Stochastic alternating algorithms for bi-objective optimization are considered when optimizing two conflicting functions for which optimization steps have to be applied separately for each function. Such algorithms consist of applying a certain number of steps of gradient or subgradient descent on each single objective at each iteration. In this paper, we show that stochastic alternating algorithms achieve a sublinear convergence rate of $\mathcal{O}(1/T)$, under strong convexity, for the determination of a minimizer of a weighted-sum of the two functions, parameterized by the number of steps applied on each of them. An extension to the convex case is presented for which the rate weakens to $\mathcal{O}(1/\sqrt{T})$. These rates are valid also in the non-smooth case. Importantly, by varying the proportion of steps applied to each function, one can determine an approximation to the Pareto front.

We consider the constrained Linear Inverse Problem (LIP), where a certain atomic norm (like the $\ell_1 $ and the Nuclear norm) is minimized subject to a quadratic constraint. Typically, such cost functions are non-differentiable which makes them not amenable to the fast optimization methods existing in practice. We propose two equivalent reformulations of the constrained LIP with improved convex regularity: (i) a smooth convex minimization problem, and (ii) a strongly convex min-max problem. These problems could be solved by applying existing acceleration based convex optimization methods which provide better $ O \big( \frac{1}{k^2} \big) $ theoretical convergence guarantee. However, to fully exploit the utility of these reformulations, we also provide a novel algorithm, to which we refer as the Fast Linear Inverse Problem Solver (FLIPS), that is tailored to solve the reformulation of the LIP. We demonstrate the performance of FLIPS on the sparse coding problem arising in image processing tasks. In this setting, we observe that FLIPS consistently outperforms the Chambolle-Pock and C-SALSA algorithms--two of the current best methods in the literature.

The celebrated FedAvg algorithm of McMahan et al. (2017) is based on three components: client sampling (CS), data sampling (DS) and local training (LT). While the first two are reasonably well understood, the third component, whose role is to reduce the number of communication rounds needed to train the model, resisted all attempts at a satisfactory theoretical explanation. Malinovsky et al. (2022) identified four distinct generations of LT methods based on the quality of the provided theoretical communication complexity guarantees. Despite a lot of progress in this area, none of the existing works were able to show that it is theoretically better to employ multiple local gradient-type steps (i.e., to engage in LT) than to rely on a single local gradient-type step only in the important heterogeneous data regime. In a recent breakthrough embodied in their ProxSkip method and its theoretical analysis, Mishchenko et al. (2022) showed that LT indeed leads to provable communication acceleration for arbitrarily heterogeneous data, thus jump-starting the $5^{\rm th}$ generation of LT methods. However, while these latest generation LT methods are compatible with DS, none of them support CS. We resolve this open problem in the affirmative. In order to do so, we had to base our algorithmic development on new algorithmic and theoretical foundations.

Identifying functional connectivity biomarkers of major depressive disorder (MDD) patients is essential to advance understanding of the disorder mechanisms and early intervention. However, due to the small sample size and the high dimension of available neuroimaging data, the performance of existing methods is often limited. Multi-site data could enhance the statistical power and sample size, while they are often subject to inter-site heterogeneity and data-sharing policies. In this paper, we propose a federated joint estimator, NOTEARS-PFL, for simultaneous learning of multiple Bayesian networks (BNs) with continuous optimization, to identify disease-induced alterations in MDD patients. We incorporate information shared between sites and site-specific information into the proposed federated learning framework to learn personalized BN structures by introducing the group fused lasso penalty. We develop the alternating direction method of multipliers, where in the local update step, the neuroimaging data is processed at each local site. Then the learned network structures are transmitted to the center for the global update. In particular, we derive a closed-form expression for the local update step and use the iterative proximal projection method to deal with the group fused lasso penalty in the global update step. We evaluate the performance of the proposed method on both synthetic and real-world multi-site rs-fMRI datasets. The results suggest that the proposed NOTEARS-PFL yields superior effectiveness and accuracy than the comparable methods.

