基于粒子群优化的差分隐私拟合框架

基于粒子群优化的差分隐私拟合框架 Framework of Model Fitting under PSO⁃Guided Differential Privacy 1 first-author 高志强，男，博士生，主要从事隐私保护方面的研究。 E-mail：1090398464@qq.com 高志强，男，博士生，主要从事隐私保护方面的研究。 E-mail：1090398464@qq.com 1090398464@qq.com 1 2 1 3 武警工程大学 乌鲁木齐校区, 新疆 乌鲁木齐 830049 武警工程大学 乌鲁木齐校区, 新疆 乌鲁木齐 830049 Urumqi Campus of Engineering University of People’s Armed Police, Urumqi 830049, Xinjiang,China Urumqi Campus of Engineering University of People’s Armed Police, Urumqi 830049, Xinjiang,China 军委联参55所, 北京 100036 军委联参55所, 北京 100036 The 55th Institute of the Central Military Commission, Beijing 100036, China The 55th Institute of the Central Military Commission, Beijing 100036, China 光大银行, 陕西 西安 710000 光大银行, 陕西 西安 710000 Everbright Bank, Xi’an 710000, Shaanxi, China Everbright Bank, Xi’an 710000, Shaanxi, China 针对监督学习中模型拟合的参数优化问题，提出基于粒子群优化的差分隐私拟合框架。以满足差分隐私的改进指数机制选择粒子群优化算法中个体最优和群体最优粒子，进而驱动模型拟合参数的全局优化，为训练数据集提供差分隐私安全保障。在改进的指数机制中，以拟合函数构造适合于粒子群优化的打分函数，通过参数向量候选集和选定集实现个体最优和群体最优参数向量的更新迭代。给出满足差分隐私的理论证明以及在回归和支持向量机模型中的具体应用。 To solve the parameter optimization problem of model fitting in supervised learning, a framework of model fitting based on PSO⁃guided differential privacy is proposed. The improved exponential mechanism in differential privacy was developed to select the personal and global best particles in particle swarm optimization algorithm, and then the global optimization of fitting parameters was promoted by the protection of differential privacy. In the improved exponential mechanism, a scoring function suitable for particle swarm optimization was constructed by fitting function. The updating iteration of personal and global best particles is realized by candidate and selected sets of parameter vectors. Finally, the theoretical proof satisfying differential privacy and its application in regression and support vector machine model were given in detail. 差分隐私 监督学习 粒子群优化 拟合函数 指数机制 differential privacy supervised learning particle swarm optimization fitting function exponential mechanism 0　 引 言 1 新科技革命的三驾马车“数据、算法、算力”不断推动着大数据、人工智能、移动互联网等技术的进步，同时，网购、移动支付、共享单车、无人驾驶等应用领域的突飞猛进，让用户切实感受到了新技术的便利，然而，Yahoo用户信息外泄、Facebook数据丑闻、Equifax大规模数据泄露等负面事件 [ 1 1 , 2 2 , 3 3 ] ，还有生活中各种不停推送的“标签广告”，以及滴滴顺风车业务“乘客-司机”的标签争议 [ 4 4 , 5 5 , 6 6 ] ，无时无刻不在提醒着我们，不可信第三方以收集个人数据形式提供的“便利”不是“免费的午餐”，利用机器学习将用户分类、打标签、做预测使得个人隐私成为“共享资源”，利益驱动下的数据滥用以及技术局限性造成隐私泄露已经成为备受关注并亟待解决的问题。 将敏感数据集完全暴露于不可信第三方和机器学习模型中，存在巨大的隐私泄露风险和安全隐患。基于差分隐私的敏感数据发布技术，具有强大的隐私保证和鲁棒性 [ 5 5 , 6 6 , 7 7 , 8 8 , 9 9 ] 。但传统差分隐私模型拟合方法存在缺陷：1） 应用范围狭小，只局限于特定方法。文献[ 10 10 ]提出针对回归模型拟合的函数机制FM，虽然在线性回归问题效果很好，但在logistic回归上引入过多噪声扰动；文献[ 11 11 ]针对经验风险最小化问题提出一系列满足差分隐私的回归模型，但是正则因子消耗了大量隐私预算导致整体精度偏低。2） 需要大量的随机扰动来满足差分隐私，导致模型性能差和复杂度高。文献[ 12 12 ]研究了差分隐私支持向量机的核函数，但其解决方案依旧是基于标准支持向量机模型，具有较高全局敏感度并引入大量噪声，针对大量数据的通用性和可用性有待提高。 本文提出满足差分隐私的监督学习模型拟合框架，在模型参数寻优过程中，引入满足差分隐私的粒子群优化算法，根据改进的指数机制，引导以参数向量形式编码的个体最优和群体最优粒子按照差分隐私的方式不断迭代学习。在每轮迭代中，依据粒子群优化算法的开采能力和快速全局收敛性能，不断修正个体最优和群体最优粒子向量，进而向全局最优方向进化。最终在保证机器学习模型可用性的同时，提高拟合框架对训练数据集的隐私保护性能。 1　 预备知识 1 1.1　监督学习模型拟合问题 2 监督学习的目标是利用数据和标签使算法模型具有学习能力，本质上，就是用拟合函数对数据集进行表征学习，直至收敛或达到预设阈值。下面给出机器学习 [ 13 13 ] 、监督学习模型拟合问题 [ 14 14 ] 的定义，以及回归 [ 15 15 ] 、支持向量机模型拟合 [ 16 16 ] 的预备知识。 定义1 （机器学习） 给定数据集 X ={ x 1 , x 2 ,…, x n }，构造从输入空间 X 到输出空间 Y 的函数映射 f ： X → Y ，通过学习和优化策略得到机器学习的最优函数映射 f ，其中输入空间包含 n 个具有 m 类属性的数据样本 x i ={ x i 1 , x i 2 ,…, x im }。 定义2 （监督学习模型拟合问题） 给定具有 n 个元组的数据集 D ∈ Γ n ，在回归、分类等经典监督学习模型中，其学习目标为利用拟合函数 f ( D , ω )，通过调整参数 ω 来最佳拟合模型 M 与敏感数据集 D ，其中，最佳参数 <math id="M1"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118510&type= 3.64066648 3.04799986 对应最优拟合 f ( D , <math id="M2"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118519&type= 3.64066648 3.04799986 )。 1） 回归模型拟合 3 在线性回归模型中，给定数据集 D ={( x 1 , y 1 ), ( x 2 , y 2 ),…,( x m , y m )}，其中， x i ={ x i 1 , x i 2 ,…, x id }, y i ∈ <math id="M3"><mi mathvariant="double-struck">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118528&type= 2.45533323 2.79399991 ，拟合目标是使得 f ( x i )= ωx i + b i ，且 f ( x i )= y i 。 