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The problem of identifying algorithmic recourse for people affected by machine learning model decisions has received much attention recently. Some recent works model user-incurred cost, which is directly linked to user satisfaction. But they assume a single global cost function that is shared across all users. This is an unrealistic assumption when users have dissimilar preferences about their willingness to act upon a feature and different costs associated with changing that feature. In this work, we formalize the notion of user-specific cost functions and introduce a new method for identifying actionable recourses for users. By default, we assume that users' cost functions are hidden from the recourse method, though our framework allows users to partially or completely specify their preferences or cost function. We propose an objective function, Expected Minimum Cost (EMC), based on two key ideas: (1) when presenting a set of options to a user, it is vital that there is at least one low-cost solution the user could adopt; (2) when we do not know the user's true cost function, we can approximately optimize for user satisfaction by first sampling plausible cost functions, then finding a set that achieves a good cost for the user in expectation. We optimize EMC with a novel discrete optimization algorithm, Cost-Optimized Local Search (COLS), which is guaranteed to improve the recourse set quality over iterations. Experimental evaluation on popular real-world datasets with simulated user costs demonstrates that our method satisfies up to 25.89 percentage points more users compared to strong baseline methods. Using standard fairness metrics, we also show that our method can provide more fair solutions across demographic groups than comparable methods, and we verify that our method is robust to misspecification of the cost function distribution.

In this work we investigate stochastic non-convex optimization problems where the objective is an expectation over smooth loss functions, and the goal is to find an approximate stationary point. The most popular approach to handling such problems is variance reduction techniques, which are also known to obtain tight convergence rates, matching the lower bounds in this case. Nevertheless, these techniques require a careful maintenance of anchor points in conjunction with appropriately selected "mega-batchsizes". This leads to a challenging hyperparameter tuning problem, that weakens their practicality. Recently, [Cutkosky and Orabona, 2019] have shown that one can employ recursive momentum in order to avoid the use of anchor points and large batchsizes, and still obtain the optimal rate for this setting. Yet, their method called STORM crucially relies on the knowledge of the smoothness, as well a bound on the gradient norms. In this work we propose STORM+, a new method that is completely parameter-free, does not require large batch-sizes, and obtains the optimal $O(1/T^{1/3})$ rate for finding an approximate stationary point. Our work builds on the STORM algorithm, in conjunction with a novel approach to adaptively set the learning rate and momentum parameters.

Due to the explosion in the size of the training datasets, distributed learning has received growing interest in recent years. One of the major bottlenecks is the large communication cost between the central server and the local workers. While error feedback compression has been proven to be successful in reducing communication costs with stochastic gradient descent (SGD), there are much fewer attempts in building communication-efficient adaptive gradient methods with provable guarantees, which are widely used in training large-scale machine learning models. In this paper, we propose a new communication-compressed AMSGrad for distributed nonconvex optimization problem, which is provably efficient. Our proposed distributed learning framework features an effective gradient compression strategy and a worker-side model update design. We prove that the proposed communication-efficient distributed adaptive gradient method converges to the first-order stationary point with the same iteration complexity as uncompressed vanilla AMSGrad in the stochastic nonconvex optimization setting. Experiments on various benchmarks back up our theory.

The problem of finding the unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low rank matrix $Y\in \mathbb{R}^{p\times n}$ that admits a sparse representation. Specifically, we consider $Y = A X\in \mathbb{R}^{p\times n}$ where the matrix $A\in \mathbb{R}^{p\times r}$ has full column rank, with $r < \min\{n,p\}$, and the matrix $X\in \mathbb{R}^{r\times n}$ is element-wise sparse. We prove that this sparse decomposition of $Y$ can be uniquely identified, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for the nonconvex optimization landscape shows that any {\em strict} local solution is close to the ground truth solution, and can be recovered by a simple data-driven initialization followed with any second order descent algorithm. At last, we corroborate these theoretical results with numerical experiments.

The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geometric rate of convergence for target measures that are nearly atomic. Furthermore, we show these algorithms perform comparably to the theoretical best possible sampling algorithm under the maximum mean discrepancy. An analysis is also conducted in a distributed setting. Our theoretical developments are supported by empirical observations on simulated data as well as a real world application.

We consider sequential optimization of an unknown function in a reproducing kernel Hilbert space. We propose a Gaussian process-based algorithm and establish its order-optimal regret performance (up to a poly-logarithmic factor). This is the first GP-based algorithm with an order-optimal regret guarantee. The proposed algorithm is rooted in the methodology of domain shrinking realized through a sequence of tree-based region pruning and refining to concentrate queries in increasingly smaller high-performing regions of the function domain. The search for high-performing regions is localized and guided by an iterative estimation of the optimal function value to ensure both learning efficiency and computational efficiency. Compared with the prevailing GP-UCB family of algorithms, the proposed algorithm reduces computational complexity by a factor of $O(T^{2d-1})$ (where $T$ is the time horizon and $d$ the dimension of the function domain).

This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then we solve the inverse problem by applying the proposed method to its equivalent nonlinear matrix equation and a preconditioner for the perturbed normal Riemannian Newton equation is also constructed. Numerical tests show the efficiency of the proposed method for solving the inverse problem.

In this paper, we design and analyze a Hybrid-High Order (HHO) approximation for a class of quasilinear elliptic problems of nonmonotone type. The proposed method has several advantages, for instance, it supports arbitrary order of approximation and general polytopal meshes. The key ingredients involve local reconstruction and high-order stabilization terms. Existence and uniqueness of the discrete solution are shown by Brouwer's fixed point theorem and contraction result. A priori error estimate is shown in discrete energy norm that shows optimal order convergence rate. Numerical experiments are performed to substantiate the theoretical results.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

We propose a geometric convexity shape prior preservation method for variational level set based image segmentation methods. Our method is built upon the fact that the level set of a convex signed distanced function must be convex. This property enables us to transfer a complicated geometrical convexity prior into a simple inequality constraint on the function. An active set based Gauss-Seidel iteration is used to handle this constrained minimization problem to get an efficient algorithm. We apply our method to region and edge based level set segmentation models including Chan-Vese (CV) model with guarantee that the segmented region will be convex. Experimental results show the effectiveness and quality of the proposed model and algorithm.