Empirical risk minimization (ERM) and distributionally robust optimization (DRO) are popular approaches for solving stochastic optimization problems that appear in operations management and machine learning. Existing generalization error bounds for these methods depend on either the complexity of the cost function or dimension of the random perturbations. Consequently, the performance of these methods can be poor for high-dimensional problems with complex objective functions. We propose a simple approach in which the distribution of random perturbations is approximated using a parametric family of distributions. This mitigates both sources of complexity; however, it introduces a model misspecification error. We show that this new source of error can be controlled by suitable DRO formulations. Our proposed parametric DRO approach has significantly improved generalization bounds over existing ERM and DRO methods and parametric ERM for a wide variety of settings. Our method is particularly effective under distribution shifts and works broadly in contextual optimization. We also illustrate the superior performance of our approach on both synthetic and real-data portfolio optimization and regression tasks.

Abstract--One of the crucial issues in federated learning is how to develop efficient optimization algorithms. Most of the current ones require full device participation and/or impose strong assumptions for convergence. Different from the widely-used gradient descentbased algorithms, in this paper, we develop an inexact alternating direction method of multipliers (ADMM), which is both computationand communication-efficient, capable of combating the stragglers' effect, and convergent under mild conditions. Furthermore, it has a high numerical performance compared with several state-of-the-art algorithms for federated learning. This idea has been extensively exploited in the [4], [5], [6], digital health [7], and mobile edge and over-theair stochastic gradient descent (SGD) algorithms, such as the computing [8], [9], [10], [11].

Due to the communication bottleneck in distributed and federated learning applications, algorithms using communication compression have attracted significant attention and are widely used in practice. Moreover, the huge number, high heterogeneity and limited availability of clients result in high client-variance. This paper addresses these two issues together by proposing compressed and client-variance reduced methods COFIG and FRECON. We prove an $O(\frac{(1+\omega)^{3/2}\sqrt{N}}{S\epsilon^2}+\frac{(1+\omega)N^{2/3}}{S\epsilon^2})$ bound on the number of communication rounds of COFIG in the nonconvex setting, where $N$ is the total number of clients, $S$ is the number of clients participating in each round, $\epsilon$ is the convergence error, and $\omega$ is the variance parameter associated with the compression operator. In case of FRECON, we prove an $O(\frac{(1+\omega)\sqrt{N}}{S\epsilon^2})$ bound on the number of communication rounds. In the convex setting, COFIG converges within $O(\frac{(1+\omega)\sqrt{N}}{S\epsilon})$ communication rounds, which, to the best of our knowledge, is also the first convergence result for compression schemes that do not communicate with all the clients in each round. We stress that neither COFIG nor FRECON needs to communicate with all the clients, and they enjoy the first or faster convergence results for convex and nonconvex federated learning in the regimes considered. Experimental results point to an empirical superiority of COFIG and FRECON over existing baselines.

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in settings with large amounts of experimental data. Unfortunately, data-driven methods often fail to discover valid laws when data is noisy or scarce. Accordingly, recent works combine regression and reasoning to eliminate formulae inconsistent with background theory. However, the problem of searching over the space of formulae consistent with background theory to find one that fits the data best is not well-solved. We propose a solution to this problem when all axioms and scientific laws are expressible via polynomial equalities and inequalities and argue that our approach is widely applicable. We further model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and prove the validity of our scientific discoveries in a principled manner using Positivestellensatz certificates. Remarkably, the optimization techniques leveraged in this paper allow our approach to run in polynomial time with fully correct background theory, or non-deterministic polynomial (NP) time with partially correct background theory. We demonstrate that some famous scientific laws, including Kepler's Third Law of Planetary Motion, the Hagen-Poiseuille Equation, and the Radiated Gravitational Wave Power equation, can be derived in a principled manner from background axioms and experimental data.

Chance constraints are frequently used to limit the probability of constraint violations in real-world optimization problems where the constraints involve stochastic components. We study chance-constrained submodular optimization problems, which capture a wide range of optimization problems with stochastic constraints. Previous studies considered submodular problems with stochastic knapsack constraints in the case where uncertainties are the same for each item that can be selected. However, uncertainty levels are usually variable with respect to the different stochastic components in real-world scenarios, and rigorous analysis for this setting is missing in the context of submodular optimization. This paper provides the first such analysis for this case, where the weights of items have the same expectation but different dispersion. We present greedy algorithms that can obtain a high-quality solution, i.e., a constant approximation ratio to the given optimal solution from the deterministic setting. In the experiments, we demonstrate that the algorithms perform effectively on several chance-constrained instances of the maximum coverage problem and the influence maximization problem.

