In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full gradient computation are bottlenecks when dealing with large-scale problems. We propose a subspace quasi-Newton method that is restricted to a deterministic subspace together with a subspace gradient based on random matrix theory. Our method does not require full gradients, let alone Hessian matrices. Yet, it achieves the same order of worst-case iteration complexity in expectation for both convex and nonconvex cases as existing subspace methods. In numerical experiments, we confirm the superiority of our algorithm in terms of computation time.

In this study, we consider an optimization problem with uncertainty dependent on decision variables, which has recently attracted attention due to its importance in machine learning and pricing applications. In this problem, the gradient of the objective function cannot be obtained explicitly because the decision-dependent distribution is unknown. Therefore, several zeroth-order methods have been proposed, which obtain noisy objective values by sampling and update the iterates. Although these existing methods have theoretical convergence for optimization problems with decision-dependent uncertainty, they require strong assumptions about the function and distribution or exhibit large variances in their gradient estimators. To overcome these issues, we propose two zeroth-order methods under mild assumptions. First, we develop a zeroth-order method with a new one-point gradient estimator including a variance reduction parameter. The proposed method updates the decision variables while adjusting the variance reduction parameter. Second, we develop a zeroth-order method with a two-point gradient estimator. There are situations where only one-point estimators can be used, but if both one-point and two-point estimators are available, it is more practical to use the two-point estimator. As theoretical results, we show the convergence of our methods to stationary points and provide the worst-case iteration and sample complexity analysis. Our simulation experiments with real data on a retail service application show that our methods output solutions with lower objective values than the conventional zeroth-order methods.

In large-scale applications, such as machine learning, it is desirable to design non-convex optimization algorithms with a high degree of parallelization. In this work, we study the adaptive complexity of finding a stationary point, which is the minimal number of sequential rounds required to achieve stationarity given polynomially many queries executed in parallel at each round. For the high-dimensional case, \emph{i.e.}, $d = \widetilde{\Omega}(\varepsilon^{-(2 + 2p)/p})$, we show that for any (potentially randomized) algorithm, there exists a function with Lipschitz $p$-th order derivatives such that the algorithm requires at least $\varepsilon^{-(p+1)/p}$ iterations to find an $\varepsilon$-stationary point. Our lower bounds are tight and show that even with $\mathrm{poly}(d)$ queries per iteration, no algorithm has better convergence rate than those achievable with one-query-per-round algorithms. In other words, gradient descent, the cubic-regularized Newton’s method, and the $p$-th order adaptive regularization method are adaptively optimal. Our proof relies upon novel analysis with the characterization of the output for the hardness potentials based on a chain-like structure with random partition. For the constant-dimensional case, \emph{i.e.}, $d = \Theta(1)$, we propose an algorithm that bridges grid search and gradient flow trapping, finding an approximate stationary point in constant iterations. Its asymptotic tightness is verified by a new lower bound on the required queries per iteration. We show there exists a smooth function such that any algorithm running with $\Theta(\log (1/\varepsilon))$ rounds requires at least $\widetilde{\Omega}((1/\varepsilon)^{(d-1)/2})$ queries per round. This lower bound is tight up to a logarithmic factor, and implies that the gradient flow trapping is adaptively optimal.

In recent years, random subspace methods have been actively studied for large-dimensional nonconvex problems. Recent subspace methods have improved theoretical guarantees such as iteration complexity and local convergence rate while reducing computational costs by deriving descent directions in randomly selected low-dimensional subspaces. This paper proposes the Random Subspace Homogenized Trust Region (RSHTR) method with the best theoretical guarantees among random subspace algorithms for nonconvex optimization. RSHTR achieves an $\varepsilon$-approximate first-order stationary point in $O(\varepsilon^{-3/2})$ iterations, converging locally at a linear rate. Furthermore, under rank-deficient conditions, RSHTR satisfies $\varepsilon$-approximate second-order necessary conditions in $O(\varepsilon^{-3/2})$ iterations and exhibits a local quadratic convergence. Experiments on real-world datasets verify the benefits of RSHTR.

