摘要翻译：
具有二元铰链损失及其变体的正则化风险最小化是许多机器学习问题的核心。正则化风险最小化的束方法(BMRM)和与之密切相关的SVMStruct被认为是解决该问题的最佳通用求解器。最近的研究表明，BMRM需要$O(1/\epsilon)$迭代才能收敛到$\epsilon$精确的解决方案。在本文的第一部分，我们利用Hadamard矩阵构造了一个正则化的风险最小化问题，并证明了这些风险率是不可提高的。然后我们展示了如何利用目标函数的结构来设计二元铰链损失的算法，该算法在$O(1/\sqrt{\epsilon}）$迭代中收敛到$\epsilon$精确解。
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英文标题：
《Lower Bounds for BMRM and Faster Rates for Training SVMs》
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作者：
Ankan Saha (1), Xinhua Zhang (2), S.V.N. Vishwanathan (3) ((1)
  University of Chicago, (2) Australian National University, NICTA, (3) Purdue
  University)
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最新提交年份：
2009
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分类信息：

一级分类：Computer Science        计算机科学
二级分类：Machine Learning        机器学习
分类描述：Papers on all aspects of machine learning research (supervised, unsupervised, reinforcement learning, bandit problems, and so on) including also robustness, explanation, fairness, and methodology. cs.LG is also an appropriate primary category for applications of machine learning methods.
关于机器学习研究的所有方面的论文（有监督的，无监督的，强化学习，强盗问题，等等），包括健壮性，解释性，公平性和方法论。对于机器学习方法的应用，CS.LG也是一个合适的主要类别。
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一级分类：Computer Science        计算机科学
二级分类：Artificial Intelligence        人工智能
分类描述：Covers all areas of AI except Vision, Robotics, Machine Learning, Multiagent Systems, and Computation and Language (Natural Language Processing), which have separate subject areas. In particular, includes Expert Systems, Theorem Proving (although this may overlap with Logic in Computer Science), Knowledge Representation, Planning, and Uncertainty in AI. Roughly includes material in ACM Subject Classes I.2.0, I.2.1, I.2.3, I.2.4, I.2.8, and I.2.11.
涵盖了人工智能的所有领域，除了视觉、机器人、机器学习、多智能体系统以及计算和语言（自然语言处理），这些领域有独立的学科领域。特别地，包括专家系统，定理证明（尽管这可能与计算机科学中的逻辑重叠），知识表示，规划，和人工智能中的不确定性。大致包括ACM学科类I.2.0、I.2.1、I.2.3、I.2.4、I.2.8和I.2.11中的材料。
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一级分类：Computer Science        计算机科学
二级分类：Data Structures and Algorithms        数据结构与算法
分类描述：Covers data structures and analysis of algorithms. Roughly includes material in ACM Subject Classes E.1, E.2, F.2.1, and F.2.2.
涵盖数据结构和算法分析。大致包括ACM学科类E.1、E.2、F.2.1和F.2.2中的材料。
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英文摘要：
  Regularized risk minimization with the binary hinge loss and its variants lies at the heart of many machine learning problems. Bundle methods for regularized risk minimization (BMRM) and the closely related SVMStruct are considered the best general purpose solvers to tackle this problem. It was recently shown that BMRM requires $O(1/\epsilon)$ iterations to converge to an $\epsilon$ accurate solution. In the first part of the paper we use the Hadamard matrix to construct a regularized risk minimization problem and show that these rates cannot be improved. We then show how one can exploit the structure of the objective function to devise an algorithm for the binary hinge loss which converges to an $\epsilon$ accurate solution in $O(1/\sqrt{\epsilon})$ iterations.
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PDF链接：
https://arxiv.org/pdf/0909.1334