The fireworks algorithm is an optimization algorithm for simulating the explosion phenomenon of fireworks. Because of its fast convergence and high precision, it is widely used in pattern recognition, optimal scheduling, and other fields. However, most of the existing research work on the fireworks algorithm is improved based on its defects, and little consideration is given to reducing the number of parameters of the fireworks algorithm. The original fireworks algorithm has too many parameters, which increases the cost of algorithm adjustment and is not conducive to engineering applications. In addition, in the fireworks population, the unselected individuals are discarded, thus causing a waste of their location information. To reduce the number of parameters of the original Fireworks Algorithm and make full use of the location information of discarded individuals, we propose a simplified version of the Fireworks Algorithm. It reduces the number of algorithm parameters by redesigning the explosion operator of the fireworks algorithm and constructs an adaptive explosion radius by using the historical optimal information to balance the local mining and global exploration capabilities. The comparative experimental results of function optimization show that the overall performance of our proposed LFWA is better than that of comparative algorithms, such as the fireworks algorithm, particle swarm algorithm, and bat algorithm.

Learn the theory of the Differential Evolution algorithm, its Python implementation and how and why it will surely help you in solving complex real-world optimization problems. This article has been written with Salvatore Guastella. Optimization is a pillar of data science. If you think about it, under the hood of each machine learning algorithms (ranging from basic linear regression to the most complex neural networks architectures), an optimization problem is solved. Moreover, in many real-world problems the goal is to find the values of one or more decision variables that minimize (or maximize) a quantity of interest while satisfying certain constraints. Few examples are given by portfolio optimization in finance, profit maximization of ad campaigns, energy efficiency in energy plants and shipment cost minimization in logistics (refer to this Medium article [1] in our Eni digiTALKS channel for an interesting example).

We present a general approach for controlling robotic systems that make and break contact with their environments. Contact-implicit model predictive control (CI-MPC) generalizes linear MPC to contact-rich settings by utilizing a bi-level planning formulation with lower-level contact dynamics formulated as time-varying linear complementarity problems (LCPs) computed using strategic Taylor approximations about a reference trajectory. These dynamics enable the upper-level planning problem to reason about contact timing and forces, and generate entirely new contact-mode sequences online. To achieve reliable and fast numerical convergence, we devise a structure-exploiting interior-point solver for these LCP contact dynamics and a custom trajectory optimizer for the tracking problem. We demonstrate real-time solution rates for CI-MPC and the ability to generate and track non-periodic behaviours in hardware experiments on a quadrupedal robot. We also show that the controller is robust to model mismatch and can respond to disturbances by discovering and exploiting new contact modes across a variety of robotic systems in simulation, including a pushbot, planar hopper, planar quadruped, and planar biped.

Stochastic alternating algorithms for bi-objective optimization are considered when optimizing two conflicting functions for which optimization steps have to be applied separately for each function. Such algorithms consist of applying a certain number of steps of gradient or subgradient descent on each single objective at each iteration. In this paper, we show that stochastic alternating algorithms achieve a sublinear convergence rate of $\mathcal{O}(1/T)$, under strong convexity, for the determination of a minimizer of a weighted-sum of the two functions, parameterized by the number of steps applied on each of them. An extension to the convex case is presented for which the rate weakens to $\mathcal{O}(1/\sqrt{T})$. These rates are valid also in the non-smooth case. Importantly, by varying the proportion of steps applied to each function, one can determine an approximation to the Pareto front.

We consider the constrained Linear Inverse Problem (LIP), where a certain atomic norm (like the $\ell_1 $ and the Nuclear norm) is minimized subject to a quadratic constraint. Typically, such cost functions are non-differentiable which makes them not amenable to the fast optimization methods existing in practice. We propose two equivalent reformulations of the constrained LIP with improved convex regularity: (i) a smooth convex minimization problem, and (ii) a strongly convex min-max problem. These problems could be solved by applying existing acceleration based convex optimization methods which provide better $ O \big( \frac{1}{k^2} \big) $ theoretical convergence guarantee. However, to fully exploit the utility of these reformulations, we also provide a novel algorithm, to which we refer as the Fast Linear Inverse Problem Solver (FLIPS), that is tailored to solve the reformulation of the LIP. We demonstrate the performance of FLIPS on the sparse coding problem arising in image processing tasks. In this setting, we observe that FLIPS consistently outperforms the Chambolle-Pock and C-SALSA algorithms--two of the current best methods in the literature.