Deep learning with noisy labels presents significant challenges. In this work, we theoretically characterize the role of label noise from a feature learning perspective. Specifically, we consider a signal-noise data distribution, where each sample comprises a label-dependent signal and label-independent noise, and rigorously analyze the training dynamics of a two-layer convolutional neural network under this data setup, along with the presence of label noise. Our analysis identifies two key stages. In Stage I, the model perfectly fits all the clean samples (i.e., samples without label noise) while ignoring the noisy ones (i.e., samples with noisy labels). During this stage, the model learns the signal from the clean samples, which generalizes well on unseen data. In Stage II, as the training loss converges, the gradient in the direction of noise surpasses that of the signal, leading to overfitting on noisy samples. Eventually, the model memorizes the noise present in the noisy samples and degrades its generalization ability. Furthermore, our analysis provides a theoretical basis for two widely used techniques for tackling label noise: early stopping and sample selection. Experiments on both synthetic and real-world setups validate our theory.

Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian manifold structure and performing optimization intrinsically on the manifold. This approach typically involves computing a search direction in the tangent space and updating variables via a retraction operation. However, as the size of the variables increases, the computational cost of the retraction can become prohibitively high, limiting the applicability of Riemannian optimization to large-scale problems. To address this challenge and enhance scalability, we propose a novel approach that restricts each update on a random submanifold, thereby significantly reducing the per-iteration complexity. We introduce two sampling strategies for selecting the random submanifolds and theoretically analyze the convergence of the proposed methods. We provide convergence results for general nonconvex functions and functions that satisfy Riemannian Polyak–Łojasiewicz condition as well as for stochastic optimization settings. Additionally, we demonstrate how our approach can be generalized to quotient manifolds derived from the orthogonal manifold. Extensive experiments verify the benefits of the proposed method, across a wide variety of problems.

The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its local optimality guarantees. In this paper, we first present conditions under which the K-means algorithm converges to a locally optimal solution. Based on this, we propose simple modifications to the K-means algorithm which ensure local optimality in both the continuous and discrete sense, with the same computational complexity as the original K-means algorithm. As the dissimilarity measure, we consider a general Bregman divergence, which is an extension of the squared Euclidean distance often used in the K-means algorithm. Numerical experiments confirm that the K-means algorithm does not always find a locally optimal solution in practice, while our proposed methods provide improved locally optimal solutions with reduced clustering loss. Our code is available at https://github.com/lmingyi/LO-K-means.

Abstract Random projection, a dimensionality reduction technique, has been found useful in recent years for reducing the size of optimization problems. In this paper, we explore the use of sparse sub-gaussian random projections to approximate semidefinite programming (SDP) problems by reducing the size of matrix variables, thereby solving the original problem with much less computational effort. We provide some theoretical bounds on the quality of the projection in terms of feasibility and optimality that explicitly depend on the sparsity parameter of the projector. We investigate the performance of the approach for semidefinite relaxations appearing in polynomial optimization, with a focus on combinatorial optimization problems. In particular, we apply our method to the semidefinite relaxations of Maxcut and Max-2-sat . We show that for large unweighted graphs, we can obtain a good bound by solving a projection of the semidefinite relaxation of Maxcut . We also explore how to apply our method to find the stability number of four classes of imperfect graphs by solving a projection of the second level of the Lasserre Hierarchy. Overall, our computational experiments show that semidefinite programming problems appearing as relaxations of combinatorial optimization problems can be approximately solved using random projections as long as the number of constraints is not too large.

Abstract In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. Using a technique from computer algebra and the second-order optimality condition, we provide a univariate representation for the set of local minimizers. In particular, for the unconstrained problem, i.e., the constraint set is $${{\,\mathrm{\mathbb {R}}\,}}^n$$ R n , the coordinates of all local minimizers can be represented by the values of n univariate polynomials at the real solutions of a univariate system containing a polynomial equation and a polynomial matrix inequality. We also develop the technique for problems with equality/inequality constraints. Based on the above technique, we design algorithms to enumerate the local minimizers and provide some experimental examples based on hybrid symbolic-numerical computations. For the case that the genericity conditions fail, at the end of the paper we propose a perturbation technique to compute approximately a global minimizer, provided that the constraint set is compact.

