Predict covariance from features using convex optimization.
problem Predicting the covariance of a Gaussian vector from another feature vector.
method A generalized linear model with convex optimization for fitting parameters.
result Predicted covariance matrices are symmetric positive definite.
Neural network approximates weakly efficient frontier of convex vector optimization problems.
problem Approximating the weakly efficient frontier of convex vector optimization problems.
method Designing a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater's condition.
result The proposed algorithm effectively approximates the true weakly efficient frontier of CVOPs, even for large problems.
Support vector regression (SVR) is one of the most popular machine learning algorithms aiming to generate the optimal regression curve through maximizing the minimal margin of selected training samples, i.e., support vectors. Recent researchers reveal that maximizing the margin distribution of whole training dataset ra…
We propose a new class of convex penalty functions, called \emph{variational Gram functions} (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as regularizers in convex optimization problems arising from hierarchical classification, m…
A new algorithm reduces communication rounds for distributed convex optimization.
problem Efficiently solving convex optimization problems in distributed systems.
method Proposes a stochastic Newton algorithm for homogeneous distributed stochastic convex optimization.
result Reduces the number and frequency of communication rounds compared to existing methods.
Optimal DP mechanisms for vector queries are found to be staircase distributions.
problem Designing optimal additive mechanisms for vector-valued queries under differential privacy.
method Reduction to radially symmetric distributions and convex rearrangement theory.
result Staircase mechanisms are optimal for any norm and cost function.
Randomized Geometric Algebra for Convex Neural Networks Optimizes Transfer Learning.
problem Training neural networks to global optimality via convex optimization.
method Randomized algorithms in Clifford's Geometric Algebra for hypercomplex vector spaces.
result Convex optimization and geometric algebra improve LLMs' robustness and reliability in transfer learning.
Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for …
The paper bounds the mean absolute error in DNN vector-to-vector regression.
problem Bounding the mean absolute error in deep neural network based vector-to-vector regression.
method Error decomposition techniques in statistical learning theory and non-convex optimization theory were used to derive upper bounds for approximation, estimation, and optimization errors.
result Theoretical upper bounds for mean absolute error in DNN vector-to-vector regression were derived and validated experimentally.
New guarantees for Group LASSO in sparse convex optimization.
problem Sparse convex optimization with vector-valued features.
method Group LASSO regularization and analysis of gradient norms.
result Group LASSO selects the same features as Orthogonal Matching Pursuit.
Paper introduces ICGNs to model convex gradients.
problem Modeling convex gradients efficiently.
method Integrates Jacobian-vector product in a neural network.
result Single layer ICGN outperforms single layer ICNN in fitting.
Neural optimal transport improves multivariate conformal prediction.
problem Multivariate quantile regression challenges and existing methods ignore joint distribution geometry.
method Combines neural optimal transport with amortized optimization for efficient training and faster inference.
result Constructs tighter and more informative predictive regions for multivariate conformal prediction.
This work explores the non-convex optimization in compressive learning and the performance of heuristics.
problem The challenge of learning from compressed representations in compressive learning.
method Numerical simulations of the non-convex optimization landscape and heuristic performance.
result Properties of the non-convex optimization landscape and heuristic performance are explored.
Neural network implementation of Brenier's polar factorization for vector fields.
problem Implementing Brenier's polar factorization theorem for vector fields using neural networks.
method Parameterizing the convex function u as an input convex neural network and estimating the measure-preserving map M. result Practical neural implementation of Brenier's polar factorization theorem.
New SGD algorithm finds critical points faster with second-order corrections.
problem Finding critical points in non-convex optimization efficiently.
method Uses Hessian-vector products to correct momentum bias in SGD.
result Finds ε-critical points in O(ε−3) time. In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in Rn+1 involving the norm of the covariant derivatives of both the difference tensor K and the Tchebychev vector field T. Our result is optimal in that, applying our recent classification for local…
A new SVM method for predicting time series labels.
problem Learning to predict labels from high-dimensional time series data.
method Extended SVM concept to continuous time series data, formulated as a convex optimization problem.
result Empirical results show the algorithm's effectiveness for analyzing long-term multivariate data.
