Online optimization with multiple budget constraints is challenging since the online decisions over a short time horizon are coupled together by strict inventory constraints. The existing manually-designed algorithms cannot achieve satisfactory average performance for this setting because they often need a large number of time steps for convergence and/or may violate the inventory constraints. In this paper, we propose a new machine learning (ML) assisted unrolling approach, called LAAU (Learning-Assisted Algorithm Unrolling), which unrolls the online decision pipeline and leverages an ML model for updating the Lagrangian multiplier online. For efficient training via backpropagation, we derive gradients of the decision pipeline over time. We also provide the average cost bounds for two cases when training data is available offline and collected online, respectively. Finally, we present numerical results to highlight that LAAU can outperform the existing baselines.

Express companies are deploying more robotic sorting systems, where mobile robots are used to sort incoming parcels by destination. In this study, we propose an integrated assignment and path-finding method for robots in such sorting systems. The method has two parts: offline and online. In the offline part, we represent the system as a traffic flow network, develop an approximate delay function using stochastic models, and solve the min-cost network flow problem. In the online part, robots are guided through the system according to the calculated optimal flow split probability. The online calculation of the method is decentralized and has linear complexity. Our method outperforms fast multi-agent path planning algorithms like prioritized planning because such algorithms lead to stochastic user equilibrium traffic assignment. In contrast, our method gives the approximated system-optimal traffic assignment. According to our simulations, our method can achieve 10%--20% higher throughput than zoning or random assignment. We also show that our method is robust even if the initial demand estimation is inaccurate.

When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called "neural collapse" phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer's classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural collapse phenomenon for normalized features. Based on an unconstrained feature model, we simplify the empirical loss function in a multi-class classification task into a nonconvex optimization problem over the Riemannian manifold by constraining all features and classifiers over the sphere. In this context, we analyze the nonconvex landscape of the Riemannian optimization problem over the product of spheres, showing a benign global landscape in the sense that the only global minimizers are the neural collapse solutions while all other critical points are strict saddles with negative curvature. Experimental results on practical deep networks corroborate our theory and demonstrate that better representations can be learned faster via feature normalization.

We study optimal transport-based distributionally robust optimization problems where a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by reshaping a prescribed reference distribution at a finite transportation cost. In this framework, we show that robustification is intimately related to various forms of variation and Lipschitz regularization even if the transportation cost function fails to be (some power of) a metric. We also derive conditions for the existence and the computability of a Nash equilibrium between the decision-maker and nature, and we demonstrate numerically that nature's Nash strategy can be viewed as a distribution that is supported on remarkably deceptive adversarial samples. Finally, we identify practically relevant classes of optimal transport-based distributionally robust optimization problems that can be addressed with efficient gradient descent algorithms even if the loss function or the transportation cost function are nonconvex (but not both at the same time).

Optimizing delivery routes for last-mile logistics service is challenging and has attracted the attention of many researchers. These problems are usually modeled and solved as variants of vehicle routing problems (VRPs) with challenging real-world constraints (e.g., time windows, precedence). However, despite many decades of solid research on solving these VRP instances, we still see significant gaps between optimized routes and the routes that are actually preferred by the practitioners. Most of these gaps are due to the difference between what's being optimized, and what the practitioners actually care about, which is hard to be defined exactly in many instances. In this paper, we propose a novel hierarchical route optimizer with learnable parameters that combines the strength of both the optimization and machine learning approaches. Our hierarchical router first solves a zone-level Traveling Salesman Problem with learnable weights on various zone-level features; with the zone visit sequence fixed, we then solve the stop-level vehicle routing problem as a Shortest Hamiltonian Path problem. The Bayesian optimization approach is then introduced to allow us to adjust the weights to be assigned to different zone features used in solving the zone-level Traveling Salesman Problem. By using a real-world delivery dataset provided by the Amazon Last Mile Routing Research Challenge, we demonstrate the importance of having both the optimization and the machine learning components. We also demonstrate how we can use route-related features to identify instances that we might have difficulty with. This paves ways to further research on how we can tackle these difficult instances.

In noisy and reverberant environments, the performance of deep learning-based speech separation methods drops dramatically because previous methods are not designed and optimized for such situations. To address this issue, we propose a multi-stage end-to-end learning method that decouples the difficult speech separation problem in noisy and reverberant environments into three sub-problems: speech denoising, separation, and de-reverberation. The probability and speed of searching for the optimal solution of the speech separation model are improved by reducing the solution space. Moreover, since the channel information of the audio sequence in the time domain is crucial for speech separation, we propose a triple-path structure capable of modeling the channel dimension of audio sequences. Experimental results show that the proposed multi-stage triple-path method can improve the performance of speech separation models at the cost of little model parameter increment.

