Performing inference in graphs is a common task within several machine learning problems, e.g., image segmentation, community detection, among others. For a given undirected connected graph, we tackle the statistical problem of exactly recovering an unknown ground-truth binary labeling of the nodes from a single corrupted observation of each edge. Such problem can be formulated as a quadratic combinatorial optimization problem over the boolean hypercube, where it has been shown before that one can (with high probability and in polynomial time) exactly recover the ground-truth labeling of graphs that have an isoperimetric number that grows with respect to the number of nodes (e.g., complete graphs, regular expanders). In this work, we apply a powerful hierarchy of relaxations, known as the sum-of-squares (SoS) hierarchy, to the combinatorial problem. Motivated by empirical evidence on the improvement in exact recoverability, we center our attention on the degree-4 SoS relaxation and set out to understand the origin of such improvement from a graph theoretical perspective. We show that the solution of the dual of the relaxed problem is related to finding edge weights of the Johnson and Kneser graphs, where the weights fulfill the SoS constraints and intuitively allow the input graph to increase its algebraic connectivity. Finally, as byproduct of our analysis, we derive a novel Cheeger-type lower bound for the algebraic connectivity of graphs with signed edge weights.

Edge federated learning (FL) is an emerging machine learning paradigm that trains a global parametric model from distributed datasets via wireless communications. This paper proposes a unit-modulus over-the-air computation (UM-AirComp) framework to facilitate efficient edge federated learning, which simultaneously uploads local model parameters and updates global model parameters via analog beamforming. The proposed framework avoids sophisticated baseband signal processing, leading to low communication delays and implementation costs. A training loss bound of UM-AirComp is derived and two low-complexity algorithms, termed penalty alternating minimization (PAM) and accelerated gradient projection (AGP), are proposed to minimize the nonconvex nonsmooth loss bound. Simulation results show that the proposed UM-AirComp framework with PAM algorithm not only achieves a smaller mean square error of model parameters' estimation, training loss, and testing error, but also requires a significantly shorter runtime than that of other benchmark schemes. Moreover, the proposed UM-AirComp framework with AGP algorithm achieves satisfactory performance while reduces the computational complexity by orders of magnitude compared with existing optimization algorithms. Finally, we demonstrate the implementation of UM-AirComp in a vehicle-to-everything autonomous driving simulation platform. It is found that autonomous driving tasks are more sensitive to model parameter errors than other tasks since the former neural networks are more sophisticated containing sparser model parameters.

Natural-gradient descent on structured parameter spaces (e.g., low-rank covariances) is computationally challenging due to complicated inverse Fisher-matrix computations. We address this issue for optimization, inference, and search problems by using \emph{local-parameter coordinates}. Our method generalizes an existing evolutionary-strategy method, recovers Newton and Riemannian-gradient methods as special cases, and also yields new tractable natural-gradient algorithms for learning flexible covariance structures of Gaussian and Wishart-based distributions. We show results on a range of applications on deep learning, variational inference, and evolution strategies. Our work opens a new direction for scalable structured geometric methods via local parameterizations.

This paper proposes a new algorithm -- the Momentum-assisted Single-timescale Stochastic Approximation (MSTSA) -- for tackling unconstrained bilevel optimization problems. We focus on bilevel problems where the lower level subproblem is strongly-convex. Unlike prior works which rely on two timescale or double loop techniques that track the optimal solution to the lower level subproblem, we design a stochastic momentum assisted gradient estimator for the upper level subproblem's updates. The latter allows us to gradually control the error in stochastic gradient updates due to inaccurate solution to the lower level subproblem. We show that if the upper objective function is smooth but possibly non-convex (resp. strongly-convex), MSTSA requires $\mathcal{O}(\epsilon^{-2})$ (resp. $\mathcal{O}(\epsilon^{-1})$) iterations (each using constant samples) to find an $\epsilon$-stationary (resp. $\epsilon$-optimal) solution. This achieves the best-known guarantees for stochastic bilevel problems. We validate our theoretical results by showing the efficiency of the MSTSA algorithm on hyperparameter optimization and data hyper-cleaning problems.