The celebrated FedAvg algorithm of McMahan et al. (2017) is based on three components: client sampling (CS), data sampling (DS) and local training (LT). While the first two are reasonably well understood, the third component, whose role is to reduce the number of communication rounds needed to train the model, resisted all attempts at a satisfactory theoretical explanation. Malinovsky et al. (2022) identified four distinct generations of LT methods based on the quality of the provided theoretical communication complexity guarantees. Despite a lot of progress in this area, none of the existing works were able to show that it is theoretically better to employ multiple local gradient-type steps (i.e., to engage in LT) than to rely on a single local gradient-type step only in the important heterogeneous data regime. In a recent breakthrough embodied in their ProxSkip method and its theoretical analysis, Mishchenko et al. (2022) showed that LT indeed leads to provable communication acceleration for arbitrarily heterogeneous data, thus jump-starting the $5^{\rm th}$ generation of LT methods. However, while these latest generation LT methods are compatible with DS, none of them support CS. We resolve this open problem in the affirmative. In order to do so, we had to base our algorithmic development on new algorithmic and theoretical foundations.

Identifying functional connectivity biomarkers of major depressive disorder (MDD) patients is essential to advance understanding of the disorder mechanisms and early intervention. However, due to the small sample size and the high dimension of available neuroimaging data, the performance of existing methods is often limited. Multi-site data could enhance the statistical power and sample size, while they are often subject to inter-site heterogeneity and data-sharing policies. In this paper, we propose a federated joint estimator, NOTEARS-PFL, for simultaneous learning of multiple Bayesian networks (BNs) with continuous optimization, to identify disease-induced alterations in MDD patients. We incorporate information shared between sites and site-specific information into the proposed federated learning framework to learn personalized BN structures by introducing the group fused lasso penalty. We develop the alternating direction method of multipliers, where in the local update step, the neuroimaging data is processed at each local site. Then the learned network structures are transmitted to the center for the global update. In particular, we derive a closed-form expression for the local update step and use the iterative proximal projection method to deal with the group fused lasso penalty in the global update step. We evaluate the performance of the proposed method on both synthetic and real-world multi-site rs-fMRI datasets. The results suggest that the proposed NOTEARS-PFL yields superior effectiveness and accuracy than the comparable methods.

Minibatch decomposition methods for empirical risk minimization are commonly analysed in a stochastic approximation setting, also known as sampling with replacement. On the other hands modern implementations of such techniques are incremental: they rely on sampling without replacement, for which available analysis are much scarcer. We provide convergence guaranties for the latter variant by analysing a versatile incremental gradient scheme. For this scheme, we consider constant, decreasing or adaptive step sizes. In the smooth setting we obtain explicit complexity estimates in terms of epoch counter. In the nonsmooth setting we prove that the sequence is attracted by solutions of optimality conditions of the problem.

Sparse optimization problems are ubiquitous in many fields such as statistics, signal/image processing and machine learning. This has led to the birth of many iterative algorithms to solve them. A powerful strategy to boost the performance of these algorithms is known as safe screening: it allows the early identification of zero coordinates in the solution, which can then be eliminated to reduce the problem's size and accelerate convergence. In this work, we extend the existing Gap Safe screening framework by relaxing the global strong-concavity assumption on the dual cost function. Instead, we exploit local regularity properties, that is, strong concavity on well-chosen subsets of the domain. The non-negativity constraint is also integrated to the existing framework. Besides making safe screening possible to a broader class of functions that includes beta-divergences (e.g., the Kullback-Leibler divergence), the proposed approach also improves upon the existing Gap Safe screening rules on previously applicable cases (e.g., logistic regression). The proposed general framework is exemplified by some notable particular cases: logistic function, beta = 1.5 and Kullback-Leibler divergences. Finally, we showcase the effectiveness of the proposed screening rules with different solvers (coordinate descent, multiplicative-update and proximal gradient algorithms) and different data sets (binary classification, hyperspectral and count data).

Fair resource allocation is one of the most important topics in communication networks. Existing solutions almost exclusively assume each user utility function is known and concave. This paper seeks to answer the following question: how to allocate resources when utility functions are unknown, even to the users? This answer has become increasingly important in the next-generation AI-aware communication networks where the user utilities are complex and their closed-forms are hard to obtain. In this paper, we provide a new solution using a distributed and data-driven bilevel optimization approach, where the lower level is a distributed network utility maximization (NUM) algorithm with concave surrogate utility functions, and the upper level is a data-driven learning algorithm to find the best surrogate utility functions that maximize the sum of true network utility. The proposed algorithm learns from data samples (utility values or gradient values) to autotune the surrogate utility functions to maximize the true network utility, so works for unknown utility functions. For the general network, we establish the nonasymptotic convergence rate of the proposed algorithm with nonconcave utility functions. The simulations validate our theoretical results and demonstrate the great effectiveness of the proposed method in a real-world network.