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Daniel P. Robinson

Daniel P. Robinson appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Machine Learningmath.OCComputer Visioneess.SPInformation Theorymath.IT

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Published work

15 published item(s)

preprint2026arXiv

A Proximal-Gradient Method for Solving Regularized Optimization Problems with General Constraints

We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step acceptance or rejection. Under various assumptions, we establish a worst-case iteration complexity result, prove that limit points are first-order KKT points, and show that manifold identification and active-set identification properties hold. Preliminary numerical experiments on a subset of the CUTEst test problems and sparse canonical correlation analysis problems demonstrate the promising performance of our approach.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Robust Server Defense Against Unreliable Clients in One-Shot Fair Collaborative Machine Learning

Collaborative machine learning (CML) enables multiple clients to train a global model jointly in a data-distributed setting. To address data privacy and communication efficiency, one-shot CML has been increasingly adopted, where clients communicate with the server only once by sharing synthetic or processed proxy data. This single-round communication, however, eliminates the possibility of iterative correction at the server, making the learning process particularly vulnerable to client unreliability. In this setting, unreliable clients, whether malicious or non-malicious, may provide biased proxy data that favors certain groups, thereby degrading the fairness of the global model and harming minority or unprivileged groups. In this work, we propose a server-side defense framework based on a bilevel optimization formulation. The proposed approach learns client-level weights to mitigate the influence of biased client proxy data while enforcing fairness constraints by using a very small trusted root dataset available at the server. Experimental results on benchmark datasets show that our method improves fairness with little accuracy loss under biased proxy data contributions from unreliable clients. Moreover, the proposed approach remains effective even when unreliable clients make up a majority of the system, consistently outperforming other existing methods.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Worst-Case Complexity of an SQP Method for Nonlinear Equality Constrained Stochastic Optimization

A worst-case complexity bound is proved for a sequential quadratic optimization (commonly known as SQP) algorithm that has been designed for solving optimization problems involving a stochastic objective function and deterministic nonlinear equality constraints. Barring additional terms that arise due to the adaptivity of the monotonically nonincreasing merit parameter sequence, the proved complexity bound is comparable to that known for the stochastic gradient algorithm for unconstrained nonconvex optimization. The overall complexity bound, which accounts for the adaptivity of the merit parameter sequence, shows that a result comparable to the unconstrained setting (with additional logarithmic factors) holds with high probability.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Subspace Acceleration Method for Minimization Involving a Group Sparsity-Inducing Regularizer

We consider the problem of minimizing an objective function that is the sum of a convex function and a group sparsity-inducing regularizer. Problems that integrate such regularizers arise in modern machine learning applications, often for the purpose of obtaining models that are easier to interpret and that have higher predictive accuracy. We present a new method for solving such problems that utilize subspace acceleration, domain decomposition, and support identification. Our analysis shows, under common assumptions, that the iterate sequence generated by our framework is globally convergent, converges to an $ε$-approximate solution in at most $O(ε^{-(1+p)})$ (respectively, $O(ε^{-(2+p)})$) iterations for all $ε$ bounded above and large enough (respectively, all $ε$ bounded above) where $p > 0$ is an algorithm parameter, and exhibits superlinear local convergence. Preliminary numerical results for the task of binary classification based on regularized logistic regression show that our approach is efficient and robust, with the ability to outperform a state-of-the-art method.

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preprint2020arXiv

Is an Affine Constraint Needed for Affine Subspace Clustering?

Subspace clustering methods based on expressing each data point as a linear combination of other data points have achieved great success in computer vision applications such as motion segmentation, face and digit clustering. In face clustering, the subspaces are linear and subspace clustering methods can be applied directly. In motion segmentation, the subspaces are affine and an additional affine constraint on the coefficients is often enforced. However, since affine subspaces can always be embedded into linear subspaces of one extra dimension, it is unclear if the affine constraint is really necessary. This paper shows, both theoretically and empirically, that when the dimension of the ambient space is high relative to the sum of the dimensions of the affine subspaces, the affine constraint has a negligible effect on clustering performance. Specifically, our analysis provides conditions that guarantee the correctness of affine subspace clustering methods both with and without the affine constraint, and shows that these conditions are satisfied for high-dimensional data. Underlying our analysis is the notion of affinely independent subspaces, which not only provides geometrically interpretable correctness conditions, but also clarifies the relationships between existing results for affine subspace clustering.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Self-Representation Based Unsupervised Exemplar Selection in a Union of Subspaces