Large language models (LLMs) have shown impressive capabilities across various tasks. However, training LLMs from scratch requires significant computational power and extensive memory capacity. Recent studies have explored low-rank structures on weights for efficient fine-tuning in terms of parameters and memory, either through low-rank adaptation or factorization. While effective for fine-tuning, low-rank structures are generally less suitable for pretraining because they restrict parameters to a low-dimensional subspace. In this work, we propose to parameterize the weights as a sum of low-rank and sparse matrices for pretraining, which we call SLTrain. The low-rank component is learned via matrix factorization, while for the sparse component, we employ a simple strategy of uniformly selecting the sparsity support at random and learning only the non-zero entries with the fixed support. While being simple, the random fixed-support sparse learning strategy significantly enhances pretraining when combined with low-rank learning. Our results show that SLTrain adds minimal extra parameters and memory costs compared to pretraining with low-rank parameterization, yet achieves substantially better performance, which is comparable to full-rank training. Remarkably, when combined with quantization and per-layer updates, SLTrain can reduce memory requirements by up to 73% when pretraining the LLaMA 7B model.

Bilevel optimization has gained prominence in various applications. In this study, we introduce a framework for solving bilevel optimization problems, where the variables in both the lower and upper levels are constrained on Riemannian manifolds. We present several hypergradient estimation strategies on manifolds and analyze their estimation errors. Furthermore, we provide comprehensive convergence and complexity analyses for the proposed hypergradient descent algorithm on manifolds. We also extend our framework to encompass stochastic bilevel optimization and incorporate the use of general retraction. The efficacy of the proposed framework is demonstrated through several applications.

Accurate outlier detection is not only a necessary preprocessing step, but can itself give important insights into the data. However, especially, for non-linear regression the detection of outliers is non-trivial, and actually ambiguous. We propose a new method that identifies outliers by finding a subset of data points T such that the marginal likelihood of all remaining data points S is maximized. Though the idea is more general, it is particular appealing for Gaussian processes regression, where the marginal likelihood has an analytic solution. While maximizing the marginal likelihood for hyper-parameter optimization is a well established non-convex optimization problem, optimizing the set of data points S is not. Indeed, even a greedy approximation is computationally challenging due to the high cost of evaluating the marginal likelihood. As a remedy, we propose an efficient projected gradient descent method with provable convergence guarantees. Moreover, we also establish the breakdown point when jointly optimizing hyper-parameters and S. For various datasets and types of outliers, our experiments demonstrate that the proposed method can improve outlier detection and robustness when compared with several popular alternatives like the student-t likelihood.

The Gaussian homotopy (GH) method is a popular approach to finding better stationary points for non-convex optimization problems by gradually reducing a parameter value $t$, which changes the problem to be solved from an almost convex one to the original target one. Existing GH-based methods repeatedly call an iterative optimization solver to find a stationary point every time $t$ is updated, which incurs high computational costs. We propose a novel single loop framework for GH methods (SLGH) that updates the parameter $t$ and the optimization decision variables at the same. Computational complexity analysis is performed on the SLGH algorithm under various situations: either a gradient or gradient-free oracle of a GH function can be obtained for both deterministic and stochastic settings. The convergence rate of SLGH with a tuned hyperparameter becomes consistent with the convergence rate of gradient descent, even though the problem to be solved is gradually changed due to $t$. In numerical experiments, our SLGH algorithms show faster convergence than an existing double loop GH method while outperforming gradient descent-based methods in terms of finding a better solution.

Many social phenomena are triggered by public opinion that is formed in the process of opinion exchange among individuals. To date, from the engineering point of view, a large body of work has been devoted to studying how to manipulate individual opinions so as to guide public opinion towards the desired state. Recently, Abebe et al. (KDD 2018) have initiated the study of the impact of interventions at the level of susceptibility rather than the interventions that directly modify individual opinions themselves. For the model, Chan et al. (The Web Conference 2019) designed a local search algorithm to find an optimal solution in polynomial time. However, it can be seen that the solution obtained by solving the above model might not be implemented in real-world scenarios. In fact, as we do not consider the amount of changes of the susceptibility, it would be too costly to change the susceptibility values for agents based on the solution.

