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In contrast to single-objective optimization (SOO), multi-objective optimization (MOO) requires an optimizer to find the Pareto frontier, a subset of feasible solutions that are not dominated by other feasible solutions. In this paper, we propose LaMOO, a novel multi-objective optimizer that learns a model from observed samples to partition the search space and then focus on promising regions that are likely to contain a subset of the Pareto frontier. The partitioning is based on the dominance number, which measures "how close" a data point is to the Pareto frontier among existing samples. To account for possible partition errors due to limited samples and model mismatch, we leverage Monte Carlo Tree Search (MCTS) to exploit promising regions while exploring suboptimal regions that may turn out to contain good solutions later. Theoretically, we prove the efficacy of learning space partitioning via LaMOO under certain assumptions. Empirically, on the HyperVolume (HV) benchmark, a popular MOO metric, LaMOO substantially outperforms strong baselines on multiple real-world MOO tasks, by up to 225% in sample efficiency for neural architecture search on Nasbench201, and up to 10% for molecular design.

We consider unconstrained optimization problems with nonsmooth and convex objective function in the form of mathematical expectation. The proposed method approximates the objective function with a sample average function by using different sample size in each iteration. The sample size is chosen in an adaptive manner based on the Inexact Restoration. The method uses line search and assumes descent directions with respect to the current approximate function. We prove the almost sure convergence under the standard assumptions. The convergence rate is also considered and the worst-case complexity of $\mathcal{O} (\varepsilon^{-2})$ is proved. Numerical results for two types of problems, machine learning hinge loss and stochastic linear complementarity problems, show the efficiency of the proposed scheme.

In this paper, we propose a new zero order optimization method called minibatch stochastic three points (MiSTP) method to solve an unconstrained minimization problem in a setting where only an approximation of the objective function evaluation is possible. It is based on the recently proposed stochastic three points (STP) method (Bergou et al., 2020). At each iteration, MiSTP generates a random search direction in a similar manner to STP, but chooses the next iterate based solely on the approximation of the objective function rather than its exact evaluations. We also analyze our method's complexity in the nonconvex and convex cases and evaluate its performance on multiple machine learning tasks.

Privacy has become a major concern in machine learning. In fact, the federated learning is motivated by the privacy concern as it does not allow to transmit the private data but only intermediate updates. However, federated learning does not always guarantee privacy-preservation as the intermediate updates may also reveal sensitive information. In this paper, we give an explicit information-theoretical analysis of a federated expectation maximization algorithm for Gaussian mixture model and prove that the intermediate updates can cause severe privacy leakage. To address the privacy issue, we propose a fully decentralized privacy-preserving solution, which is able to securely compute the updates in each maximization step. Additionally, we consider two different types of security attacks: the honest-but-curious and eavesdropping adversary models. Numerical validation shows that the proposed approach has superior performance compared to the existing approach in terms of both the accuracy and privacy level.

The notion of omnipredictors (Gopalan, Kalai, Reingold, Sharan and Wieder ITCS 2021), suggested a new paradigm for loss minimization. Rather than learning a predictor based on a known loss function, omnipredictors can easily be post-processed to minimize any one of a rich family of loss functions compared with the loss of a class $C$. It has been shown that such omnipredictors exist and are implied (for all convex and Lipschitz loss functions) by the notion of multicalibration from the algorithmic fairness literature. Nevertheless, it is often the case that the action selected must obey some additional constraints (such as capacity or parity constraints). In itself, the original notion of omnipredictors does not apply in this well-motivated and heavily studied the context of constrained loss minimization. In this paper, we introduce omnipredictors for constrained optimization and study their complexity and implications. The notion that we introduce allows the learner to be unaware of the loss function that will be later assigned as well as the constraints that will be later imposed, as long as the subpopulations that are used to define these constraints are known. The paper shows how to obtain omnipredictors for constrained optimization problems, relying on appropriate variants of multicalibration. For some interesting constraints and general loss functions and for general constraints and some interesting loss functions, we show how omnipredictors are implied by a variant of multicalibration that is similar in complexity to standard multicalibration. We demonstrate that in the general case, standard multicalibration is insufficient and show that omnipredictors are implied by multicalibration with respect to a class containing all the level sets of hypotheses in $C$. We also investigate the implications when the constraints are group fairness notions.