RHPSVM improves SVM performance with robust loss function.
problem Outliers and resampling instability in SVM models.
method RHPSVM uses a rescaled Huberized pinball loss function.
result RHPSVM outperforms existing SVM models in noisy and small-sample scenarios.
Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
Robust support vector machine (RSVM) has been shown to perform remarkably well to improve the generalization performance of support vector machine under the noisy environment. Unfortunately, in order to handle the non-convexity induced by ramp loss in RSVM, existing RSVM solvers often adopt the DC programming framework…
VOGP efficiently identifies Pareto optimal solutions in black-box vector optimization.
problem Black-box vector optimization with incomplete order relations.
method VOGP is an adaptive elimination algorithm using Gaussian process bandits.
result VOGP achieves theoretical guarantees with sample complexity bounds.
VOPy optimizes multiple objectives with flexible cone-based ordering.
problem Optimizing multiple objectives with partial order constraints.
method Flexible cone-based ordering, modular architecture, integration of existing and novel methods.
result Advances black-box vector optimization in noisy, discrete, or limited budget settings.
We present a distributionally robust formulation of a stochastic optimization problem for non-i.i.d vector autoregressive data. We use the Wasserstein distance to define robustness in the space of distributions and we show, using duality theory, that the problem is equivalent to a finite convex-concave saddle point pro…
New methods for sketching non-PSD matrices improve regression and optimization tasks.
problem Efficiently handling non-PSD matrices in computations.
method Developed novel matrix sketching techniques for non-PSD and complex matrices.
result Improved performance in convex and non-convex optimization, regression, and vector-matrix-vector queries.
Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of sub…
We propose a rank-k variant of the classical Frank-Wolfe algorithm to solve convex optimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation (1-SVD) in Frank-Wolfe with a top-k singular-vector computation (k-SVD), which can be done by repeatedly applying 1-SVD k times. …
We consider distributed online convex optimization problems, where the distributed system consists of various computing units connected through a time-varying communication graph. In each time step, each computing unit selects a constrained vector, experiences a loss equal to an arbitrary convex function evaluated at t…
The area under the ROC curve (AUC) is a widely used performance measure in machine learning. Increasingly, however, in several applications, ranging from ranking to biometric screening to medicine, performance is measured not in terms of the full area under the ROC curve, but in terms of the \emph{partial} area under t…
NNLMs optimize poorly for word probabilities due to embedding space structure.
problem NNLMs assign suboptimal probabilities to some words.
method Analyzed the inductive bias of NNLMs and the structure of word embeddings.
result Words on the convex hull have bounded probability, affecting others.
New insights into CE dynamics reveal how Hadamard initialization simplifies softmax.
problem Understanding the dynamics of cross-entropy training loss in deep learning.
method Analyzing a two-layer linear neural network with standard-basis vectors as inputs.
result Gradient flow on cross-entropy converges to neural collapse geometry, proving global convergence.
For incomplete preference relations that are represented by multiple priors and/or multiple -- possibly multivariate -- utility functions, we define a certainty equivalent as well as the utility buy and sell prices and indifference price bounds as set-valued functions of the claim. Furthermore, we motivate and introduc…
Tick-by-tick liquidity provision aims to maximize fees and reserves.
problem Maximizing fees and reserves in concentrated liquidity.
method Convex optimization for tick-level liquidity provision.
result Concentrating liquidity around current price is not always best.
New method finds all Nash equilibria via vector optimization.
problem Finding all Nash equilibria in games.
method Formulate vector optimization problem to find Pareto optimal solutions.
result Characterize set of all Nash equilibria as Pareto optimal solutions.
Paper develops exact convex optimization for neural networks with polynomial activations.
problem Training two-layer neural networks with nonlinear polynomial activations.
method Exact convex optimization using semidefinite programming.
result Global optimization of neural networks is polynomial-time computable.
