A Framework for Bilevel Optimization on Riemannian Manifolds

Bilevel optimization has seen an increasing presence in various domains of applications. In this work, we propose a framework for solving bilevel optimization problems where variables of both lower and upper level problems are constrained on Riemannian manifolds. We provide several hypergradient estimation strategies on manifolds and study their estimation error. We provide convergence and complexity analysis for the proposed hypergradient descent algorithm on manifolds. We also extend the developments to stochastic bilevel optimization and to the use of general retraction. We showcase the utility of the proposed framework on various applications.

Convergence Error Analysis of Reflected Gradient Langevin Dynamics for Globally Optimizing Non-Convex Constrained Problems

Non-convex optimization problems have various important applications, whereas many algorithms have been proven only to converge to stationary points. Meanwhile, gradient Langevin dynamics (GLD) and its variants have attracted increasing attention as a framework to provide theoretical convergence guarantees for a global solution in non-convex settings. The studies on GLD initially treated unconstrained convex problems and very recently expanded to convex constrained non-convex problems by Lamperski (2021). In this work, we can deal with non-convex problems with some kind of non-convex feasible region. This work analyzes reflected gradient Langevin dynamics (RGLD), a global optimization algorithm for smoothly constrained problems, including non-convex constrained ones, and derives a convergence rate to a solution with $\epsilon$-sampling error. The convergence rate is faster than the one given by Lamperski (2021) for convex constrained cases. Our proofs exploit the Poisson equation to effectively utilize the reflection for the faster convergence rate.

A Gradient Method for Multilevel Optimization

Although application examples of multilevel optimization have already been discussed since the '90s, 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 with $n T$ variables 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.

BODAME: Bilevel Optimization for Defense Against Model Extraction

Model extraction attacks have become serious issues for service providers using machine learning. We consider an adversarial setting to prevent model extraction under the assumption that attackers will make their best guess on the service provider's model using query accesses, and propose to build a surrogate model that significantly keeps away the predictions of the attacker's model from those of the true model. We formulate the problem as a non-convex constrained bilevel optimization problem and show that for kernel models, it can be transformed into a non-convex 1-quadratically constrained quadratic program with a polynomial-time algorithm to find the global optimum. Moreover, we give a tractable transformation and an algorithm for more complicated models that are learned by using stochastic gradient descent-based algorithms. Numerical experiments show that the surrogate model performs well compared with existing defense models when the difference between the attacker's and service provider's distributions is large. We also empirically confirm the generalization ability of the surrogate model.

Theory and Algorithms for Shapelet-based Multiple-Instance Learning

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 are 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 of the 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 with reasonable computational time.

Convex Fairness Constrained Model Using Causal Effect Estimators

Recent years have seen much research on fairness in machine learning. Here, mean difference (MD) or demographic parity is one of the most popular measures of fairness. However, MD quantifies not only discrimination but also explanatory bias which is the difference of outcomes justified by explanatory features. In this paper, we devise novel models, called FairCEEs, which remove discrimination while keeping explanatory bias. The models are based on estimators of causal effect utilizing propensity score analysis. We prove that FairCEEs with the squared loss theoretically outperform a naive MD constraint model. We provide an efficient algorithm for solving FairCEEs in regression and binary classification tasks. In our experiment on synthetic and real-world data in these two tasks, FairCEEs outperformed an existing model that considers explanatory bias in specific cases.

Multiple-Instance Learning by Boosting Infinitely Many Shapelet-based Classifiers

We propose a new formulation of Multiple-Instance Learning (MIL). In typical MIL settings, a unit of data is given as a set of instances called a bag and the goal is to find a good classifier of bags based on similarity from a single or finitely many "shapelets" (or patterns), where the similarity of the bag from a shapelet is the maximum similarity of instances in the bag. Classifiers based on a single shapelet are not sufficiently strong for certain applications. Additionally, previous work with multiple shapelets has heuristically chosen some of the instances as shapelets with no theoretical guarantee of its generalization ability. Our formulation provides a richer class of the final classifiers based on infinitely many shapelets. We provide an efficient algorithm for the new formulation, in addition to generalization bound. Our empirical study demonstrates that our approach is effective not only for MIL tasks but also for Shapelet Learning for time-series classification.

Robust Bayesian Model Selection for Variable Clustering with the Gaussian Graphical Model

Variable clustering is important for explanatory analysis. However, only few dedicated methods for variable clustering with the Gaussian graphical model have been proposed. Even more severe, small insignificant partial correlations due to noise can dramatically change the clustering result when evaluating for example with the Bayesian Information Criteria (BIC). In this work, we try to address this issue by proposing a Bayesian model that accounts for negligible small, but not necessarily zero, partial correlations. Based on our model, we propose to evaluate a variable clustering result using the marginal likelihood. To address the intractable calculation of the marginal likelihood, we propose two solutions: one based on a variational approximation, and another based on MCMC. Experiments on simulated data shows that the proposed method is similarly accurate as BIC in the no noise setting, but considerably more accurate when there are noisy partial correlations. Furthermore, on real data the proposed method provides clustering results that are intuitively sensible, which is not always the case when using BIC or its extensions.

A successive difference-of-convex approximation method for a class of nonconvex nonsmooth optimization problems

We consider a class of nonconvex nonsmooth optimization problems whose objective is the sum of a smooth function and a finite number of nonnegative proper closed possibly nonsmooth functions (whose proximal mappings are easy to compute), some of which are further composed with linear maps. This kind of problems arises naturally in various applications when different regularizers are introduced for inducing simultaneous structures in the solutions. Solving these problems, however, can be challenging because of the coupled nonsmooth functions: the corresponding proximal mapping can be hard to compute so that standard first-order methods such as the proximal gradient algorithm cannot be applied efficiently. In this paper, we propose a successive difference-of-convex approximation method for solving this kind of problems. In this algorithm, we approximate the nonsmooth functions by their Moreau envelopes in each iteration. Making use of the simple observation that Moreau envelopes of nonnegative proper closed functions are continuous {\em difference-of-convex} functions, we can then approximately minimize the approximation function by first-order methods with suitable majorization techniques. These first-order methods can be implemented efficiently thanks to the fact that the proximal mapping of {\em each} nonsmooth function is easy to compute. Under suitable assumptions, we prove that the sequence generated by our method is bounded and any accumulation point is a stationary point of the objective. We also discuss how our method can be applied to concrete applications such as nonconvex fused regularized optimization problems and simultaneously structured matrix optimization problems, and illustrate the performance numerically for these two specific applications.