Nonconvex and nonsmooth optimization problems are important and challenging for statistics and machine learning. In this paper, we propose Projected Proximal Gradient Descent (PPGD) which solves a class of nonconvex and nonsmooth optimization problems, where the nonconvexity and nonsmoothness come from a nonsmooth regularization term which is nonconvex but piecewise convex. In contrast with existing convergence analysis of accelerated PGD methods for nonconvex and nonsmooth problems based on the Kurdyka-\L{}ojasiewicz (K\L{}) property, we provide a new theoretical analysis showing local fast convergence of PPGD. It is proved that PPGD achieves a fast convergence rate of $\cO(1/k^2)$ when the iteration number $k \ge k_0$ for a finite $k_0$ on a class of nonconvex and nonsmooth problems under mild assumptions, which is locally Nesterov's optimal convergence rate of first-order methods on smooth and convex objective function with Lipschitz continuous gradient. Experimental results demonstrate the effectiveness of PPGD.

Model predictive control (MPC) achieves stability and constraint satisfaction for general nonlinear systems, but requires computationally expensive online optimization. This paper studies approximations of such MPC controllers via neural networks (NNs) to achieve fast online evaluation. We propose safety augmentation that yields deterministic guarantees for convergence and constraint satisfaction despite approximation inaccuracies. We approximate the entire input sequence of the MPC with NNs, which allows us to verify online if it is a feasible solution to the MPC problem. We replace the NN solution by a safe candidate based on standard MPC techniques whenever it is infeasible or has worse cost. Our method requires a single evaluation of the NN and forward integration of the input sequence online, which is fast to compute on resource-constrained systems. The proposed control framework is illustrated on three non-linear MPC benchmarks of different complexity, demonstrating computational speedups orders of magnitudes higher than online optimization. In the examples, we achieve deterministic safety through the safety-augmented NNs, where naive NN implementation fails.

A key challenge in sequential decision making is optimizing systems safely under partial information. While much of the literature has focused on the cases of either partially known states or partially known dynamics, it is further exacerbated in cases where both states and dynamics are partially known. Computing heparin doses for patients fits this paradigm since the concentration of heparin in the patient cannot be measured directly and the rates at which patients metabolize heparin vary greatly between individuals. While many proposed solutions are model free, they require complex models and have difficulty ensuring safety. However, if some of the structure of the dynamics is known, a model based approach can be leveraged to provide safe policies. In this paper we propose such a framework to address the challenge of optimizing personalized heparin doses. We use a predictive model parameterized individually by patient to predict future therapeutic effects. We then leverage this model using a scenario generation based approach that is capable of ensuring patient safety. We validate our models with numerical experiments by comparing the predictive capabilities of our model against existing machine learning techniques and demonstrating how our dosing algorithm can treat patients in a simulated ICU environment.

Hyper-parameters optimization (HPO) is vital for machine learning models. Besides model accuracy, other tuning intentions such as model training time and energy consumption are also worthy of attention from data analytic service providers. Hence, it is essential to take both model hyperparameters and system parameters into consideration to execute cross-layer multi-objective hyperparameter auto-tuning. Towards this challenging target, we propose HyperTuner in this paper. To address the formulated high-dimensional black-box multi-objective optimization problem, HyperTuner first conducts multi-objective parameter importance ranking with its MOPIR algorithm and then leverages the proposed ADUMBO algorithm to find the Pareto-optimal configuration set. During each iteration, ADUMBO selects the most promising configuration from the generated Pareto candidate set via maximizing a new well-designed metric, which can adaptively leverage the uncertainty as well as the predicted mean across all the surrogate models along with the iteration times. We evaluate HyperTuner on our local distributed TensorFlow cluster and experimental results show that it is always able to find a better Pareto configuration front superior in both convergence and diversity compared with the other four baseline algorithms. Besides, experiments with different training datasets, different optimization objectives and different machine learning platforms verify that HyperTuner can well adapt to various data analytic service scenarios.

Federated learning is a machine learning training paradigm that enables clients to jointly train models without sharing their own localized data. However, the implementation of federated learning in practice still faces numerous challenges, such as the large communication overhead due to the repetitive server-client synchronization and the lack of adaptivity by SGD-based model updates. Despite that various methods have been proposed for reducing the communication cost by gradient compression or quantization, and the federated versions of adaptive optimizers such as FedAdam are proposed to add more adaptivity, the current federated learning framework still cannot solve the aforementioned challenges all at once. In this paper, we propose a novel communication-efficient adaptive federated learning method (FedCAMS) with theoretical convergence guarantees. We show that in the nonconvex stochastic optimization setting, our proposed FedCAMS achieves the same convergence rate of $O(\frac{1}{\sqrt{TKm}})$ as its non-compressed counterparts. Extensive experiments on various benchmarks verify our theoretical analysis.