White box adversarial perturbations are generated via iterative optimization algorithms most often by minimizing an adversarial loss on a $\ell_p$ neighborhood of the original image, the so-called distortion set. Constraining the adversarial search with different norms results in disparately structured adversarial examples. Here we explore several distortion sets with structure-enhancing algorithms. These new structures for adversarial examples might provide challenges for provable and empirical robust mechanisms. Because adversarial robustness is still an empirical field, defense mechanisms should also reasonably be evaluated against differently structured attacks. Besides, these structured adversarial perturbations may allow for larger distortions size than their $\ell_p$ counter-part while remaining imperceptible or perceptible as natural distortions of the image. We will demonstrate in this work that the proposed structured adversarial examples can significantly bring down the classification accuracy of adversarialy trained classifiers while showing low $\ell_2$ distortion rate. For instance, on ImagNet dataset the structured attacks drop the accuracy of adversarial model to near zero with only 50\% of $\ell_2$ distortion generated using white-box attacks like PGD. As a byproduct, our finding on structured adversarial examples can be used for adversarial regularization of models to make models more robust or improve their generalization performance on datasets which are structurally different.

In this paper, we introduce a Model-based Algorithm Tuning Engine, namely MATE, where the parameters of an algorithm are represented as expressions of the features of a target optimisation problem. In contrast to most static (feature-independent) algorithm tuning engines such as irace and SPOT, our approach aims to derive the best parameter configuration of a given algorithm for a specific problem, exploiting the relationships between the algorithm parameters and the features of the problem. We formulate the problem of finding the relationships between the parameters and the problem features as a symbolic regression problem and we use genetic programming to extract these expressions in a human-readable form. For the evaluation, we apply our approach to the configuration of the (1 1) EA and RLS algorithms for the One-Max, LeadingOnes, BinValue and Jump optimisation problems, where the theoretically optimal algorithm parameters to the problems are available as functions of the features of the problems. Our study shows that the found relationships typically comply with known theoretical results - this demonstrates (1) the potential of model-based parameter tuning as an alternative to existing static algorithm tuning engines, and (2) its potential to discover relationships between algorithm performance and instance features in human-readable form.

We consider combinatorial semi-bandits with uncorrelated Gaussian rewards. In this article, we propose the first method, to the best of our knowledge, that enables to compute the solution of the Graves-Lai optimization problem in polynomial time for many combinatorial structures of interest. In turn, this immediately yields the first known approach to implement asymptotically optimal algorithms in polynomial time for combinatorial semi-bandits.

We study multi-marginal optimal transport, a generalization of optimal transport that allows us to define discrepancies between multiple measures. It provides a framework to solve multi-task learning problems and to perform barycentric averaging. However, multi-marginal distances between multiple measures are typically challenging to compute because they require estimating a transport plan with $N^P$ variables. In this paper, we address this issue in the following way: 1) we efficiently solve the one-dimensional multi-marginal Monge-Wasserstein problem for a classical cost function in closed form, and 2) we propose a higher-dimensional multi-marginal discrepancy via slicing and study its generalized metric properties. We show that computing the sliced multi-marginal discrepancy is massively scalable for a large number of probability measures with support as large as $10^7$ samples. Our approach can be applied to solving problems such as barycentric averaging, multi-task density estimation and multi-task reinforcement learning.

Meta learning with multiple objectives can be formulated as a Multi-Objective Bi-Level optimization Problem (MOBLP) where the upper-level subproblem is to solve several possible conflicting targets for the meta learner. However, existing studies either apply an inefficient evolutionary algorithm or linearly combine multiple objectives as a single-objective problem with the need to tune combination weights. In this paper, we propose a unified gradient-based Multi-Objective Meta Learning (MOML) framework and devise the first gradient-based optimization algorithm to solve the MOBLP by alternatively solving the lower-level and upper-level subproblems via the gradient descent method and the gradient-based multi-objective optimization method, respectively. Theoretically, we prove the convergence properties of the proposed gradient-based optimization algorithm. Empirically, we show the effectiveness of the proposed MOML framework in several meta learning problems, including few-shot learning, neural architecture search, domain adaptation, and multi-task learning.

Fairness has emerged as an important concern in automated decision-making in recent years, especially when these decisions affect human welfare. In this work, we study fairness in temporally extended decision-making settings, specifically those formulated as Markov Decision Processes (MDPs). Our proposed notion of fairness ensures that each state's long-term visitation frequency is more than a specified fraction. In an average-reward MDP (AMDP) setting, we formulate the problem as a bilinear saddle point program and, for a generative model, solve it using a Stochastic Mirror Descent (SMD) based algorithm. The proposed solution guarantees a simultaneous approximation on the expected average-reward and the long-term state-visitation frequency. We validate our theoretical results with experiments on synthetic data.