Although application examples of multilevel optimization have already been discussed since the 1990s, the development of solution methods was almost limited to bilevel cases due to the difficulty of the problem. In recent years, in machine learning, Franceschi et al. have proposed a method for solving bilevel optimization problems by replacing their lower-level problems with the $T$ steepest descent update equations with some prechosen iteration number $T$. In this paper, we have developed a gradient-based algorithm for multilevel optimization with $n$ levels based on their idea and proved that our reformulation asymptotically converges to the original multilevel problem. As far as we know, this is one of the first algorithms with some theoretical guarantee for multilevel optimization. Numerical experiments show that a trilevel hyperparameter learning model considering data poisoning produces more stable prediction results than an existing bilevel hyperparameter learning model in noisy data settings.

This paper focuses on stochastic proximal gradient methods for optimizing a smooth non-convex loss function with a non-smooth non-convex regularizer and convex constraints. To the best of our knowledge we present the first non-asymptotic convergence bounds for this class of problem. We present two simple stochastic proximal gradient algorithms, for general stochastic and finite-sum optimization problems. In a numerical experiment we compare our algorithms with the current state-of-the-art deterministic algorithm and find our algorithms to exhibit superior convergence.

We propose a bilevel optimization strategy for selecting the best hyperparameter value for the nonsmooth $\ell_p$ regularizer with $0 [abs][pdf][bib] © JMLR 2021. (edit, beta) Mastodon

We propose a new formulation of multiple-instance learning (MIL), in which a unit of data consists of a set of instances called a bag. The goal is to find a good classifier of bags based on the similarity with a "shapelet" (or pattern), where the similarity of a bag with a shapelet is the maximum similarity of instances in the bag. In previous work, some of the training instances have been chosen as shapelets with no theoretical justification. In our formulation, we use all possible, and thus infinitely many, shapelets, resulting in a richer class of classifiers. We show that the formulation is tractable, that is, it can be reduced through linear programming boosting (LPBoost) to difference of convex (DC) programs of finite (actually polynomial) size. Our theoretical result also gives justification to the heuristics of some previous work. The time complexity of the proposed algorithm highly depends on the size of the set of all instances in the training sample. To apply to the data containing a large number of instances, we also propose a heuristic option of the algorithm without the loss of the theoretical guarantee. Our empirical study demonstrates that our algorithm uniformly works for shapelet learning tasks on time-series classification and various MIL tasks with comparable accuracy to the existing methods. Moreover, we show that the proposed heuristics allow us to achieve the result in reasonable computational time.

In this paper, we present an alternating gradient descent algorithm for estimating parameters of a spherical Gaussian mixture model by the method of moments (AGD-MoM). We formulate the problem as a constrained optimisation problem which simultaneously matches the third order moments from the data, represented as a tensor, and the second order moment, which is the empirical covariance matrix . We derive the necessary gradients (and second derivatives), and use them to implement alternating gradient search to estimate the parameters of the model. We show that the proposed method is applicable in both a batch as well as in a streaming (online) setting. Using synthetic and benchmark datasets, we demonstrate empirically that the proposed algorithm outperforms the more classical algorithms like Expectation Maximisation and variational Bayes.

Our work focuses on stochastic gradient methods for optimizing a smooth non-convex loss function with a non-smooth non-convex regularizer. Research on this class of problem is quite limited, and until recently no non-asymptotic convergence results have been reported. We present two simple stochastic gradient algorithms, for finite-sum and general stochastic optimization problems, which have superior convergence complexities compared to the current state-of-the-art. We also compare our algorithms’ performance in practice for empirical risk minimization.