Finding a small set of representatives from an unlabeled dataset is a core problem in a broad range of applications such as dataset summarization and information extraction. Classical exemplar selection methods such as $k$-medoids work under the assumption that the data points are close to a few cluster centroids, and cannot handle the case where data lie close to a union of subspaces. This paper proposes a new exemplar selection model that searches for a subset that best reconstructs all data points as measured by the $\ell_1$ norm of the representation coefficients. Geometrically, this subset best covers all the data points as measured by the Minkowski functional of the subset. To solve our model efficiently, we introduce a farthest first search algorithm that iteratively selects the worst represented point as an exemplar. When the dataset is drawn from a union of independent subspaces, our method is able to select sufficiently many representatives from each subspace. We further develop an exemplar based subspace clustering method that is robust to imbalanced data and efficient for large scale data. Moreover, we show that a classifier trained on the selected exemplars (when they are labeled) can correctly classify the rest of the data points.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization

Sequential quadratic optimization algorithms are proposed for solving smooth nonlinear optimization problems with equality constraints. The main focus is an algorithm proposed for the case when the constraint functions are deterministic, and constraint function and derivative values can be computed explicitly, but the objective function is stochastic. It is assumed in this setting that it is intractable to compute objective function and derivative values explicitly, although one can compute stochastic function and gradient estimates. As a starting point for this stochastic setting, an algorithm is proposed for the deterministic setting that is modeled after a state-of-the-art line-search SQP algorithm, but uses a stepsize selection scheme based on Lipschitz constants (or adaptively estimated Lipschitz constants) in place of the line search. This sets the stage for the proposed algorithm for the stochastic setting, for which it is assumed that line searches would be intractable. Under reasonable assumptions, convergence (resp.,~convergence in expectation) from remote starting points is proved for the proposed deterministic (resp.,~stochastic) algorithm. The results of numerical experiments demonstrate the practical performance of our proposed techniques.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

What is the Largest Sparsity Pattern that Can Be Recovered by 1-Norm Minimization?

Much of the existing literature in sparse recovery is concerned with the following question: given a sparsity pattern and a corresponding regularizer, derive conditions on the dictionary under which exact recovery is possible. In this paper, we study the opposite question: given a dictionary and the 1-norm regularizer, find the largest sparsity pattern that can be recovered. We show that such a pattern is described by a mathematical object called a "maximum abstract simplicial complex", and provide two different characterizations of this object: one based on extreme points and the other based on vectors of minimal support. In addition, we show how this new framework is useful in the study of sparse recovery problems when the dictionary takes the form of a graph incidence matrix or a partial discrete Fourier transform. In case of incidence matrices, we show that the largest sparsity pattern that can be recovered is determined by the set of simple cycles of the graph. As a byproduct, we show that standard sparse recovery can be certified in polynomial time, although this is known to be NP-hard for general matrices. In the case of the partial discrete Fourier transform, our characterization of the largest sparsity pattern that can be recovered requires the unknown signal to be real and its dimension to be a prime number.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Basis Pursuit and Orthogonal Matching Pursuit for Subspace-preserving Recovery: Theoretical Analysis

Given an overcomplete dictionary $A$ and a signal $b = Ac^*$ for some sparse vector $c^*$ whose nonzero entries correspond to linearly independent columns of $A$, classical sparse signal recovery theory considers the problem of whether $c^*$ can be recovered as the unique sparsest solution to $b = A c$. It is now well-understood that such recovery is possible by practical algorithms when the dictionary $A$ is incoherent or restricted isometric. In this paper, we consider the more general case where $b$ lies in a subspace $\mathcal{S}_0$ spanned by a subset of linearly dependent columns of $A$, and the remaining columns are outside of the subspace. In this case, the sparsest representation may not be unique, and the dictionary may not be incoherent or restricted isometric. The goal is to have the representation $c$ correctly identify the subspace, i.e. the nonzero entries of $c$ should correspond to columns of $A$ that are in the subspace $\mathcal{S}_0$. Such a representation $c$ is called subspace-preserving, a key concept that has found important applications for learning low-dimensional structures in high-dimensional data. We present various geometric conditions that guarantee subspace-preserving recovery. Among them, the major results are characterized by the covering radius and the angular distance, which capture the distribution of points in the subspace and the similarity between points in the subspace and points outside the subspace, respectively. Importantly, these conditions do not require the dictionary to be incoherent or restricted isometric. By establishing that the subspace-preserving recovery problem and the classical sparse signal recovery problem are equivalent under common assumptions on the latter, we show that several of our proposed conditions are generalizations of some well-known conditions in the sparse signal recovery literature.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A Reduced-Space Algorithm for Minimizing $\ell_1$-Regularized Convex Functions

We present a new method for minimizing the sum of a differentiable convex function and an $\ell_1$-norm regularizer. The main features of the new method include: $(i)$ an evolving set of indices corresponding to variables that are predicted to be nonzero at a solution (i.e., the support); $(ii)$ a reduced-space subproblem defined in terms of the predicted support; $(iii)$ conditions that determine how accurately each subproblem must be solved, which allow for Newton, Newton-CG, and coordinate-descent techniques to be employed; $(iv)$ a computationally practical condition that determines when the predicted support should be updated; and $(v)$ a reduced proximal gradient step that ensures sufficient decrease in the objective function when it is decided that variables should be added to the predicted support. We prove a convergence guarantee for our method and demonstrate its efficiency on a large set of model prediction problems.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering

