This paper proposes a new mathematical paradigm to analyze discrete-time mean-field games. It is shown that finding Nash equilibrium solutions for a general class of discrete-time mean-field games is equivalent to solving an optimization problem with bounded variables and simple convex constraints, called MF-OMO. This equivalence framework enables finding multiple (and possibly all) Nash equilibrium solutions of mean-field games by standard algorithms. For instance, projected gradient descent is shown to be capable of retrieving all possible Nash equilibrium solutions when there are finitely many of them, with proper initializations. Moreover, analyzing mean-field games with linear rewards and mean-field independent dynamics is reduced to solving a finite number of linear programs, hence solvable in finite time. This framework does not rely on the contractive and the monotone assumptions and the uniqueness of the Nash equilibrium.

We consider the problem of allocating multiple indivisible items to a set of networked agents to maximize the social welfare subject to network externalities. Here, the social welfare is given by the sum of agents' utilities and externalities capture the effect that one user of an item has on the item's value to others. We first provide a general formulation that captures some of the existing models as a special case. We then show that the social welfare maximization problem benefits some nice diminishing or increasing marginal return properties. That allows us to devise polynomial-time approximation algorithms using the Lovasz extension and multilinear extension of the objective functions. Our principled approach recovers or improves some of the existing algorithms and provides a simple and unifying framework for maximizing social welfare subject to network externalities.

The sparse modeling is an evident manifestation capturing the parsimony principle just described, and sparse models are widespread in statistics, physics, information sciences, neuroscience, computational mathematics, and so on. In statistics the many applications of sparse modeling span regression, classification tasks, graphical model selection, sparse M-estimators and sparse dimensionality reduction. It is also particularly effective in many statistical and machine learning areas where the primary goal is to discover predictive patterns from data which would enhance our understanding and control of underlying physical, biological, and other natural processes, beyond just building accurate outcome black-box predictors. Common examples include selecting biomarkers in biological procedures, finding relevant brain activity locations which are predictive about brain states and processes based on fMRI data, and identifying network bottlenecks best explaining end-to-end performance. Moreover, the research and applications of efficient recovery of high-dimensional sparse signals from a relatively small number of observations, which is the main focus of compressed sensing or compressive sensing, have rapidly grown and became an extremely intense area of study beyond classical signal processing. Likewise interestingly, sparse modeling is directly related to various artificial vision tasks, such as image denoising, segmentation, restoration and superresolution, object or face detection and recognition in visual scenes, and action recognition. In this manuscript, we provide a brief introduction of the basic theory underlying sparse representation and compressive sensing, and then discuss some methods for recovering sparse solutions to optimization problems in effective way, together with some applications of sparse recovery in a machine learning problem known as sparse dictionary learning.

Memory-aware network scheduling is becoming increasingly important for deep neural network (DNN) inference on resource-constrained devices. However, due to the complex cell-level and network-level topologies, memory-aware scheduling becomes very challenging. While previous algorithms all suffer from poor scalability, in this paper, we propose an efficient memory-aware scheduling framework based on iterative computation graph optimization. Our framework features an iterative graph fusion algorithm that simplifies the computation graph while preserving the scheduling optimality. We further propose an integer linear programming formulation together with topology-aware variable pruning to schedule the simplified graph efficiently. We evaluate our method against prior-art algorithms on different networks and demonstrate that our method outperforms existing techniques in all the benchmarks, reducing the peak memory footprint by 13.4%, and achieving better scalability for networks with complex network-level topologies.

This yields an equivalent flat system that is completely described by integrator dynamics. It There is an increasing demand to extend the boundaries is significantly easier to generate control trajectories in of autonomy in cyber-physical systems (CPS) using the flat space, wherein the trajectories can be exactly experimental testbeds (see: Rubenstein et al. (2012); mapped back to the original coordinate system. Differentially Jang et al. (2019); Beaver et al. (2020); Chalaki et al. flat systems have garnered significant interest (2022)) and outdoor experiments (see: Vásárhelyi et al. since their introduction by Fliess et al. (1995), and it has (2018); Mahbub and Malikopoulos (2020); Chalaki et al. been shown that generating trajectories in the flat space (2022)). As CPS achieve higher autonomy levels, they can reduce computational time by at least an order of will be forced into complicated interactions with other magnitude (e.g., see: Petit et al. (2001)). Differentially agents and the surrounding environment (Malikopoulos flat systems are closely related to feedback linearizable et al., 2021; Beaver and Malikopoulos, 2021; Oh et al., systems (Lévine, 2007); however, the standard control 2017). These autonomous agents must be able to react techniques for flat systems are distinct from feedback quickly to their environment and re-plan efficient trajectories.

