Greedy algorithms have long been a workhorse for learning graphical models, and more broadly for learning statistical models with sparse structure. In the context of learning directed acyclic graphs, greedy algorithms are popular despite their worst-case exponential runtime. In practice, however, they are very efficient. We provide new insight into this phenomenon by studying a general greedy score-based algorithm for learning DAGs. Unlike edge-greedy algorithms such as the popular GES and hill-climbing algorithms, our approach is vertex-greedy and requires at most a polynomial number of score evaluations. We then show how recent polynomial-time algorithms for learning DAG models are a special case of this algorithm, thereby illustrating how these order-based algorithms can be rigourously interpreted as score-based algorithms. This observation suggests new score functions and optimality conditions based on the duality between Bregman divergences and exponential families, which we explore in detail. Explicit sample and computational complexity bounds are derived. Finally, we provide extensive experiments suggesting that this algorithm indeed optimizes the score in a variety of settings.

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. Multi-objective optimization (MOO) has been extensively used in many practical scenarios involving trade-offs between multiple objectives. For example, in automobile design (Chang, 2015), we must maximize the performance of the engine while simultaneously minimizing emissions and fuel consumption.

Recent research on fair regression focused on developing new fairness notions and approximation methods as target variables and even the sensitive attribute are continuous in the regression setting. However, all previous fair regression research assumed the training data and testing data are drawn from the same distributions. This assumption is often violated in real world due to the sample selection bias between the training and testing data. In this paper, we develop a framework for fair regression under sample selection bias when dependent variable values of a set of samples from the training data are missing as a result of another hidden process. Our framework adopts the classic Heckman model for bias correction and the Lagrange duality to achieve fairness in regression based on a variety of fairness notions. Heckman model describes the sample selection process and uses a derived variable called the Inverse Mills Ratio (IMR) to correct sample selection bias. We use fairness inequality and equality constraints to describe a variety of fairness notions and apply the Lagrange duality theory to transform the primal problem into the dual convex optimization. For the two popular fairness notions, mean difference and mean squared error difference, we derive explicit formulas without iterative optimization, and for Pearson correlation, we derive its conditions of achieving strong duality. We conduct experiments on three real-world datasets and the experimental results demonstrate the approach's effectiveness in terms of both utility and fairness metrics.

In decentralized machine learning, workers compute model updates on their local data. Because the workers only communicate with few neighbors without central coordination, these updates propagate progressively over the network. This paradigm enables distributed training on networks without all-to-all connectivity, helping to protect data privacy as well as to reduce the communication cost of distributed training in data centers. A key challenge, primarily in decentralized deep learning, remains the handling of differences between the workers' local data distributions. To tackle this challenge, we introduce the RelaySum mechanism for information propagation in decentralized learning. RelaySum uses spanning trees to distribute information exactly uniformly across all workers with finite delays depending on the distance between nodes. In contrast, the typical gossip averaging mechanism only distributes data uniformly asymptotically while using the same communication volume per step as RelaySum. We prove that RelaySGD, based on this mechanism, is independent of data heterogeneity and scales to many workers, enabling highly accurate decentralized deep learning on heterogeneous data. Our code is available at http://github.com/epfml/relaysgd.

We propose a learning framework for graph kernels, which is theoretically grounded on regularizing optimal transport. This framework provides a novel optimal transport distance metric, namely Regularized Wasserstein (RW) discrepancy, which can preserve both features and structure of graphs via Wasserstein distances on features and their local variations, local barycenters and global connectivity. Two strongly convex regularization terms are introduced to improve the learning ability. One is to relax an optimal alignment between graphs to be a cluster-to-cluster mapping between their locally connected vertices, thereby preserving the local clustering structure of graphs. The other is to take into account node degree distributions in order to better preserve the global structure of graphs. We also design an efficient algorithm to enable a fast approximation for solving the optimization problem. Theoretically, our framework is robust and can guarantee the convergence and numerical stability in optimization. We have empirically validated our method using 12 datasets against 16 state-of-the-art baselines. The experimental results show that our method consistently outperforms all state-of-the-art methods on all benchmark databases for both graphs with discrete attributes and graphs with continuous attributes.