The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box, and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a doubly nonnegative relaxation of the POP, and then solves the COP by applying the bisection and projection method. The COP is expressed with a linear objective function and constraints described as a single hyperplane and two cones, which are the Cartesian product of positive semidefinite cones and a polyhedral cone induced from the BBC constraints. BBCPOP aims to compute a tight lower bound for the optimal value of a large-scale POP in the class that is beyond the comfort zone of existing software packages. The robustness, reliability, and efficiency of BBCPOP are demonstrated in comparison to the state-of-the-art software SDP package SDPNAL+ on randomly generated sparse POPs of degree 2 and 3 with up to a few thousands variables, and ones of degree from 5 to 8 with up to a few hundred variables. Numerical results on BBC-constrained POPs that arise from quadratic assignment problems are also reported. The software package BBCPOP is available at https://sites.google.com/site/bbcpop1/.

Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph G = (V, E) with an edge-weight vector w = (We)e∈E, we aim to find a subset of vertices S that maximizes the density, i.e., w(S) / |S|, where w(S) is the sum of the weights of the edges in the subgraph induced by S. Although the densest subgraph problem is one of the most well-studied optimization problems for dense subgraph discovery, there is an implicit strong assumption; it is assumed that the weights of all the edges are known exactly as input. In real-world applications, there are often cases where we have only uncertain information of the edge weights. In this study, we provide a framework for dense subgraph discovery under the uncertainty of edge weights. Specifically, we address such an uncertainty issue using the theory of robust optimization. First, we formulate our fundamental problem, the robust densest subgraph problem, and present a simple algorithm. We then formulate the robust densest subgraph problem with sampling oracle that models dense subgraph discovery using an edge-weight sampling oracle, and present an algorithm with a strong theoretical performance guarantee. Computational experiments using both synthetic graphs and popular real-world graphs demonstrate the effectiveness of our proposed algorithms.

The unfairness of a regressor is evaluated by measuring the correlation between the estimator and the sensitive attribute (e.g., race, gender, age), and the coefficient of determination (CoD) is a natural extension of the correlation coefficient when more than one sensitive attribute exists. As is well known, there is a trade-off between fairness and accuracy of a regressor, which implies a perfectly fair optimizer does not always yield a useful prediction. Taking this into consideration, we optimize the accuracy of the estimation subject to a user-defined level of fairness. However, a fairness level as a constraint induces a nonconvexity of the feasible region, which disables the use of an off-the-shelf convex optimizer. Despite such nonconvexity, we show an exact solution is available by using tools of global optimization theory. Furthermore, we propose a nonlinear extension of the method by kernel representation. Unlike most of existing fairness-aware machine learning methods, our method allows us to deal with numeric and multiple sensitive attributes.

Motivated by online advertising, we study a multiple-play multi-armed bandit problem with position bias that involves several slots and the latter slots yield fewer rewards. We characterize the hardness of the problem by deriving an asymptotic regret bound. We propose the Permutation Minimum Empirical Divergence (PMED) algorithm and derive its asymptotically optimal regret bound. Because of the uncertainty of the position bias, the optimal algorithm for such a problem requires non-convex optimizations that are different from usual partial monitoring and semi-bandit problems. We propose a cutting-plane method and related bi-convex relaxation for these optimizations by using auxiliary variables.

Density ratio estimation is a vital tool in both machine learning and statistical community. However, due to the unbounded nature of density ratio, the estimation proceudre can be vulnerable to corrupted data points, which often pushes the estimated ratio toward infinity. In this paper, we present a robust estimator which automatically identifies and trims outliers. The proposed estimator has a convex formulation, and the global optimum can be obtained via subgradient descent. We analyze the parameter estimation error of this estimator under high-dimensional settings. Experiments are conducted to verify the effectiveness of the estimator.

Binary classification is the problem of predicting the class a given sample belongs to. To achieve a good prediction performance, it is important to find a suitable model for a given dataset. However, it is often time consuming and impractical for practitioners to try various classification models because each model employs a different formulation and algorithm. The difficulty can be mitigated if we have a unified formulation and an efficient universal algorithmic framework for various classification models to expedite the comparison of performance of different models for a given dataset. In this paper, we present a unified formulation of various classification models (including $C$-SVM, $\ell_2$-SVM, $\nu$-SVM, MM-FDA, MM-MPM, logistic regression, distance weighted discrimination) and develop a general optimization algorithm based on an accelerated proximal gradient (APG) method for the formulation. We design various techniques such as backtracking line search and adaptive restarting strategy in order to speed up the practical convergence of our method. We also give a theoretical convergence guarantee for the proposed fast APG algorithm. Numerical experiments show that our algorithm is stable and highly competitive to specialized algorithms designed for specific models (e.g., sequential minimal optimization (SMO) for SVM).

