterbi EM algorithm. We provide empirical improvements on six language pairs under two metrics and show that the prior theoretically and empirically helps to mitigate the hubness problem. We also demonstrate how previous work may be viewed as a similarly fashioned latent-variable model, albeit with a different prior.

A minimum dominating set in a graph is a minimum set of vertices such that every vertex of the graph either belongs to it, or is adjacent to one vertex of this set. This mathematical object is of high relevance in a number of applications related to social networks analysis, design of wireless networks, coding theory, and data mining, among many others. When vertex weights are given, minimizing the total weight of the dominating set gives rise to a problem variant known as the minimum weight dominating set problem. To solve this problem, we introduce a hybrid matheuristic combining a tabu search with an integer programming solver. The latter is used to solve subproblems in which only a fraction of the decision variables, selected relatively to the search history, are left free while the others are fixed. Moreover, we introduce an adaptive penalty to promote the exploration of intermediate infeasible solutions during the search, enhance the algorithm with perturbations and node elimination procedures, and exploit richer neighborhood classes. Extensive experimental analyses on a variety of instance classes demonstrate the good performance of the algorithm, and the contribution of each component in the success of the search is analyzed.