Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these methods. However, little can be claimed about their optimality as no lower bound is known, except for a few special \emph{smooth non-convex} cases. In this paper, we make the first attempt to establish lower complexity bounds of FOMs for solving a class of composite non-convex non-smooth optimization with linear constraints. Assuming two different first-order oracles, we establish lower complexity bounds of FOMs to produce a (near) $\epsilon$-stationary point of a problem (and its reformulation) in the considered problem class, for any given tolerance $\epsilon>0$. In addition, we present an inexact proximal gradient (IPG) method by using the more relaxed one of the two assumed first-order oracles. The oracle complexity of the proposed IPG, to find a (near) $\epsilon$-stationary point of the considered problem and its reformulation, matches our established lower bounds up to a logarithmic factor. Therefore, our lower complexity bounds and the proposed IPG method are almost non-improvable.

This paper studies a new problem, \emph{active learning with partial labels} (ALPL). In this setting, an oracle annotates the query samples with partial labels, relaxing the oracle from the demanding accurate labeling process. To address ALPL, we first build an intuitive baseline that can be seamlessly incorporated into existing AL frameworks. Though effective, this baseline is still susceptible to the \emph{overfitting}, and falls short of the representative partial-label-based samples during the query process. Drawing inspiration from human inference in cognitive science, where accurate inferences can be explicitly derived from \emph{counter-examples} (CEs), our objective is to leverage this human-like learning pattern to tackle the \emph{overfitting} while enhancing the process of selecting representative samples in ALPL. Specifically, we construct CEs by reversing the partial labels for each instance, and then we propose a simple but effective WorseNet to directly learn from this complementary pattern. By leveraging the distribution gap between WorseNet and the predictor, this adversarial evaluation manner could enhance both the performance of the predictor itself and the sample selection process, allowing the predictor to capture more accurate patterns in the data. Experimental results on five real-world datasets and four benchmark datasets show that our proposed method achieves comprehensive improvements over ten representative AL frameworks, highlighting the superiority of WorseNet. The source code will be available at \url{https://github.com/Ferenas/APLL}.

We propose a method for learning decision-makers' behavior in routing problems using Inverse Optimization (IO). The IO framework falls into the supervised learning category and builds on the premise that the target behavior is an optimizer of an unknown cost function. This cost function is to be learned through historical data, and in the context of routing problems, can be interpreted as the routing preferences of the decision-makers. In this view, the main contributions of this study are to propose an IO methodology with a hypothesis function, loss function, and stochastic first-order algorithm tailored to routing problems. We further test our IO approach in the Amazon Last Mile Routing Research Challenge, where the goal is to learn models that replicate the routing preferences of human drivers, using thousands of real-world routing examples. Our final IO-learned routing model achieves a score that ranks 2nd compared with the 48 models that qualified for the final round of the challenge. Our results showcase the flexibility and real-world potential of the proposed IO methodology to learn from decision-makers' decisions in routing problems.

The evolutionary diversity optimization aims at finding a diverse set of solutions which satisfy some constraint on their fitness. In the context of multi-objective optimization this constraint can require solutions to be Pareto-optimal. In this paper we study how the GSEMO algorithm with additional diversity-enhancing heuristic optimizes a diversity of its population on a bi-objective benchmark problem OneMinMax, for which all solutions are Pareto-optimal. We provide a rigorous runtime analysis of the last step of the optimization, when the algorithm starts with a population with a second-best diversity, and prove that it finds a population with optimal diversity in expected time $O(n^2)$, when the problem size $n$ is odd. For reaching our goal, we analyse the random walk of the population, which reflects the frequency of changes in the population and their outcomes.

Even nowadays, where Deep Learning (DL) has achieved state-of-the-art performance in a wide range of research domains, accelerating training and building robust DL models remains a challenging task. To this end, generations of researchers have pursued to develop robust methods for training DL architectures that can be less sensitive to weight distributions, model architectures and loss landscapes. However, such methods are limited to adaptive learning rate optimizers, initialization schemes, and clipping gradients without investigating the fundamental rule of parameters update. Although multiplicative updates have contributed significantly to the early development of machine learning and hold strong theoretical claims, to best of our knowledge, this is the first work that investigate them in context of DL training acceleration and robustness. In this work, we propose an optimization framework that fits to a wide range of optimization algorithms and enables one to apply alternative update rules. To this end, we propose a novel multiplicative update rule and we extend their capabilities by combining it with a traditional additive update term, under a novel hybrid update method. We claim that the proposed framework accelerates training, while leading to more robust models in contrast to traditionally used additive update rule and we experimentally demonstrate their effectiveness in a wide range of task and optimization methods. Such tasks ranging from convex and non-convex optimization to difficult image classification benchmarks applying a wide range of traditionally used optimization methods and Deep Neural Network (DNN) architectures.