In this paper, we consider a team of mobile robots executing simultaneously multiple behaviors by different subgroups, while maintaining global and subgroup line-of-sight (LOS) network connectivity that minimally constrains the original multi-robot behaviors. The LOS connectivity between pairwise robots is preserved when two robots stay within the limited communication range and their LOS remains occlusion-free from static obstacles while moving. By using control barrier functions (CBF) and minimum volume enclosing ellipsoids (MVEE), we first introduce the LOS connectivity barrier certificate (LOS-CBC) to characterize the state-dependent admissible control space for pairwise robots, from which their resulting motion will keep the two robots LOS connected over time. We then propose the Minimum Line-of-Sight Connectivity Constraint Spanning Tree (MLCCST) as a step-wise bilevel optimization framework to jointly optimize (a) the minimum set of LOS edges to actively maintain, and (b) the control revision with respect to a nominal multi-robot controller due to LOS connectivity maintenance. As proved in the theoretical analysis, this allows the robots to improvise the optimal composition of LOS-CBC control constraints that are least constraining around the nominal controllers, and at the same time enforce the global and subgroup LOS connectivity through the resulting preserved set of pairwise LOS edges. The framework thus leads to robots staying as close to their nominal behaviors, while exhibiting dynamically changing LOS-connected network topology that provides the greatest flexibility for the existing multi-robot tasks in real time. We demonstrate the effectiveness of our approach through simulations with up to 64 robots.

We consider minimizing the average of a very large number of smooth and possibly non-convex functions. This optimization problem has deserved much attention in the past years due to the many applications in different fields, the most challenging being training Machine Learning models. Widely used approaches for solving this problem are mini-batch gradient methods which, at each iteration, update the decision vector moving along the gradient of a mini-batch of the component functions. We consider the Incremental Gradient (IG) and the Random reshuffling (RR) methods which proceed in cycles, picking batches in a fixed order or by reshuffling the order after each epoch. Convergence properties of these schemes have been proved under different assumptions, usually quite strong. We aim to define ease-controlled modifications of the IG/RR schemes, which require a light additional computational effort and can be proved to converge under very weak and standard assumptions. In particular, we define two algorithmic schemes, monotone or non-monotone, in which the IG/RR iteration is controlled by using a watchdog rule and a derivative-free line search that activates only sporadically to guarantee convergence. The two schemes also allow controlling the updating of the stepsize used in the main IG/RR iteration, avoiding the use of preset rules. We prove convergence under the lonely assumption of Lipschitz continuity of the gradients of the component functions and perform extensive computational analysis using Deep Neural Architectures and a benchmark of datasets. We compare our implementation with both full batch gradient methods and online standard implementation of IG/RR methods, proving that the computational effort is comparable with the corresponding online methods and that the control on the learning rate may allow faster decrease.

We introduce a Python open-source library for $\mathcal{X}$-armed bandit and online blackbox optimization named PyXAB. PyXAB contains the implementations for more than 10 $\mathcal{X}$-armed bandit algorithms, such as HOO, StoSOO, HCT, and the most recent works GPO and VHCT. PyXAB also provides the most commonly-used synthetic objectives to evaluate the performance of different algorithms and the various choices of the hierarchical partitions on the parameter space. The online documentation for PyXAB includes clear instructions for installation, straight-forward examples, detailed feature descriptions, and a complete reference of the API. PyXAB is released under the MIT license in order to encourage both academic and industrial usage. The library can be directly installed from PyPI with its source code available at https://github.com/WilliamLwj/PyXAB

We consider a generic min-max multi-objective bilevel optimization problem with applications in robust machine learning such as representation learning and hyperparameter optimization. We design MORBiT, a novel single-loop gradient descent-ascent bilevel optimization algorithm, to solve the generic problem and present a novel analysis showing that MORBiT converges to the first-order stationary point at a rate of $\widetilde{\mathcal{O}}(n^{1/2} K^{-2/5})$ for a class of weakly convex problems with $n$ objectives upon $K$ iterations of the algorithm. Our analysis utilizes novel results to handle the non-smooth min-max multi-objective setup and to obtain a sublinear dependence in the number of objectives $n$. Experimental results on robust representation learning and robust hyperparameter optimization showcase (i) the advantages of considering the min-max multi-objective setup, and (ii) convergence properties of the proposed MORBiT. Our code is at https://github.com/minimario/MORBiT.