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Hyperparameter optimization (HPO) is a necessary step to ensure the best possible performance of Machine Learning (ML) algorithms. Several methods have been developed to perform HPO; most of these are focused on optimizing one performance measure (usually an error-based measure), and the literature on such single-objective HPO problems is vast. Recently, though, algorithms have appeared which focus on optimizing multiple conflicting objectives simultaneously. This article presents a systematic survey of the literature published between 2014 and 2020 on multi-objective HPO algorithms, distinguishing between metaheuristic-based algorithms, metamodel-based algorithms, and approaches using a mixture of both. We also discuss the quality metrics used to compare multi-objective HPO procedures and present future research directions.

Matching on covariates is a well-established framework for estimating causal effects in observational studies. The principal challenge in these settings stems from the often high-dimensional structure of the problem. Many methods have been introduced to deal with this challenge, with different advantages and drawbacks in computational and statistical performance and interpretability. Moreover, the methodological focus has been on matching two samples in binary treatment scenarios, but a dedicated method that can optimally balance samples across multiple treatments has so far been unavailable. This article introduces a natural optimal matching method based on entropy-regularized multimarginal optimal transport that possesses many useful properties to address these challenges. It provides interpretable weights of matched individuals that converge at the parametric rate to the optimal weights in the population, can be efficiently implemented via the classical iterative proportional fitting procedure, and can even match several treatment arms simultaneously. It also possesses demonstrably excellent finite sample properties.

Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the filter graph distance. It is an optimal transport based distance which drives graph comparison through the probability distribution of filtered graph signals. This creates a highly flexible distance, capable of prioritising different spectral information in observed graphs, offering a wide range of choices for a comparison metric. We tackle the problem of graph alignment by computing graph permutations that minimise our new filter distances, which implicitly solves the graph comparison problem. We then propose a new approximate cost function that circumvents many computational difficulties inherent to graph comparison and permits the exploitation of fast algorithms such as mirror gradient descent, without grossly sacrificing the performance. We finally propose a novel algorithm derived from a stochastic version of mirror gradient descent, which accommodates the non-convexity of the alignment problem, offering a good trade-off between performance accuracy and speed. The experiments on graph alignment and classification show that the flexibility gained through filter graph distances can have a significant impact on performance, while the difference in speed offered by the approximation cost makes the framework applicable in practical settings.

We present an approach to solving hard geometric optimization problems in the RANSAC framework. The hard minimal problems arise from relaxing the original geometric optimization problem into a minimal problem with many spurious solutions. Our approach avoids computing large numbers of spurious solutions. We design a learning strategy for selecting a starting problem-solution pair that can be numerically continued to the problem and the solution of interest. We demonstrate our approach by developing a RANSAC solver for the problem of computing the relative pose of three calibrated cameras, via a minimal relaxation using four points in each view. On average, we can solve a single problem in under 70 $\mu s.$ We also benchmark and study our engineering choices on the very familiar problem of computing the relative pose of two calibrated cameras, via the minimal case of five points in two views.

Multi-marginal optimal transport (MOT) is a generalization of optimal transport to multiple marginals. Optimal transport has evolved into an important tool in many machine learning applications, and its multi-marginal extension opens up for addressing new challenges in the field of machine learning. However, the usage of MOT has been largely impeded by its computational complexity which scales exponentially in the number of marginals. Fortunately, in many applications, such as barycenter or interpolation problems, the cost function adheres to structures, which has recently been exploited for developing efficient computational methods. In this work we derive computational bounds for these methods. With $m$ marginal distributions supported on $n$ points, we provide a $ \mathcal{\tilde O}(d(G)m n^2\epsilon^{-2})$ bound for a $\epsilon$-accuracy when the problem is associated with a tree with diameter $d(G)$. For the special case of the Wasserstein barycenter problem, which corresponds to a star-shaped tree, our bound is in alignment with the existing complexity bound for it.

In this work we propose a batch version of the Greenkhorn algorithm for multimarginal regularized optimal transport problems. Our framework is general enough to cover, as particular cases, some existing algorithms like Sinkhorn and Greenkhorn algorithm for the bi-marginal setting, and (greedy) MultiSinkhorn for multimarginal optimal transport. We provide a complete convergence analysis, which is based on the properties of the iterative Bregman projections (IBP) method with greedy control. Global linear rate of convergence and explicit bound on the iteration complexity are obtained. When specialized to above mentioned algorithms, our results give new insights and/or improve existing ones.

Modeling non-Euclidean data is drawing attention along with the unprecedented successes of deep neural networks in diverse fields. In particular, symmetric positive definite (SPD) matrix is being actively studied in computer vision, signal processing, and medical image analysis, thanks to its ability to learn appropriate statistical representations. However, due to its strong constraints, it remains challenging for optimization problems or inefficient computation costs, especially, within a deep learning framework. In this paper, we propose to exploit a diffeomorphism mapping between Riemannian manifolds and a Cholesky space, by which it becomes feasible not only to efficiently solve optimization problems but also to reduce computation costs greatly. Further, in order for dynamics modeling in time series data, we devise a continuous manifold learning method by integrating a manifold ordinary differential equation and a gated recurrent neural network in a systematic manner. It is noteworthy that because of the nice parameterization of matrices in a Cholesky space, it is straightforward to train our proposed network with Riemannian geometric metrics equipped. We demonstrate through experiments that the proposed model can be efficiently and reliably trained as well as outperform existing manifold methods and state-of-the-art methods in two classification tasks: action recognition and sleep staging classification.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.