We propose a simple yet effective multiple kernel clustering algorithm, termed simple multiple kernel k-means (SimpleMKKM). It extends the widely used supervised kernel alignment criterion to multi-kernel clustering. Our criterion is given by an intractable minimization-maximization problem in the kernel coefficient and clustering partition matrix. To optimize it, we re-formulate the problem as a smooth minimization one, which can be solved efficiently using a reduced gradient descent algorithm. We theoretically analyze the performance of SimpleMKKM in terms of its clustering generalization error. Comprehensive experiments on 11 benchmark datasets demonstrate that SimpleMKKM outperforms state of the art multi-kernel clustering alternatives.

In this paper, we consider the Byzantine-robust stochastic optimization problem defined over decentralized static and time-varying networks, where the agents collaboratively minimize the summation of expectations of stochastic local cost functions, but some of the agents are unreliable due to data corruptions, equipment failures or cyber-attacks. The unreliable agents, which are called as Byzantine agents thereafter, can send faulty values to their neighbors and bias the optimization process. Our key idea to handle the Byzantine attacks is to formulate a total variation (TV) norm-penalized approximation of the Byzantine-free problem, where the penalty term forces the local models of regular agents to be close, but also allows the existence of outliers from the Byzantine agents. A stochastic subgradient method is applied to solve the penalized problem. We prove that the proposed method reaches a neighborhood of the Byzantine-free optimal solution, and the size of neighborhood is determined by the number of Byzantine agents and the network topology. Numerical experiments corroborate the theoretical analysis, as well as demonstrate the robustness of the proposed method to Byzantine attacks and its superior performance comparing to existing methods.

Bayesian optimization methods have been successfully applied to black box optimization problems that are expensive to evaluate. In this paper, we adapt the so-called super effcient global optimization algorithm to solve more accurately mixed constrained problems. The proposed approach handles constraints by means of upper trust bound, the latter encourages exploration of the feasible domain by combining the mean prediction and the associated uncertainty function given by the Gaussian processes. On top of that, a refinement procedure, based on a learning rate criterion, is introduced to enhance the exploitation and exploration trade-off. We show the good potential of the approach on a set of numerical experiments. Finally, we present an application to conceptual aircraft configuration upon which we show the superiority of the proposed approach compared to a set of the state-of-the-art black box optimization solvers. Keywords: Global Optimization, Mixed Constrained Optimization, Black box optimization, Bayesian Optimization, Gaussian Process.

Gravitational Search Algorithm (GSA) and Particle Swarm Optimization (PSO) are nature-inspired, swarm-based optimization algorithms respectively. Though they have been widely used for single-objective optimization since their inception, they suffer from premature convergence. Even though the hybrids of GSA and PSO perform much better, the problem remains. Hence, to solve this issue we have proposed a fuzzy mutation model for two hybrid versions of PSO and GSA - Gravitational Particle Swarm (GPS) and PSOGSA. The developed algorithms are called Mutation based GPS (MGPS) and Mutation based PSOGSA (MPSOGSA). The mutation operator is based on a fuzzy model where the probability of mutation has been calculated based on the closeness of particle to population centroid and improvement in the particle value. We have evaluated these two new algorithms on 23 benchmark functions of three categories (unimodal, multi-modal and multi-modal with fixed dimension). The experimental outcome shows that our proposed model outperforms their corresponding ancestors, MGPS outperforms GPS 13 out of 23 times (56.52%) and MPSOGSA outperforms PSOGSA 17 times out of 23 (73.91 %). We have also compared our results against those of recent optimization algorithms such as Sine Cosine Algorithm (SCA), Opposition-Based SCA, and Volleyball Premier League Algorithm (VPL). In addition, we have applied our proposed algorithms on some classic engineering design problems and the outcomes are satisfactory. The related codes of the proposed algorithms can be found in this link: Fuzzy-Mutation-Embedded-Hybrids-of-GSA-and-PSO.

We study differentially private (DP) algorithms for stochastic convex optimization: the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. (2019) has established the optimal bound on the excess population loss achievable given $n$ samples. Unfortunately, their algorithm achieving this bound is relatively inefficient: it requires $O(\min\{n^{3/2}, n^{5/2}/d\})$ gradient computations, where $d$ is the dimension of the optimization problem. We describe two new techniques for deriving DP convex optimization algorithms both achieving the optimal bound on excess loss and using $O(\min\{n, n^2/d\})$ gradient computations. In particular, the algorithms match the running time of the optimal non-private algorithms. The first approach relies on the use of variable batch sizes and is analyzed using the privacy amplification by iteration technique of Feldman et al. (2018). The second approach is based on a general reduction to the problem of localizing an approximately optimal solution with differential privacy. Such localization, in turn, can be achieved using existing (non-private) uniformly stable optimization algorithms. As in the earlier work, our algorithms require a mild smoothness assumption. We also give a linear-time algorithm achieving the optimal bound on the excess loss for the strongly convex case, as well as a faster algorithm for the non-smooth case.