We introduce the algorithm Bayesian Optimization (BO) with Fictitious Play (BOFiP) for the optimization of high dimensional black box functions. BOFiP decomposes the original, high dimensional, space into several sub-spaces defined by non-overlapping sets of dimensions. These sets are randomly generated at the start of the algorithm, and they form a partition of the dimensions of the original space. BOFiP searches the original space with alternating BO, within sub-spaces, and information exchange among sub-spaces, to update the sub-space function evaluation. The basic idea is to distribute the high dimensional optimization across low dimensional sub-spaces, where each sub-space is a player in an equal interest game. At each iteration, BO produces approximate best replies that update the players belief distribution. The belief update and BO alternate until a stopping condition is met. High dimensional problems are common in real applications, and several contributions in the BO literature have highlighted the difficulty in scaling to high dimensions due to the computational complexity associated to the estimation of the model hyperparameters. Such complexity is exponential in the problem dimension, resulting in substantial loss of performance for most techniques with the increase of the input dimensionality. We compare BOFiP to several state-of-the-art approaches in the field of high dimensional black box optimization. The numerical experiments show the performance over three benchmark objective functions from 20 up to 1000 dimensions. A neural network architecture design problem is tested with 42 up to 911 nodes in 6 up to 92 layers, respectively, resulting into networks with 500 up to 10,000 weights. These sets of experiments empirically show that BOFiP outperforms its competitors, showing consistent performance across different problems and increasing problem dimensionality.

It is difficult to optimize a giant model with deep and wider layers. Similar to most optimization algorithms, training a deep model with gradient method (SGD-like Algorithms) has disadvantages such as easy to fall into local minima or saddle point and slow convergence speed. There have been a lot of researches on the improvement of the gradient method, and a considerable part of these researches focus on how to refine the search direction while keeping the iteration cost as low as possible to accelerate the convergence of the algorithm[10, 11, 12, 13, 14, 15, 16]. These improvements for the search direction are roughly divided into two categories. One is the momentum method[11] based on the principles of physics and the corresponding improved algorithms[12, 20, 21], the momentum method avoids excessive swing amplitude of the search track by retaining part of the potential energy of the original track to accelerate the convergence.

Traceless Genetic Programming (TGP) is a Genetic Programming (GP) variant that is used in cases where the focus is rather the output of the program than the program itself. The main difference between TGP and other GP techniques is that TGP does not explicitly store the evolved computer programs. Two genetic operators are used in conjunction with TGP: crossover and insertion. In this paper, we shall focus on how to apply TGP for solving multi-objective optimization problems which are quite unusual for GP. Each TGP individual stores the output of a computer program (tree) representing a point in the search space. Numerical experiments show that TGP is able to solve very fast and very well the considered test problems.

An agent's functionality is largely determined by its design, i.e., skeletal structure and joint attributes (e.g., length, size, strength). However, finding the optimal agent design for a given function is extremely challenging since the problem is inherently combinatorial and the design space is prohibitively large. Additionally, it can be costly to evaluate each candidate design which requires solving for its optimal controller. To tackle these problems, our key idea is to incorporate the design procedure of an agent into its decision-making process. Specifically, we learn a conditional policy that, in an episode, first applies a sequence of transform actions to modify an agent's skeletal structure and joint attributes, and then applies control actions under the new design. To handle a variable number of joints across designs, we use a graph-based policy where each graph node represents a joint and uses message passing with its neighbors to output joint-specific actions. Using policy gradient methods, our approach enables first-order optimization of agent design and control as well as experience sharing across different designs, which improves sample efficiency tremendously. Experiments show that our approach, Transform2Act, outperforms prior methods significantly in terms of convergence speed and final performance. Notably, Transform2Act can automatically discover plausible designs similar to giraffes, squids, and spiders. Our project website is at https://sites.google.com/view/transform2act.

Multi-task learning (MTL), which aims to improve performance by learning multiple tasks simultaneously, inherently presents an optimization challenge due to multiple objectives. Hence, multi-objective optimization (MOO) approaches have been proposed for multitasking problems. Recent MOO methods approximate multiple optimal solutions (Pareto front) with a single unified model, which is collectively referred to as Pareto front learning (PFL). In this paper, we show that PFL can be re-formulated into another MOO problem with multiple objectives, each of which corresponds to different preference weights for the tasks. We leverage an evolutionary algorithm (EA) to propose a method for PFL called self-evolutionary optimization (SEO) by directly maximizing the hypervolume. By using SEO, the neural network learns to approximate the Pareto front conditioned on multiple hyper-parameters that drastically affect the hypervolume. Then, by generating a population of approximations simply by inferencing the network, the hyper-parameters of the network can be optimized by EA. Utilizing SEO for PFL, we also introduce self-evolutionary Pareto networks (SEPNet), enabling the unified model to approximate the entire Pareto front set that maximizes the hypervolume. Extensive experimental results confirm that SEPNet can find a better Pareto front than the current state-of-the-art methods while minimizing the increase in model size and training cost.