Resource allocation is the assignment of resources to activities that must be executed in a business process at a particular moment at run-time. While resource allocation is well-studied in other fields, such as manufacturing, there exist only a few methods in business process management. Existing methods are not suited for application in large business processes or focus on optimizing resource allocation for a single case rather than for all cases combined. To fill this gap, this paper proposes two learning-based methods for resource allocation in business processes: a deep reinforcement learning-based approach and a score-based value function approximation approach. The two methods are compared against existing heuristics in a set of scenarios that represent typical business process structures and on a complete network that represents a realistic business process. The results show that our learning-based methods outperform or are competitive with common heuristics in most scenarios and outperform heuristics in the complete network.

Describing systems in terms of choices and their resulting costs and rewards offers the promise of freeing algorithm designers and programmers from specifying how those choices should be made; in implementations, the choices can be realized by optimization techniques and, increasingly, by machine-learning methods. We study this approach from a programming-language perspective. We define two small languages that support decision-making abstractions: one with choices and rewards, and the other additionally with probabilities. We give both operational and denotational semantics. In the case of the second language we consider three denotational semantics, with varying degrees of correlation between possible program values and expected rewards. The operational semantics combine the usual semantics of standard constructs with optimization over spaces of possible execution strategies. The denotational semantics, which are compositional, rely on the selection monad, to handle choice, augmented with an auxiliary monad to handle other effects, such as rewards or probability. We establish adequacy theorems that the two semantics coincide in all cases. We also prove full abstraction at base types, with varying notions of observation in the probabilistic case corresponding to the various degrees of correlation. We present axioms for choice combined with rewards and probability, establishing completeness at base types for the case of rewards without probability.

Zeroth order optimization is a central topic in optimization and related fields. Algorithms for zeroth order optimization find important real-world applications, since often times in practice, we cannot directly access the derivatives of the objective function. To optimize the function in such scenarios, one can estimate the gradient/Hessian first and deploy first/second order algorithms with the estimated derivatives. Previously, many authors have considered this problem. Yet stochastic zeroth order methods for Łojasiewicz functions have not been carefully investigated (See Section 2 for more discussion).

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable PDEconstrained layer that can be incorporated into any NN architecture. Our method leverages differentiable optimization and the implicit function theorem to effectively enforce physical constraints. Inspired by dictionary learning, our model learns a family of functions, each of which defines a mapping from PDE parameters to PDE solutions. At inference time, the model finds an optimal linear combination of the functions in the learned family by solving a PDE-constrained optimization problem. Our method provides continuous solutions over the domain of interest that accurately satisfy desired physical constraints. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective. Methods based on neural networks (NNs) have shown promise in recent years for physics-based problems (Raissi et al., 2019; Li et al., 2020; Lu et al., 2021a; Li et al., 2021). Current NN methods use two main training approaches to solve Equation 1. The first approach is strictly supervised learning, and the NN is trained on PDE solution data using a regression loss (Lu et al., 2021a; Li et al., 2020). In this case, the feasibility problem only appears through the data; it does not appear explicitly in the training algorithm. The second approach (Raissi et al., 2019) aims to solve the feasibility problem in Equation 1 by considering the relaxation, min E This second approach does not require access to any PDE solution data. These two approaches have also been combined by having both a data fitting loss and the PDE residual loss (Li et al., 2021).

Manufacturing advanced materials and products with a specific property or combination of properties is often warranted. To achieve that it is crucial to find out the optimum recipe or processing conditions that can generate the ideal combination of these properties. Most of the time, a sufficient number of experiments are needed to generate a Pareto front. However, manufacturing experiments are usually costly and even conducting a single experiment can be a time-consuming process. So, it's critical to determine the optimal location for data collection to gain the most comprehensive understanding of the process. Sequential learning is a promising approach to actively learn from the ongoing experiments, iteratively update the underlying optimization routine, and adapt the data collection process on the go. This paper presents a novel data-driven Bayesian optimization framework that utilizes sequential learning to efficiently optimize complex systems with multiple conflicting objectives. Additionally, this paper proposes a novel metric for evaluating multi-objective data-driven optimization approaches. This metric considers both the quality of the Pareto front and the amount of data used to generate it. The proposed framework is particularly beneficial in practical applications where acquiring data can be expensive and resource intensive. To demonstrate the effectiveness of the proposed algorithm and metric, the algorithm is evaluated on a manufacturing dataset. The results indicate that the proposed algorithm can achieve the actual Pareto front while processing significantly less data. It implies that the proposed data-driven framework can lead to similar manufacturing decisions with reduced costs and time.