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We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual $L^2(\Omega)$ norm regularization term with a constant regularization parameter $\varrho$ is replaced by a suitable representation of the energy norm in $H^{-1}(\Omega)$ involving a variable, mesh-dependent regularization parameter $\varrho(x)$. It turns out that the error between the computed finite element state $\widetilde{u}_{\varrho h}$ and the desired state $\bar{u}$ (target) is optimal in the $L^2(\Omega)$ norm provided that $\varrho(x)$ behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm $\| \widetilde{u}_{\varrho h} - \bar{u}\|_{L^2(\Omega)}$ between the finite element state $\widetilde{u}_{\varrho h}$ and the target $\bar{u}$. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.

In the Internet of Things (IoT) environment, edge computing can be initiated at anytime and anywhere. However, in an IoT, edge computing sessions are often ephemeral, i.e., they last for a short period of time and can often be discontinued once the current application usage is completed or the edge devices leave the system due to factors such as mobility. Therefore, in this paper, the problem of ephemeral edge computing in an IoT is studied by considering scenarios in which edge computing operates within a limited time period. To this end, a novel online framework is proposed in which a source edge node offloads its computing tasks from sensors within an area to neighboring edge nodes for distributed task computing, within the limited period of time of an ephemeral edge computing system. The online nature of the framework allows the edge nodes to optimize their task allocation and decide on which neighbors to use for task processing, even when the tasks are revealed to the source edge node in an online manner, and the information on future task arrivals is unknown. The proposed framework essentially maximizes the number of computed tasks by jointly considering the communication and computation latency. To solve the problem, an online greedy algorithm is proposed and solved by using the primal-dual approach. Since the primal problem provides an upper bound of the original dual problem, the competitive ratio of the online approach is analytically derived as a function of the task sizes and the data rates of the edge nodes. Simulation results show that the proposed online algorithm can achieve a near-optimal task allocation with an optimality gap that is no higher than 7.1% compared to the offline, optimal solution with complete knowledge of all tasks.

Experienced users often have useful knowledge and intuition in solving real-world optimization problems. User knowledge can be formulated as inter-variable relationships to assist an optimization algorithm in finding good solutions faster. Such inter-variable interactions can also be automatically learned from high-performing solutions discovered at intermediate iterations in an optimization run - a process called innovization. These relations, if vetted by the users, can be enforced among newly generated solutions to steer the optimization algorithm towards practically promising regions in the search space. Challenges arise for large-scale problems where the number of such variable relationships may be high. This paper proposes an interactive knowledge-based evolutionary multi-objective optimization (IK-EMO) framework that extracts hidden variable-wise relationships as knowledge from evolving high-performing solutions, shares them with users to receive feedback, and applies them back to the optimization process to improve its effectiveness. The knowledge extraction process uses a systematic and elegant graph analysis method which scales well with number of variables. The working of the proposed IK-EMO is demonstrated on three large-scale real-world engineering design problems. The simplicity and elegance of the proposed knowledge extraction process and achievement of high-performing solutions quickly indicate the power of the proposed framework. The results presented should motivate further such interaction-based optimization studies for their routine use in practice.

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schr\"odinger framework to solve the Liouville equation for the HJE, since it can be recast as the semiclassical limit of the Wigner transform of the Schr\"odinger equation. Comparsion between the Schr\"odinger and the Liouville framework will also be made.

Efficient inference is often possible in a streaming context using Rao-Blackwellized particle filters (RBPFs), which exactly solve inference problems when possible and fall back on sampling approximations when necessary. While RBPFs can be implemented by hand to provide efficient inference, the goal of streaming probabilistic programming is to automatically generate such efficient inference implementations given input probabilistic programs. In this work, we propose semi-symbolic inference, a technique for executing probabilistic programs using a runtime inference system that automatically implements Rao-Blackwellized particle filtering. To perform exact and approximate inference together, the semi-symbolic inference system manipulates symbolic distributions to perform exact inference when possible and falls back on approximate sampling when necessary. This approach enables the system to implement the same RBPF a developer would write by hand. To ensure this, we identify closed families of distributions -- such as linear-Gaussian and finite discrete models -- on which the inference system guarantees exact inference. We have implemented the runtime inference system in the ProbZelus streaming probabilistic programming language. Despite an average $1.6\times$ slowdown compared to the state of the art on existing benchmarks, our evaluation shows that speedups of $3\times$-$87\times$ are obtainable on a new set of challenging benchmarks we have designed to exploit closed families.

Neural Architecture Search (NAS) has significantly improved productivity in the design and deployment of neural networks (NN). As NAS typically evaluates multiple models by training them partially or completely, the improved productivity comes at the cost of significant carbon footprint. To alleviate this expensive training routine, zero-shot/cost proxies analyze an NN at initialization to generate a score, which correlates highly with its true accuracy. Zero-cost proxies are currently designed by experts conducting multiple cycles of empirical testing on possible algorithms, data-sets, and neural architecture design spaces. This lowers productivity and is an unsustainable approach towards zero-cost proxy design as deep learning use-cases diversify in nature. Additionally, existing zero-cost proxies fail to generalize across neural architecture design spaces. In this paper, we propose a genetic programming framework to automate the discovery of zero-cost proxies for neural architecture scoring. Our methodology efficiently discovers an interpretable and generalizable zero-cost proxy that gives state of the art score-accuracy correlation on all data-sets and search spaces of NASBench-201 and Network Design Spaces (NDS). We believe that this research indicates a promising direction towards automatically discovering zero-cost proxies that can work across network architecture design spaces, data-sets, and tasks.

Click-through rate (CTR) prediction plays a critical role in recommender systems and online advertising. The data used in these applications are multi-field categorical data, where each feature belongs to one field. Field information is proved to be important and there are several works considering fields in their models. In this paper, we proposed a novel approach to model the field information effectively and efficiently. The proposed approach is a direct improvement of FwFM, and is named as Field-matrixed Factorization Machines (FmFM, or $FM^2$). We also proposed a new explanation of FM and FwFM within the FmFM framework, and compared it with the FFM. Besides pruning the cross terms, our model supports field-specific variable dimensions of embedding vectors, which acts as soft pruning. We also proposed an efficient way to minimize the dimension while keeping the model performance. The FmFM model can also be optimized further by caching the intermediate vectors, and it only takes thousands of floating-point operations (FLOPs) to make a prediction. Our experiment results show that it can out-perform the FFM, which is more complex. The FmFM model's performance is also comparable to DNN models which require much more FLOPs in runtime.

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.

To address the sparsity and cold start problem of collaborative filtering, researchers usually make use of side information, such as social networks or item attributes, to improve recommendation performance. This paper considers the knowledge graph as the source of side information. To address the limitations of existing embedding-based and path-based methods for knowledge-graph-aware recommendation, we propose Ripple Network, an end-to-end framework that naturally incorporates the knowledge graph into recommender systems. Similar to actual ripples propagating on the surface of water, Ripple Network stimulates the propagation of user preferences over the set of knowledge entities by automatically and iteratively extending a user's potential interests along links in the knowledge graph. The multiple "ripples" activated by a user's historically clicked items are thus superposed to form the preference distribution of the user with respect to a candidate item, which could be used for predicting the final clicking probability. Through extensive experiments on real-world datasets, we demonstrate that Ripple Network achieves substantial gains in a variety of scenarios, including movie, book and news recommendation, over several state-of-the-art baselines.