Multiple-objective optimization (MOO) aims to simultaneously optimize multiple conflicting objectives and has found important applications in machine learning, such as minimizing classification loss and discrepancy in treating different populations for fairness. At optimality, further optimizing one objective will necessarily harm at least another objective, and decision-makers need to comprehensively explore multiple optima (called Pareto front) to pinpoint one final solution. We address the efficiency of finding the Pareto front. First, finding the front from scratch using stochastic multi-gradient descent (SMGD) is expensive with large neural networks and datasets. We propose to explore the Pareto front as a manifold from a few initial optima, based on a predictor-corrector method. Second, for each exploration step, the predictor solves a large-scale linear system that scales quadratically in the number of model parameters and requires one backpropagation to evaluate a second-order Hessian-vector product per iteration of the solver. We propose a Gauss-Newton approximation that only scales linearly, and that requires only first-order inner-product per iteration. This also allows for a choice between the MINRES and conjugate gradient methods when approximately solving the linear system. The innovations make predictor-corrector possible for large networks. Experiments on multi-objective (fairness and accuracy) misinformation detection tasks show that 1) the predictor-corrector method can find Pareto fronts better than or similar to SMGD with less time; and 2) the proposed first-order method does not harm the quality of the Pareto front identified by the second-order method, while further reduce running time.

We study the problem of high-dimensional sparse linear regression in a distributed setting under both computational and communication constraints. Specifically, we consider a star topology network whereby several machines are connected to a fusion center, with whom they can exchange relatively short messages. Each machine holds noisy samples from a linear regression model with the same unknown sparse $d$-dimensional vector of regression coefficients $\theta$. The goal of the fusion center is to estimate the vector $\theta$ and its support using few computations and limited communication at each machine. In this work, we consider distributed algorithms based on Orthogonal Matching Pursuit (OMP) and theoretically study their ability to exactly recover the support of $\theta$. We prove that under certain conditions, even at low signal-to-noise-ratios where individual machines are unable to detect the support of $\theta$, distributed-OMP methods correctly recover it with total communication sublinear in $d$. In addition, we present simulations that illustrate the performance of distributed OMP-based algorithms and show that they perform similarly to more sophisticated and computationally intensive methods, and in some cases even outperform them.

Safe reinforcement learning (RL) studies problems where an intelligent agent has to not only maximize reward but also avoid exploring unsafe areas. In this study, we propose CUP, a novel policy optimization method based on Constrained Update Projection framework that enjoys rigorous safety guarantee. Central to our CUP development is the newly proposed surrogate functions along with the performance bound. Compared to previous safe RL methods, CUP enjoys the benefits of 1) CUP generalizes the surrogate functions to generalized advantage estimator (GAE), leading to strong empirical performance. 2) CUP unifies performance bounds, providing a better understanding and interpretability for some existing algorithms; 3) CUP provides a non-convex implementation via only first-order optimizers, which does not require any strong approximation on the convexity of the objectives. To validate our CUP method, we compared CUP against a comprehensive list of safe RL baselines on a wide range of tasks. Experiments show the effectiveness of CUP both in terms of reward and safety constraint satisfaction. We have opened the source code of CUP at //github.com/RL-boxes/Safe-RL/tree/ main/CUP.

Despite the success of large-scale empirical risk minimization (ERM) at achieving high accuracy across a variety of machine learning tasks, fair ERM is hindered by the incompatibility of fairness constraints with stochastic optimization. We consider the problem of fair classification with discrete sensitive attributes and potentially large models and data sets, requiring stochastic solvers. Existing in-processing fairness algorithms are either impractical in the large-scale setting because they require large batches of data at each iteration or they are not guaranteed to converge. In this paper, we develop the first stochastic in-processing fairness algorithm with guaranteed convergence. For demographic parity, equalized odds, and equal opportunity notions of fairness, we provide slight variations of our algorithm--called FERMI--and prove that each of these variations converges in stochastic optimization with any batch size. Empirically, we show that FERMI is amenable to stochastic solvers with multiple (non-binary) sensitive attributes and non-binary targets, performing well even with minibatch size as small as one. Extensive experiments show that FERMI achieves the most favorable tradeoffs between fairness violation and test accuracy across all tested setups compared with state-of-the-art baselines for demographic parity, equalized odds, equal opportunity. These benefits are especially significant with small batch sizes and for non-binary classification with large number of sensitive attributes, making FERMI a practical fairness algorithm for large-scale problems.

The aim of this work is to develop a fully-distributed algorithmic framework for training graph convolutional networks (GCNs). The proposed method is able to exploit the meaningful relational structure of the input data, which are collected by a set of agents that communicate over a sparse network topology. After formulating the centralized GCN training problem, we first show how to make inference in a distributed scenario where the underlying data graph is split among different agents. Then, we propose a distributed gradient descent procedure to solve the GCN training problem. The resulting model distributes computation along three lines: during inference, during back-propagation, and during optimization. Convergence to stationary solutions of the GCN training problem is also established under mild conditions. Finally, we propose an optimization criterion to design the communication topology between agents in order to match with the graph describing data relationships. A wide set of numerical results validate our proposal. To the best of our knowledge, this is the first work combining graph convolutional neural networks with distributed optimization.