Distance metric learning (DML), which learns a distance metric from labeled "similar" and "dissimilar" data pairs, is widely utilized. Recently, several works investigate orthogonality-promoting regularization (OPR), which encourages the projection vectors in DML to be close to being orthogonal, to achieve three effect…
Maximizes determinant of vector sums under matroid constraints.
problem Finding a basis in a matroid that maximizes the determinant of vector sums.
method New approximation algorithm with guarantees depending only on vector dimension.
result Significant improvement in approximation guarantees for various matroids.
This work shows neural networks can solve non-convex constraints problems.
problem Training neural networks under non-convex constraints.
method Project stochastic gradient descent with no-regret analysis of online learning.
result Overparameterized neural networks achieve near-optimal and near-feasible solutions.
New algorithm finds approximate stationary points in non-convex optimization.
problem Finding approximate stationary points in non-convex stochastic optimization.
method Design of an algorithm using O(ε−3) stochastic gradient and Hessian-vector products. result Optimal rate of O(ε−3) for finding ε-approximate stationary points, matching lower bounds. We study the convergence properties of the VR-PCA algorithm introduced by \cite{shamir2015stochastic} for fast computation of leading singular vectors. We prove several new results, including a formal analysis of a block version of the algorithm, and convergence from random initialization. We also make a few observatio…
Paper constructs L2 estimates for flat vector bundles and generalizes Prékopa's theorem.
problem Constructing L2 estimates for flat vector bundles. method Using Hörmander's L2-estimate for the operator d on a flat vector bundle over a p-convex Riemannian manifold. result Generalizes Prékopa's theorem in convex analysis.
Robust SVM optimization in Banach spaces tackles classification uncertainty.
problem Binary classification in Banach spaces with uncertainty.
method Generalization of SVM results to Banach spaces, Representer Theorem, strong duality, Nash equilibrium formulation.
result Generalization of SVM results to Banach spaces, including Representer Theorem and strong duality.
Develops iso-Riemannian optimization for data manifolds.
problem Challenges in performing optimization on learned data manifolds.
method Introduces iso-connection and iso-Riemannian descent algorithm.
result Demonstrates efficient solutions to inverse problems on learned data manifolds.
Partial convexification improves tractability of low-rank spectral optimization problems.
problem Minimizing linear objectives subject to matrix inequalities and low-rank constraints.
method Partial convexification of the domain set, deriving rank bounds, and developing a column generation algorithm.
result The partial convexification LSOP-R is equivalent to the original LSOP under certain conditions and yields high-quality solutions.
The Hessian-vector product has been utilized to find a second-order stationary solution with strong complexity guarantee (e.g., almost linear time complexity in the problem's dimensionality). In this paper, we propose to further reduce the number of Hessian-vector products for faster non-convex optimization. Previous a…
New method recovers signals from noisy indirect data, even when noise is uncertain.
problem Recovering signals from indirect observations with uncertain noise.
method Polyhedral estimates, incorporating convex optimization.
result Presumably good estimates can be constructed for ellitope signal sets.
We consider a discriminative learning (regression) problem, whereby the regression function is a convex combination of k linear classifiers. Existing approaches are based on the EM algorithm, or similar techniques, without provable guarantees. We develop a simple method based on spectral techniques and a `mirroring' tr…
Study invariant minimizers in convex functions under amenable groups.
problem Finding invariant minimizers in convex functions invariant under amenable groups.
method Analyze smallest closed invariant convex subsets and apply to invariant optimality problem.
result Clarifies relations between equivariant neural networks and statistical theorems.
Adam optimizer converges to zeros of a new vector field, not just gradient zeros.
problem Prove convergence rates for Adam optimizer in simple quadratic optimization problems.
method Introduced Adam vector field to analyze Adam optimizer's convergence.
result Established optimal convergence rates for Adam optimizer.