PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. For doing that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modelling parts, and the worst-case analysis is performed numerically via a standard solver.
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We present an easily accessible, object oriented code (written exclusively in Matlab) for finite element simulations in 2D. The object oriented programming paradigm allows for fast implementation of higher-order FEM on triangular meshes for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.
We consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as the Subsampled Newton and Newton Sketch, which can efficiently construct stochastic Hessian estimates for many tasks. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme enjoys local $Q$-superlinear convergence with a non-asymptotic rate of $(\Upsilon\sqrt{\log (t)/t}\,)^{t}$, where $\Upsilon$ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the iteration, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still enjoys a superlinear convergence~rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.
We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.
We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in $\tilde{O}(D+\sqrt{n})$ rounds in the standard CONGEST model (where $n$ is the network size and $D$ is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as ad hoc wireless and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the \emph{sleeping model} [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round -- \emph{sleeping} or \emph{awake} (unlike the traditional model where nodes are always awake). Only the rounds in which a node is \emph{awake} are counted, while \emph{sleeping} rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the \emph{awake complexity} of a distributed algorithm, the worst-case number of rounds any node is awake. We present deterministic and randomized distributed MST algorithms that have an \emph{optimal} awake complexity of $O(\log n)$ time with a matching lower bound. We also show that our randomized awake-optimal algorithm has essentially the best possible round complexity by presenting a lower bound of $\tilde{\Omega}(n)$ on the product of the awake and round complexity of any distributed algorithm (including randomized) that outputs an MST, where $\tilde{\Omega}$ hides a $1/(\text{polylog } n)$ factor.
Given two strings $T$ and $S$ and a set of strings $P$, for each string $p \in P$, consider the unique substrings of $T$ that have $p$ as their prefix and $S$ as their suffix. Two problems then come to mind; the first problem being the counting of such substrings, and the second problem being the problem of listing all such substrings. In this paper, we describe linear-time, linear-space suffix tree-based algorithms for both problems. More specifically, we describe an $O(|T| + |P|)$ time algorithm for the counting problem, and an $O(|T| + |P| + \#(ans))$ time algorithm for the listing problem, where $\#(ans)$ refers to the number of strings being listed in total, and $|P|$ refers to the total length of the strings in $P$. We also consider the reversed version of the problems, where one prefix condition string and multiple suffix condition strings are given instead, and similarly describe linear-time, linear-space algorithms to solve them.
We propose a stochastic conditional gradient method (CGM) for minimizing convex finite-sum objectives formed as a sum of smooth and non-smooth terms. Existing CGM variants for this template either suffer from slow convergence rates, or require carefully increasing the batch size over the course of the algorithm's execution, which leads to computing full gradients. In contrast, the proposed method, equipped with a stochastic average gradient (SAG) estimator, requires only one sample per iteration. Nevertheless, it guarantees fast convergence rates on par with more sophisticated variance reduction techniques. In applications we put special emphasis on problems with a large number of separable constraints. Such problems are prevalent among semidefinite programming (SDP) formulations arising in machine learning and theoretical computer science. We provide numerical experiments on matrix completion, unsupervised clustering, and sparsest-cut SDPs.
One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.
Enforcing fairness in private federated learning via the modified method of differential multipliers
Federated learning with differential privacy, or private federated learning, provides a strategy to train machine learning models while respecting users' privacy. However, differential privacy can disproportionately degrade the performance of the models on under-represented groups, as these parts of the distribution are difficult to learn in the presence of noise. Existing approaches for enforcing fairness in machine learning models have considered the centralized setting, in which the algorithm has access to the users' data. This paper introduces an algorithm to enforce group fairness in private federated learning, where users' data does not leave their devices. First, the paper extends the modified method of differential multipliers to empirical risk minimization with fairness constraints, thus providing an algorithm to enforce fairness in the central setting. Then, this algorithm is extended to the private federated learning setting. The proposed algorithm, \texttt{FPFL}, is tested on a federated version of the Adult dataset and an "unfair" version of the FEMNIST dataset. The experiments on these datasets show how private federated learning accentuates unfairness in the trained models, and how FPFL is able to mitigate such unfairness.
Sufficient dimension reduction (SDR) is a successful tool in regression models. It is a feasible method to solve and analyze the nonlinear nature of the regression problems. This paper introduces the \textbf{itdr} R package that provides several functions based on integral transformation methods to estimate the SDR subspaces in a comprehensive and user-friendly manner. In particular, the \textbf{itdr} package includes the Fourier method (FM) and the convolution method (CM) of estimating the SDR subspaces such as the central mean subspace (CMS) and the central subspace (CS). In addition, the \textbf{itdr} package facilitates the recovery of the CMS and the CS by using the iterative Hessian transformation (IHT) method and the Fourier transformation approach for inverse dimension reduction method (invFM), respectively. Moreover, the use of the package is illustrated by three datasets. \textcolor{black}{Furthermore, this is the first package that implements integral transformation methods to estimate SDR subspaces. Hence, the \textbf{itdr} package may provide a huge contribution to research in the SDR field.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
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