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Register a new userPrimal-Dual Rates and Certificates
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Added by
Martin Jaggi
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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We propose an algorithm-independent framework to equip existing optimization methods with primal-dual certificates. Such certificates and corresponding rate of convergence guarantees are important for practitioners to diagnose progress, in particular in machine learning applications. We obtain new primal-dual convergence rates, e.g., for the Lasso as well as many L1, Elastic Net, group Lasso and TV-regularized problems. The theory applies to any norm-regularized generalized linear model. Our approach provides efficiently computable duality gaps which are globally defined, without modifying the original problems in the region of interest.
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We propose a variant of the Frank-Wolfe algorithm for solving a class of sparse/low-rank optimization problems. Our formulation includes Elastic Net, regularized SVMs and phase retrieval as special cases. The proposed Primal-Dual Block Frank-Wolfe algorithm reduces the per-iteration cost while maintaining linear convergence rate. The per iteration cost of our method depends on the structural complexity of the solution (i.e. sparsity/low-rank) instead of the ambient dimension. We empirically show that our algorithm outperforms the state-of-the-art methods on (multi-class) classification tasks.
Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrices. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses such as the least-squares loss. In this article, we propose a first-order primal-dual algorithm for non-negative decomposition problems (where one factor is fixed) with the KL divergence, based on the Chambolle-Pock algorithm. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic examples, face recognition or music source separation datasets, it is either faster than existing algorithms, or leads to improved local optima, or both.
Previous studies on stochastic primal-dual algorithms for solving min-max problems with faster convergence heavily rely on the bilinear structure of the problem, which restricts their applicability to a narrowed range of problems. The main contribution of this paper is the design and analysis of new stochastic primal-dual algorithms that use a mixture of stochastic gradient updates and a logarithmic number of deterministic dual updates for solving a family of convex-concave problems with no bilinear structure assumed. Faster convergence rates than $O(1/sqrt{T})$ with $T$ being the number of stochastic gradient updates are established under some mild conditions of involved functions on the primal and the dual variable. For example, for a family of problems that enjoy a weak strong convexity in terms of the primal variable and has a strongly concave function of the dual variable, the convergence rate of the proposed algorithm is $O(1/T)$. We also investigate the effectiveness of the proposed algorithms for learning robust models and empirical AUC maximization.
Let $G$ be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching $B$ is popular if $B$ does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if $G$ admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that approximately popular branchings always exist.
The problem of learning to generalize to unseen classes during training, known as few-shot classification, has attracted considerable attention. Initialization based methods, such as the gradient-based model agnostic meta-learning (MAML), tackle the few-shot learning problem by learning to fine-tune. The goal of these approaches is to learn proper model initialization, so that the classifiers for new classes can be learned from a few labeled examples with a small number of gradient update steps. Few shot meta-learning is well-known with its fast-adapted capability and accuracy generalization onto unseen tasks. Learning fairly with unbiased outcomes is another significant hallmark of human intelligence, which is rarely touched in few-shot meta-learning. In this work, we propose a Primal-Dual Fair Meta-learning framework, namely PDFM, which learns to train fair machine learning models using only a few examples based on data from related tasks. The key idea is to learn a good initialization of a fair models primal and dual parameters so that it can adapt to a new fair learning task via a few gradient update steps. Instead of manually tuning the dual parameters as hyperparameters via a grid search, PDFM optimizes the initialization of the primal and dual parameters jointly for fair meta-learning via a subgradient primal-dual approach. We further instantiate examples of bias controlling using mean difference and decision boundary covariance as fairness constraints to each task for supervised regression and classification, respectively. We demonstrate the versatility of our proposed approach by applying our approach to various real-world datasets. Our experiments show substantial improvements over the best prior work for this setting.
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