40 pages, 8 figures; the NIPS 2011 submission "The Fast Convergence of Boosting" is a brief presentation of the primary results; compared with the JMLR version, this arXiv version has hyperref and some formatting tweaks
Boosting combines weak learners into a predictor with low empirical risk. Its dual constructs a high entropy distribution upon which weak learners and training labels are uncorrelated. This manuscript studies this primal-dual relationship under a broad family of losses, including the exponential loss of AdaBoost and the logistic loss, revealing:
- Weak learnability aids the whole loss family: for any ε>0, O(ln(1/ε)) iterations suffice to produce a predictor with empirical risk ε-close to the infimum;
- The circumstances granting the existence of an empirical risk minimizer may be characterized in terms of the primal and dual problems, yielding a new proof of the known rate O(ln(1/ε));
- Arbitrary instances may be decomposed into the above two, granting rate O(1/ε), with a matching lower bound provided for the logistic loss.
Stochastic Primal Dual Coordinate Method with Non-Uniform Sampling Based on Optimality Violations
Nagoya Institute of Technology（名古屋工业大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所革新智能综合研究中心）；National Institute for Materials Science（国立材料科学研究所）
The University of Iowa（爱荷华大学）；Nanjing University（南京大学）
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 s…
In this work, we propose a deep neural network architecture motivated by primal-dual splitting methods from convex optimization. We show theoretically that there exists a close relation between the derived architecture and residual networks, and further investigate this connection in numerical experiments. Moreover, we demonstrate how our approach can be used to unroll optimization algorithms for certain problems with hard constraints. Using the example of speech dequantization, we show that our method can outperform classical splitting methods when both are applied to the same task.
EPFL（洛桑联邦理工学院）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎综合理工学院）
We introduce a randomly extrapolated primal-dual coordinate descent method that adapts to sparsity of the data matrix and the favorable structures of the objective function. Our method updates only a subset of primal and dual variables with sparse data, and it uses large step sizes with dense data, retaining the benefits of the specific methods designed for each case. In addition to adapting to sparsity, our method attains fast convergence guarantees in favorable cases \textit{without any modifications}. In particular, we prove linear convergence under metric subregularity, which applies to st…
Tilburg University（蒂尔堡大学）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
We give tight bounds on the relation between the primal and dual of various combinatorial dimensions, such as the pseudo-dimension and fat-shattering dimension, for multi-valued function classes. These dimensional notions play an important role in the area of learning theory. We first review some (folklore) results that bound the dual dimension of a function class in terms of its primal, and after that give (almost) matching lower bounds. In particular, we give an appropriate generalization to multi-valued function classes of a well-known bound due to Assouad (1983), that relates the primal an…
WeBank AI Group（微众银行AI集团）；Tencent Inc.（腾讯公司）；University of California, Davis（加州大学戴维斯分校）；The Hong Kong University of Science and Technology（香港科技大学）
Dual averaging-type methods are widely used in industrial machine learning applications due to their ability to promoting solution structure (e.g., sparsity) efficiently. In this paper, we propose a novel accelerated dual-averaging primal-dual algorithm for minimizing a composite convex function. We also derive a stochastic version of the proposed method which solves empirical risk minimization, and its advantages on handling sparse data are demonstrated both theoretically and empirically.
We prove complexity bounds for the primal-dual algorithm with random extrapolation and coordinate descent (PURE-CD), which has been shown to obtain good practical performance for solving convex-concave min-max problems with bilinear coupling. Our complexity bounds either match or improve the best-known results in the literature for both dense and sparse (strongly)-convex-(strongly)-concave problems.
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Federated sampling algorithms have recently gained great popularity in the community of machine learning and statistics. This paper studies variants of such algorithms called Error Feedback Langevin algorithms (ELF). In particular, we analyze the combinations of EF21 and EF21-P with the federated Langevin Monte-Carlo. We propose three algorithms: P-ELF, D-ELF, and B-ELF that use, respectively, primal, dual, and bidirectional compressors. We analyze the proposed methods under Log-Sobolev inequality and provide non-asymptotic convergence guarantees.