Quality Diversity (QD) algorithms such as MAP-Elites are a class of optimisation techniques that attempt to find a set of high-performing points from an objective function while enforcing behavioural diversity of the points over one or more interpretable, user chosen, feature functions. In this paper we propose the Bayesian Optimisation of Elites (BOP-Elites) algorithm that uses techniques from Bayesian Optimisation to explicitly model both quality and diversity with Gaussian Processes. By considering user defined regions of the feature space as 'niches' our task is to find the optimal solution in each niche. We propose a novel acquisition function to intelligently choose new points that provide the highest expected improvement to the ensemble problem of identifying the best solution in every niche. In this way each function evaluation enriches our modelling and provides insight to the whole problem, naturally balancing exploration and exploitation of the search space. The resulting algorithm is very effective in identifying the parts of the search space that belong to a niche in feature space, and finding the optimal solution in each niche. It is also significantly more sample efficient than simpler benchmark approaches. BOP-Elites goes further than existing QD algorithms by quantifying the uncertainty around our predictions and offering additional illumination of the search space through surrogate models.

When training end-to-end learned models for lossy compression, one has to balance the rate and distortion losses. This is typically done by manually setting a tradeoff parameter $\beta$, an approach called $\beta$-VAE. Using this approach it is difficult to target a specific rate or distortion value, because the result can be very sensitive to $\beta$, and the appropriate value for $\beta$ depends on the model and problem setup. As a result, model comparison requires extensive per-model $\beta$-tuning, and producing a whole rate-distortion curve (by varying $\beta$) for each model to be compared. We argue that the constrained optimization method of Rezende and Viola, 2018 is a lot more appropriate for training lossy compression models because it allows us to obtain the best possible rate subject to a distortion constraint. This enables pointwise model comparisons, by training two models with the same distortion target and comparing their rate. We show that the method does manage to satisfy the constraint on a realistic image compression task, outperforms a constrained optimization method based on a hinge-loss, and is more practical to use for model selection than a $\beta$-VAE.

Given a set of dissimilarity measurements amongst data points, determining what metric representation is most "consistent" with the input measurements or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. Existing methods are restricted to specific kinds of metrics or small problem sizes because of the large number of metric constraints in such problems. In this paper, we provide an active set algorithm, Project and Forget, that uses Bregman projections, to solve metric constrained problems with many (possibly exponentially) inequality constraints. We provide a theoretical analysis of \textsc{Project and Forget} and prove that our algorithm converges to the global optimal solution and that the $L_2$ distance of the current iterate to the optimal solution decays asymptotically at an exponential rate. We demonstrate that using our method we can solve large problem instances of three types of metric constrained problems: general weight correlation clustering, metric nearness, and metric learning; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes.

The canonical solution methodology for finite constrained Markov decision processes (CMDPs), where the objective is to maximize the expected infinite-horizon discounted rewards subject to the expected infinite-horizon discounted costs constraints, is based on convex linear programming. In this brief, we first prove that the optimization objective in the dual linear program of a finite CMDP is a piece-wise linear convex function (PWLC) with respect to the Lagrange penalty multipliers. Next, we propose a novel two-level Gradient-Aware Search (GAS) algorithm which exploits the PWLC structure to find the optimal state-value function and Lagrange penalty multipliers of a finite CMDP. The proposed algorithm is applied in two stochastic control problems with constraints: robot navigation in a grid world and solar-powered unmanned aerial vehicle (UAV)-based wireless network management. We empirically compare the convergence performance of the proposed GAS algorithm with binary search (BS), Lagrangian primal-dual optimization (PDO), and Linear Programming (LP). Compared with benchmark algorithms, it is shown that the proposed GAS algorithm converges to the optimal solution faster, does not require hyper-parameter tuning, and is not sensitive to initialization of the Lagrange penalty multiplier.

Generative Adversarial Networks (GANs) is a novel class of deep generative models which has recently gained significant attention. GANs learns complex and high-dimensional distributions implicitly over images, audio, and data. However, there exists major challenges in training of GANs, i.e., mode collapse, non-convergence and instability, due to inappropriate design of network architecture, use of objective function and selection of optimization algorithm. Recently, to address these challenges, several solutions for better design and optimization of GANs have been investigated based on techniques of re-engineered network architectures, new objective functions and alternative optimization algorithms. To the best of our knowledge, there is no existing survey that has particularly focused on broad and systematic developments of these solutions. In this study, we perform a comprehensive survey of the advancements in GANs design and optimization solutions proposed to handle GANs challenges. We first identify key research issues within each design and optimization technique and then propose a new taxonomy to structure solutions by key research issues. In accordance with the taxonomy, we provide a detailed discussion on different GANs variants proposed within each solution and their relationships. Finally, based on the insights gained, we present the promising research directions in this rapidly growing field.