State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. $\ell_2$ and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$, $\ell_2$ and nuclear norm regularizations offer a balance between the subspace-preserving and connectedness properties, but this comes at the cost of increased computational complexity. This paper studies the geometry of the elastic net regularizer (a mixture of the $\ell_1$ and $\ell_2$ norms) and uses it to derive a provably correct and scalable active set method for finding the optimal coefficients. Our geometric analysis also provides a theoretical justification and a geometric interpretation for the balance between the connectedness (due to $\ell_2$ regularization) and subspace-preserving (due to $\ell_1$ regularization) properties for elastic net subspace clustering. Our experiments show that the proposed active set method not only achieves state-of-the-art clustering performance, but also efficiently handles large-scale datasets.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit

Subspace clustering methods based on $\ell_1$, $\ell_2$ or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions (e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, $\ell_2$ and nuclear norm regularization provide efficient closed form solutions, but require very strong assumptions to guarantee a subspace-preserving affinity, e.g., independent subspaces and uncorrupted data. In this paper we study a subspace clustering method based on orthogonal matching pursuit. We show that the method is both computationally efficient and guaranteed to give a subspace-preserving affinity under broad conditions. Experiments on synthetic data verify our theoretical analysis, and applications in handwritten digit and face clustering show that our approach achieves the best trade off between accuracy and efficiency.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Trading-Off Cost of Deployment Versus Accuracy in Learning Predictive Models

Predictive models are finding an increasing number of applications in many industries. As a result, a practical means for trading-off the cost of deploying a model versus its effectiveness is needed. Our work is motivated by risk prediction problems in healthcare. Cost-structures in domains such as healthcare are quite complex, posing a significant challenge to existing approaches. We propose a novel framework for designing cost-sensitive structured regularizers that is suitable for problems with complex cost dependencies. We draw upon a surprising connection to boolean circuits. In particular, we represent the problem costs as a multi-layer boolean circuit, and then use properties of boolean circuits to define an extended feature vector and a group regularizer that exactly captures the underlying cost structure. The resulting regularizer may then be combined with a fidelity function to perform model prediction, for example. For the challenging real-world application of risk prediction for sepsis in intensive care units, the use of our regularizer leads to models that are in harmony with the underlying cost structure and thus provide an excellent prediction accuracy versus cost tradeoff.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

A Flexible ADMM Algorithm for Big Data Applications

We present a flexible Alternating Direction Method of Multipliers (F-ADMM) algorithm for solving optimization problems involving a strongly convex objective function that is separable into $n \geq 2$ blocks, subject to (non-separable) linear equality constraints. The F-ADMM algorithm uses a Gauss-Seidel scheme to update blocks of variables, and a regularization term is added to each of the subproblems arising within F-ADMM. We prove, under common assumptions, that F-ADMM is globally convergent. We also present a special case of F-ADMM that is partially parallelizable, which makes it attractive in a big data setting. In particular, we partition the data into groups, so that each group consists of multiple blocks of variables. By applying F-ADMM to this partitioning of the data, and using a specific regularization matrix, we obtain a hybrid ADMM (H-ADMM) algorithm: the grouped data is updated in a Gauss-Seidel fashion, and the blocks within each group are updated in a Jacobi manner. Convergence of H-ADMM follows directly from the convergence properties of F-ADMM. Also, a special case of H-ADMM can be applied to functions that are convex, rather than strongly convex. We present numerical experiments to demonstrate the practical advantages of this algorithm.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Adaptive Augmented Lagrangian Methods: Algorithms and Practical Numerical Experience

In this paper, we consider augmented Lagrangian (AL) algorithms for solving large-scale nonlinear optimization problems that execute adaptive strategies for updating the penalty parameter. Our work is motivated by the recently proposed adaptive AL trust region method by Curtis, Jiang, and Robinson [Math. Prog., DOI: 10.1007/s10107-014-0784-y, 2013]. The first focal point of this paper is a new variant of the approach that employs a line search rather than a trust region strategy, where a critical algorithmic feature for the line search strategy is the use of convexified piecewise quadratic models of the AL function for computing the search directions. We prove global convergence guarantees for our line search algorithm that are on par with those for the previously proposed trust region method. A second focal point of this paper is the practical performance of the line search and trust region algorithm variants in Matlab software, as well as that of an adaptive penalty parameter updating strategy incorporated into the Lancelot software. We test these methods on problems from the CUTEst and COPS collections, as well as on challenging test problems related to optimal power flow. Our numerical experience suggests that the adaptive algorithms outperform traditional AL methods in terms of efficiency and reliability. As with traditional AL algorithms, the adaptive methods are matrix-free and thus represent a viable option for solving extreme-scale problems.

0 citations0 reviewsNo code linkedOpen access