Nonconvex variants of support vector machines (SVMs) have been developed for various purposes. For example, robust SVMs attain robustness to outliers by using a nonconvex loss function, while extended [Formula: see text]-SVM (E[Formula: see text]-SVM) extends the range of the hyperparameter by introducing a nonconvex constraint. Here, we consider an extended robust support vector machine (ER-SVM), a robust variant of E[Formula: see text]-SVM. ER-SVM combines two types of nonconvexity from robust SVMs and E[Formula: see text]-SVM. Because of the two nonconvexities, the existing algorithm we proposed needs to be divided into two parts depending on whether the hyperparameter value is in the extended range or not. The algorithm also heuristically solves the nonconvex problem in the extended range. In this letter, we propose a new, efficient algorithm for ER-SVM. The algorithm deals with two types of nonconvexity while never entailing more computations than either E[Formula: see text]-SVM or robust SVM, and it finds a critical point of ER-SVM. Furthermore, we show that ER-SVM includes the existing robust SVMs as special cases. Numerical experiments confirm the effectiveness of integrating the two nonconvexities.

In this paper we consider robust classifications and show equivalence between the regularized classifications. In general, robust classifications are used to create a classifier robust to data by taking into account the uncertainty of the data. Our result shows that regularized classifications inherit robustness and provide reason on why some regularized classifications tend to be robust against data. Although most robust classification problems assume that every uncertain data lie within an identical bounded set, this paper considers a generalized model where the sizes of the bounded sets are different for each data. These models can be transformed into regularized classification models where the penalties for each data are assigned according to their losses. We see that considering such models opens up for new applications. For an example, we show that this robust classification technique can be used for Imbalanced Data Learning. We conducted experimentation with actual data and compared it with other IDL algorithms such as Cost Sensitive SVMs. This is a novel usage for the robust classification scheme and encourages it to be a suitable candidate for imbalanced data learning.

BACKGROUND: In high-throughput experimental biology, it is widely acknowledged that while expression levels measured at the levels of transcriptome and the corresponding proteome do not, in general, correlate well, messenger RNA levels are used as convenient proxies for protein levels. Our interest is in developing data-driven computational models that can bridge the gap between these two levels of measurement at which different mechanisms of regulation may act on different molecular species causing any observed lack of correlations. To this end, we build data-driven predictors of protein levels using mRNA levels and known proxies of translation efficiencies as covariates. Previous work showed that in such a setting, outliers with respect to the model are reliable candidates for post-translational regulation. RESULTS: Here, we present and compare two novel formulations of deriving a protein concentration predictor from which outliers may be extracted in a systematic manner. The first approach, outlier rejecting regression, allows explicit specification of a certain fraction of the data as outliers. In a regression setting, this is a non-convex optimization problem which we solve by deriving a difference of convex functions algorithm (DCA). With post-translationally regulated proteins, one expects their concentrations to be affected primarily by disruption of protein stability. Our second algorithm exploits this observation by minimizing an asymmetric loss using quantile regression and extracts outlier proteins whose measured concentrations are lower than what a genome-wide regression would predict. We validate the two approaches on a dataset of yeast transcriptome and proteome. Functional annotation check on detected outliers demonstrate that the methods are able to identify post-translationally regulated genes with high statistical confidence.