Ensuring safety is important for the practical deployment of reinforcement learning (RL). Various challenges must be addressed, such as handling stochasticity in the environments, providing rigorous guarantees of persistent state-wise safety satisfaction, and avoiding overly conservative behaviors that sacrifice performance. We propose a new framework, Reachability Estimation for Safe Policy Optimization (RESPO), for safety-constrained RL in general stochastic settings. In the feasible set where there exist violation-free policies, we optimize for rewards while maintaining persistent safety. Outside this feasible set, our optimization produces the safest behavior by guaranteeing entrance into the feasible set whenever possible with the least cumulative discounted violations. We introduce a class of algorithms using our novel reachability estimation function to optimize in our proposed framework and in similar frameworks such as those concurrently handling multiple hard and soft constraints. We theoretically establish that our algorithms almost surely converge to locally optimal policies of our safe optimization framework. We evaluate the proposed methods on a diverse suite of safe RL environments from Safety Gym, PyBullet, and MuJoCo, and show the benefits in improving both reward performance and safety compared with state-of-the-art baselines.

With distributed machine learning being a prominent technique for large-scale machine learning tasks, communication complexity has become a major bottleneck for speeding up training and scaling up machine numbers. In this paper, we propose a new technique named Common randOm REconstruction(CORE), which can be used to compress the information transmitted between machines in order to reduce communication complexity without other strict conditions. Especially, our technique CORE projects the vector-valued information to a low-dimensional one through common random vectors and reconstructs the information with the same random noises after communication. We apply CORE to two distributed tasks, respectively convex optimization on linear models and generic non-convex optimization, and design new distributed algorithms, which achieve provably lower communication complexities. For example, we show for linear models CORE-based algorithm can encode the gradient vector to $\mathcal{O}(1)$-bits (against $\mathcal{O}(d)$), with the convergence rate not worse, preceding the existing results.

Population-based evolutionary algorithms are often considered when approaching computationally expensive black-box optimization problems. They employ a selection mechanism to choose the best solutions from a given population after comparing their objective values, which are then used to generate the next population. This iterative process explores the solution space efficiently, leading to improved solutions over time. However, these algorithms require a large number of evaluations to provide a quality solution, which might be computationally expensive when the evaluation cost is high. In some cases, it is possible to replace the original objective function with a less accurate approximation of lower cost. This introduces a trade-off between the evaluation cost and its accuracy. In this paper, we propose a technique capable of choosing an appropriate approximate function cost during the execution of the optimization algorithm. The proposal finds the minimum evaluation cost at which the solutions are still properly ranked, and consequently, more evaluations can be computed in the same amount of time with minimal accuracy loss. An experimental section on four very different problems reveals that the proposed approach can reach the same objective value in less than half of the time in certain cases.

Linear scalarization, i.e., combining all loss functions by a weighted sum, has been the default choice in the literature of multi-task learning (MTL) since its inception. In recent years, there is a surge of interest in developing Specialized Multi-Task Optimizers (SMTOs) that treat MTL as a multi-objective optimization problem. However, it remains open whether there is a fundamental advantage of SMTOs over scalarization. In fact, heated debates exist in the community comparing these two types of algorithms, mostly from an empirical perspective. To approach the above question, in this paper, we revisit scalarization from a theoretical perspective. We focus on linear MTL models and study whether scalarization is capable of fully exploring the Pareto front. Our findings reveal that, in contrast to recent works that claimed empirical advantages of scalarization, scalarization is inherently incapable of full exploration, especially for those Pareto optimal solutions that strike the balanced trade-offs between multiple tasks. More concretely, when the model is under-parametrized, we reveal a multi-surface structure of the feasible region and identify necessary and sufficient conditions for full exploration. This leads to the conclusion that scalarization is in general incapable of tracing out the Pareto front. Our theoretical results partially answer the open questions in Xin et al. (2021), and provide a more intuitive explanation on why scalarization fails beyond non-convexity. We additionally perform experiments on a real-world dataset using both scalarization and state-of-the-art SMTOs. The experimental results not only corroborate our theoretical findings, but also unveil the potential of SMTOs in finding balanced solutions, which cannot be achieved by scalarization.

Multiphysics simulations that involve multiple coupled physical phenomena quickly become computationally expensive. This imposes challenges for practitioners aiming to find optimal configurations for these problems satisfying multiple objectives, as optimization algorithms often require querying the simulation many times. This paper presents a methodological framework for training, self-optimizing, and self-organizing surrogate models to approximate and speed up Multiphysics simulations. We generate two real-world tabular datasets, which we make publicly available, and show that surrogate models can be trained on relatively small amounts of data to approximate the underlying simulations accurately. We conduct extensive experiments combining four machine learning and deep learning algorithms with two optimization algorithms and a comprehensive evaluation strategy. Finally, we evaluate the performance of our combined training and optimization pipeline by verifying the generated Pareto-optimal results using the ground truth simulations. We also employ explainable AI techniques to analyse our surrogates and conduct a preselection strategy to determine the most relevant features in our real-world examples. This approach lets us understand the underlying problem and identify critical partial dependencies.