We consider the convex-concave saddle point problem $\min_{x}\max_{y} f(x)+y^\top A x-g(y)$ where $f$ is smooth and convex and $g$ is smooth and strongly convex. We prove that if the coupling matrix $A$ has full column rank, the vanilla primal-dual gradient method can achieve linear convergence even if $f$ is not strongly convex. Our result generalizes previous work which either requires $f$ and $g$ to be quadratic functions or requires proximal mappings for both $f$ and $g$. We adopt a novel analysis technique that in each iteration uses a "ghost" update as a reference, and show that the iter…
We develop a parameterized Primal-Dual $π$ Learning method based on deep neural networks for Markov decision process with large state space and off-policy reinforcement learning. In contrast to the popular Q-learning and actor-critic methods that are based on successive approximations to the nonlinear Bellman equation, our method makes primal-dual updates to the policy and value functions utilizing the fundamental linear Bellman duality. Naive parametrization of the primal-dual $π$ learning method using deep neural networks would encounter two major challenges: (1) each update requires computi…
We study structured nonsmooth convex finite-sum optimization that appears widely in machine learning applications, including support vector machines and least absolute deviation. For the primal-dual formulation of this problem, we propose a novel algorithm called \emph{Variance Reduction via Primal-Dual Accelerated Dual Averaging (\vrpda)}. In the nonsmooth and general convex setting, \vrpda~has the overall complexity $O(nd\log\min \{1/ε, n\} + d/ε)$ in terms of the primal-dual gap, where $n$ denotes the number of samples, $d$ the dimension of the primal variables, and $ε$ the desired accuracy…
In this paper, we consider optimizing a smooth, convex, lower semicontinuous function in Riemannian space with constraints. To solve the problem, we first convert it to a dual problem and then propose a general primal-dual algorithm to optimize the primal and dual variables iteratively. In each optimization iteration, we employ a proximal operator to search optimal solution in the primal space. We prove convergence of the proposed algorithm and show its non-asymptotic convergence rate. By utilizing the proposed primal-dual optimization technique, we propose a novel metric learning algorithm wh…
Constrained Markov Decision Process (CMDP) is a natural framework for reinforcement learning tasks with safety constraints, where agents learn a policy that maximizes the long-term reward while satisfying the constraints on the long-term cost. A canonical approach for solving CMDPs is the primal-dual method which updates parameters in primal and dual spaces in turn. Existing methods for CMDPs only use on-policy data for dual updates, which results in sample inefficiency and slow convergence. In this paper, we propose a policy search method for CMDPs called Accelerated Primal-Dual Optimization …
We introduce Primal-Dual Wasserstein GAN, a new learning algorithm for building latent variable models of the data distribution based on the primal and the dual formulations of the optimal transport (OT) problem. We utilize the primal formulation to learn a flexible inference mechanism and to create an optimal approximate coupling between the data distribution and the generative model. In order to learn the generative model, we use the dual formulation and train the decoder adversarially through a critic network that is regularized by the approximate coupling obtained from the primal. Unlike p…
University Paris-Saclay（巴黎萨克雷大学）；Sorbonne University（巴黎索邦大学）；IBISC（IBISC（需注意：IBISC是巴黎萨克雷大学下属研究机构，此处按原文提取））；CNRS（法国国家科学研究中心）；LIP6（LIP6（巴黎索邦大学下属实验室，此处按原文提取））
The domain of online algorithms with predictions has been extensively studied for different applications such as scheduling, caching (paging), clustering, ski rental, etc. Recently, Bamas et al., aiming for an unified method, have provided a primal-dual framework for linear covering problems. They extended the online primal-dual method by incorporating predictions in order to achieve a performance beyond the worst-case case analysis. In this paper, we consider this research line and present a framework to design algorithms with predictions for non-linear packing problems. We illustrate the app…
In this paper, the Primal-Dual UNet for sparse view CT reconstruction is modified to be applicable to cone beam projections and perform reconstructions of entire volumes instead of slices. Experiments show that the PSNR of the proposed method is increased by 10dB compared to the direct FDK reconstruction and almost 3dB compared to the modified original Primal-Dual Network when using only 23 projections. The presented network is not optimized wrt. memory consumption or hyperparameters but merely serves as a proof of concept and is limited to low resolution projections and volumes.