The Fisher discriminant analysis (FDA) is a common technique for binary classification. A parametrized extension, which we call the extended FDA, has been introduced from the viewpoint of robust optimization. In this work, we first give a new probabilistic interpretation of the extended FDA. We then develop algorithms for solving an optimization problem that arises from the extended FDA: computing the distance between a point and the surface of an ellipsoid. We solve this problem via the KKT points, which we show are obtained by solving a generalized eigenvalue problem. We speed up the algorithm by taking advantage of the matrix structure and proving that a globally optimal solution is a KKT point with the smallest Lagrange multiplier, which can be computed efficiently as the leftmost eigenvalue. Numerical experiments illustrate the efficiency and effectiveness of the extended FDA model combined with our algorithm.

Stochastic gradient descent has been advanced as a computationally efficient method for large-scale problems. In classification problems, many proposed linear support vector machines are very effective. However, they assume that the data is already in memory which might be not always the case. Recent work suggests a classical method that divides such a problem into smaller blocks then solves the sub-problems iteratively. We show that a simple modification of shrinking the dataset early will produce significant saving in computation and memory. We further find that on problems larger than previously considered, our approach is able to reach solutions on top-end desktop machines while competing methods cannot.

Financial risk measures have been used recently in machine learning. For example, ν-support vector machine ν-SVM) minimizes the conditional value at risk (CVaR) of margin distribution. The measure is popular in finance because of the subadditivity property, but it is very sensitive to a few outliers in the tail of the distribution. We propose a new classification method, extended robust SVM (ER-SVM), which minimizes an intermediate risk measure between the CVaR and value at risk (VaR) by expecting that the resulting model becomes less sensitive than ν-SVM to outliers. We can regard ER-SVM as an extension of robust SVM, which uses a truncated hinge loss. Numerical experiments imply the ER-SVM's possibility of achieving a better prediction performance with proper parameter setting.

When a probabilistic model and its prior are given, Bayesian learning offers inference with automatic parameter tuning. However, Bayesian learning is often obstructed by computational difficulty: the rigorous Bayesian learning is intractable in many models, and its variational Bayesian (VB) approximation is prone to suffer from local minima. In this paper, we overcome this difficulty for low-rank subspace clustering (LRSC) by providing an exact global solver and its efficient approximation. LRSC extracts a low-dimensional structure of data by embedding samples into the union of low-dimensional subspaces, and its variational Bayesian variant has shown good performance. We first prove a key property that the VB-LRSC model is highly redundant. Thanks to this property, the optimization problem of VB-LRSC can be separated into small subproblems, each of which has only a small number of unknown variables. Our exact global solver relies on another key property that the stationary condition of each subproblem is written as a set of polynomial equations, which is solvable with the homotopy method. For further computational efficiency, we also propose an efficient approximate variant, of which the stationary condition can be written as a polynomial equation with a single variable. Experimental results show the usefulness of our approach.

A wide variety of machine learning algorithms such as the support vector machine (SVM), minimax probability machine (MPM), and Fisher discriminant analysis (FDA) exist for binary classification. The purpose of this letter is to provide a unified classification model that includes these models through a robust optimization approach. This unified model has several benefits. One is that the extensions and improvements intended for SVMs become applicable to MPM and FDA, and vice versa. For example, we can obtain nonconvex variants of MPM and FDA by mimicking Perez-Cruz, Weston, Hermann, and Schölkopf's (2003) extension from convex ν-SVM to nonconvex Eν-SVM. Another benefit is to provide theoretical results concerning these learning methods at once by dealing with the unified model. We give a statistical interpretation of the unified classification model and prove that the model is a good approximation for the worst-case minimization of an expected loss with respect to the uncertain probability distribution. We also propose a nonconvex optimization algorithm that can be applied to nonconvex variants of existing learning methods and show promising numerical results.

The ν-support vector classification (ν-SVC) algorithm was shown to work well and provide intuitive interpretations, e.g., the parameter ν roughly specifies the fraction of support vectors. Although ν corresponds to a fraction, it cannot take the entire range between 0 and 1 in its original form. This problem was settled by a non-convex extension of ν-SVC and the extended method was experimentally shown to generalize better than original ν-SVC. However, its good generalization performance and convergence properties of the optimization algorithm have not been studied yet. In this paper, we provide new theoretical insights into these issues and propose a novel ν-SVC algorithm that has guaranteed generalization performance and convergence properties. 1.