Safety is one of the fundamental challenges in control theory. Recently, multi-step optimal control problems for discrete-time dynamical systems were formulated to enforce stability, while subject to input constraints as well as safety-critical requirements using discrete-time control barrier functions within a model predictive control (MPC) framework. Existing work usually focus on the feasibility or the safety for the optimization problem, and the majority of the existing work restrict the discussions to relative-degree one control barrier functions. Additionally, the real-time computation is challenging when a large horizon is considered in the MPC problem for relative-degree one or high-order control barrier functions. In this paper, we propose a framework that solves the safety-critical MPC problem in an iterative optimization, which is applicable for any relative-degree control barrier functions. In the proposed formulation, the nonlinear system dynamics as well as the safety constraints modeled as discrete-time high-order control barrier functions (DHOCBF) are linearized at each time step. Our formulation is generally valid for any control barrier function with an arbitrary relative-degree. The advantages of fast computational performance with safety guarantee are analyzed and validated with numerical results.

We study the problem of best-item identification from choice-based feedback. In this problem, a company sequentially and adaptively shows display sets to a population of customers and collects their choices. The objective is to identify the most preferred item with the least number of samples and at a high confidence level. We propose an elimination-based algorithm, namely Nested Elimination (NE), which is inspired by the nested structure implied by the information-theoretic lower bound. NE is simple in structure, easy to implement, and has a strong theoretical guarantee for sample complexity. Specifically, NE utilizes an innovative elimination criterion and circumvents the need to solve any complex combinatorial optimization problem. We provide an instance-specific and non-asymptotic bound on the expected sample complexity of NE. We also show NE achieves high-order worst-case asymptotic optimality. Finally, numerical experiments from both synthetic and real data corroborate our theoretical findings.

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. The Hessian of this loss at low-rank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights.

This paper considers the problem of understanding the behavior of a general class of accelerated gradient methods on smooth nonconvex functions. Motivated by some recent works that have proposed effective algorithms, based on Polyak's heavy ball method and the Nesterov accelerated gradient method, to achieve convergence to a local minimum of nonconvex functions, this work proposes a broad class of Nesterov-type accelerated methods and puts forth a rigorous study of these methods encompassing the escape from saddle-points and convergence to local minima through a both asymptotic and a non-asymptotic analysis. In the asymptotic regime, this paper answers an open question of whether Nesterov's accelerated gradient method (NAG) with variable momentum parameter avoids strict saddle points almost surely. This work also develops two metrics of asymptotic rate of convergence and divergence, and evaluates these two metrics for several popular standard accelerated methods such as the NAG, and Nesterov's accelerated gradient with constant momentum (NCM) near strict saddle points. In the local regime, this work provides an analysis that leads to the "linear" exit time estimates from strict saddle neighborhoods for trajectories of these accelerated methods as well the necessary conditions for the existence of such trajectories. Finally, this work studies a sub-class of accelerated methods that can converge in convex neighborhoods of nonconvex functions with a near optimal rate to a local minima and at the same time this sub-class offers superior saddle-escape behavior compared to that of NAG.

The performance of data fusion and tracking algorithms often depends on parameters that not only describe the sensor system, but can also be task-specific. While for the sensor system tuning these variables is time-consuming and mostly requires expert knowledge, intrinsic parameters of targets under track can even be completely unobservable until the system is deployed. With state-of-the-art sensor systems growing more and more complex, the number of parameters naturally increases, necessitating the automatic optimization of the model variables. In this paper, the parameters of an interacting multiple model (IMM) filter are optimized solely using measurements, thus without necessity for any ground-truth data. The resulting method is evaluated through an ablation study on simulated data, where the trained model manages to match the performance of a filter parametrized with ground-truth values.