Minibatch decomposition methods for empirical risk minimization are commonly analysed in a stochastic approximation setting, also known as sampling with replacement. On the other hands modern implementations of such techniques are incremental: they rely on sampling without replacement, for which available analysis are much scarcer. We provide convergence guaranties for the latter variant by analysing a versatile incremental gradient scheme. For this scheme, we consider constant, decreasing or adaptive step sizes. In the smooth setting we obtain explicit complexity estimates in terms of epoch counter. In the nonsmooth setting we prove that the sequence is attracted by solutions of optimality conditions of the problem.

Sparse optimization problems are ubiquitous in many fields such as statistics, signal/image processing and machine learning. This has led to the birth of many iterative algorithms to solve them. A powerful strategy to boost the performance of these algorithms is known as safe screening: it allows the early identification of zero coordinates in the solution, which can then be eliminated to reduce the problem's size and accelerate convergence. In this work, we extend the existing Gap Safe screening framework by relaxing the global strong-concavity assumption on the dual cost function. Instead, we exploit local regularity properties, that is, strong concavity on well-chosen subsets of the domain. The non-negativity constraint is also integrated to the existing framework. Besides making safe screening possible to a broader class of functions that includes beta-divergences (e.g., the Kullback-Leibler divergence), the proposed approach also improves upon the existing Gap Safe screening rules on previously applicable cases (e.g., logistic regression). The proposed general framework is exemplified by some notable particular cases: logistic function, beta = 1.5 and Kullback-Leibler divergences. Finally, we showcase the effectiveness of the proposed screening rules with different solvers (coordinate descent, multiplicative-update and proximal gradient algorithms) and different data sets (binary classification, hyperspectral and count data).

Fair resource allocation is one of the most important topics in communication networks. Existing solutions almost exclusively assume each user utility function is known and concave. This paper seeks to answer the following question: how to allocate resources when utility functions are unknown, even to the users? This answer has become increasingly important in the next-generation AI-aware communication networks where the user utilities are complex and their closed-forms are hard to obtain. In this paper, we provide a new solution using a distributed and data-driven bilevel optimization approach, where the lower level is a distributed network utility maximization (NUM) algorithm with concave surrogate utility functions, and the upper level is a data-driven learning algorithm to find the best surrogate utility functions that maximize the sum of true network utility. The proposed algorithm learns from data samples (utility values or gradient values) to autotune the surrogate utility functions to maximize the true network utility, so works for unknown utility functions. For the general network, we establish the nonasymptotic convergence rate of the proposed algorithm with nonconcave utility functions. The simulations validate our theoretical results and demonstrate the great effectiveness of the proposed method in a real-world network.

We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle we obtain a local-to-global paradigm, namely solving a local, i.e., a one time-step robust optimization problem leads to an optimizer of the global (i.e. infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the S&P 500. We present two different types of ambiguity sets; one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

Graph matching can be formalized as a combinatorial optimization problem, where there are corresponding relationships between pairs of nodes that can be represented as edges. This problem becomes challenging when there are potential ambiguities present due to nodes and edges with high similarity, and there is a need to find accurate results for similar content matching. In this paper, we introduce a novel end-to-end neural network that can map the linear assignment problem into a high-dimensional space augmented with node-level relative position information, which is crucial for improving the method's performance for similar content matching. Our model constructs the anchor set for the relative position of nodes and then aggregates the feature information of the target node and each anchor node based on a measure of relative position. It then learns the node feature representation by integrating the topological structure and the relative position information, thus realizing the linear assignment between the two graphs. To verify the effectiveness and generalizability of our method, we conduct graph matching experiments, including cross-category matching, on different real-world datasets. Comparisons with different baselines demonstrate the superiority of our method. Our source code is available under https://github.com/anonymous.

Existing gradient-based optimization methods update parameters locally, in a direction that minimizes the loss function. We study a different approach, symmetry teleportation, that allows parameters to travel a large distance on the loss level set, in order to improve the convergence speed in subsequent steps. Teleportation exploits symmetries in the loss landscape of optimization problems. We derive loss-invariant group actions for test functions in optimization and multi-layer neural networks, and prove a necessary condition for teleportation to improve convergence rate. We also show that our algorithm is closely related to second order methods. Experimentally, we show that teleportation improves the convergence speed of gradient descent and AdaGrad for several optimization problems including test functions, multi-layer regressions, and MNIST classification.