Reduced-order models (ROM) are popular in online motion planning due to their simplicity. A good ROM for control captures critical task-relevant aspects of the full dynamics while remaining low dimensional. However, planning within the reduced-order space unavoidably constrains the full model, and hence we sacrifice the full potential of the robot. In the community of legged locomotion, this has lead to a search for better model extensions, but many of these extensions require human intuition, and there has not existed a principled way of evaluating the model performance and discovering new models. In this work, we propose a model optimization algorithm that automatically synthesizes reduced-order models, optimal with respect to a user-specified distribution of tasks and corresponding cost functions. To demonstrate our work, we optimized models for a bipedal robot Cassie. We show in simulation that the optimal ROM reduces the cost of Cassie's joint torques by up to 23% and increases its walking speed by up to 54%. We also show hardware result that the real robot walks on flat ground with 10% lower torque cost. All videos and code can be found at https://sites.google.com/view/ymchen/research/optimal-rom.

Computing the Jacobian of the solution of an optimization problem is a central problem in machine learning, with applications in hyperparameter optimization, meta-learning, optimization as a layer, and dataset distillation, to name a few. Unrolled differentiation is a popular heuristic that approximates the solution using an iterative solver and differentiates it through the computational path. This work provides a non-asymptotic convergence-rate analysis of this approach on quadratic objectives for gradient descent and the Chebyshev method. We show that to ensure convergence of the Jacobian, we can either 1) choose a large learning rate leading to a fast asymptotic convergence but accept that the algorithm may have an arbitrarily long burn-in phase or 2) choose a smaller learning rate leading to an immediate but slower convergence. We refer to this phenomenon as the curse of unrolling. Finally, we discuss open problems relative to this approach, such as deriving a practical update rule for the optimal unrolling strategy and making novel connections with the field of Sobolev orthogonal polynomials.

Permutation matrices play a key role in matching and assignment problems across the fields, especially in computer vision and robotics. However, memory for explicitly representing permutation matrices grows quadratically with the size of the problem, prohibiting large problem instances. In this work, we propose to tackle the curse of dimensionality of large permutation matrices by approximating them using low-rank matrix factorization, followed by a nonlinearity. To this end, we rely on the Kissing number theory to infer the minimal rank required for representing a permutation matrix of a given size, which is significantly smaller than the problem size. This leads to a drastic reduction in computation and memory costs, e.g., up to $3$ orders of magnitude less memory for a problem of size $n=20000$, represented using $8.4\times10^5$ elements in two small matrices instead of using a single huge matrix with $4\times 10^8$ elements. The proposed representation allows for accurate representations of large permutation matrices, which in turn enables handling large problems that would have been infeasible otherwise. We demonstrate the applicability and merits of the proposed approach through a series of experiments on a range of problems that involve predicting permutation matrices, from linear and quadratic assignment to shape matching problems.

In this paper we consider inverse problems that are mathematically ill-posed. That is, given some (noisy) data, there is more than one solution that approximately fits the data. In recent years, deep neural techniques that find the most appropriate solution, in the sense that it contains a-priori information, were developed. However, they suffer from several shortcomings. First, most techniques cannot guarantee that the solution fits the data at inference. Second, while the derivation of the techniques is inspired by the existence of a valid scalar regularization function, such techniques do not in practice rely on such a function, and therefore veer away from classical variational techniques. In this work we introduce a new family of neural regularizers for the solution of inverse problems. These regularizers are based on a variational formulation and are guaranteed to fit the data. We demonstrate their use on a number of highly ill-posed problems, from image deblurring to limited angle tomography.

Today's cloud data centers are often distributed geographically to provide robust data services. But these geo-distributed data centers (GDDCs) have a significant associated environmental impact due to their increasing carbon emissions and water usage, which needs to be curtailed. Moreover, the energy costs of operating these data centers continue to rise. This paper proposes a novel framework to co-optimize carbon emissions, water footprint, and energy costs of GDDCs, using a hybrid workload management framework called SHIELD that integrates machine learning guided local search with a decomposition-based evolutionary algorithm. Our framework considers geographical factors and time-based differences in power generation/use, costs, and environmental impacts to intelligently manage workload distribution across GDDCs and data center operation. Experimental results show that SHIELD can realize 34.4x speedup and 2.1x improvement in Pareto Hypervolume while reducing the carbon footprint by up to 3.7x, water footprint by up to 1.8x, energy costs by up to 1.3x, and a cumulative improvement across all objectives (carbon, water, cost) of up to 4.8x compared to the state-of-the-art.