其求解可转化为均方误差的最小化问题 <math display="block" id="M4"><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118538&type= 15.40933418 3.81000018 <math display="block" id="M5"><munder><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo>,</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo></mrow></munder><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>m</mi></mrow></mover><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118548&type= 37.16866684 8.46666718 当数据集 D 由多个属性构成，即得到多元线性回归 f ( x i )= ω T x i + b i （2） 当用线性回归模型的预测结果去逼近函数 <math id="M6"><mi mathvariant="bold-italic">y</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">x</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118565&type= 27.43200111 8.38199997 时，得到logistic回归 <math display="block" id="M7"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">y</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>|</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118573&type= 22.09799957 3.72533321 <math display="block" id="M8"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">x</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118583&type= 19.89666748 8.38199997 其求解目标为 <math display="block" id="M9"><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118591&type= 10.07533360 3.81000018 <math display="block" id="M10"><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118600&type= 4.82599974 3.55599999 <math display="block" id="M11"><munder><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo>,</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo></mrow></munder><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>m</mi></mrow></mover><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118608&type= 69.51133728 8.46666718 2） 支持向量机模型拟合给定数据集 D ={( x 1 , y 1 ), ( x 2 , y 2 ),…, ( x m , y m )}，其中， y i ∈{-1,+1}，支持向量机模型拟合的目标是找到具有最大间隔的划分超平面，即通过调节参数，使得支持向量的间隔最大，其求解目标函数为 <math display="block" id="M12"><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118619&type= 10.07533360 3.81000018 <math display="block" id="M13"><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118628&type= 4.82599974 3.55599999 <math display="block" id="M14"><munder><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo>,</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo></mrow></munder><mo>-</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>α</mi><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>m</mi></mrow></mover><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118638&type= 74.25267029 8.46666718 从上述模型拟合问题的目标函数可以看出，传统监督学习中，通过训练数据集的迭代，调整模型拟合参数 ω ，最佳模型参数 <math id="M15"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118647&type= 3.64066648 3.04799986 使得拟合函数 f ( D , <math id="M16"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118655&type= 3.64066648 3.04799986 )最大化，进而得到模型对数据集的表征学习。然而，可能涉及敏感数据集的模型拟合存在着不可预知的隐私泄露风险，这也就是本文的研究重点与价值所在。 1.2　差分隐私 2 差分隐私保证了攻击者在具有最大化背景知识条件下敏感数据的安全性。隐私保护强度由隐私预算 ε 控制，隐私预算 ε 越小，保护强度越大，但噪声扰动也就越大。 定义3 （兄弟数据集） [ 5 5 ] 同域 ϰ 中数据集 D 和 <math id="M17"><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118664&type= 3.64066648 3.30199981 为兄弟数据集，当且仅当 <math id="M18"><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><mo>-</mo><mi mathvariant="bold-italic">D</mi><mi>'</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118672&type= 20.82799911 4.91066647 ，即两数据集只相差一条数据。 定义4 （差分隐私） [ 5 5 ] 给定兄弟数据集 D 、 <math id="M19"><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118679&type= 3.64066648 3.30199981 和随机函数 f 及其输出域 O ，函数 f 为 ε ⁃差分隐私，当且仅当输出概率 P [ ]满足 <math display="block" id="M20"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">P</mi><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="bold-italic">O</mi><mo stretchy="false">]</mo><mo>≤</mo></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>·</mo><mi mathvariant="bold-italic">P</mi><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="bold-italic">O</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118688&type= 50.03800201 9.48266697 差分隐私的主要实现机制包括Laplace机制和指数机制。指数机制主要应用在非数值型属性的选择方面。 差分隐私中全局敏感度是度量数据集变化对函数输出影响的重要参数，定义如下： 定义5 （全局敏感度） [ 6 6 ] 给定兄弟数据集 D ， <math id="M21"><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118694&type= 3.64066648 3.30199981 ，函数 A 的敏感度 <math display="block" id="M22"><mi mathvariant="normal">Δ</mi><mi>A</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118700&type= 9.65199947 2.79399991 <math display="block" id="M23"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><mi>A</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">D</mi></mrow></mfenced><mo>-</mo><mi>A</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118706&type= 38.01533508 4.91066647 定理1 （指数机制） [ 7 7 ] 给定分类型数据集 Ω ，类别 ω ∈ Ω 服从打分函数 Q ( ω )，当且仅当类别 <math id="M24"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118711&type= 3.64066648 3.04799986 被选择输出的概率 P ( <math id="M25"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118719&type= 3.64066648 3.04799986 )服从参数为Δ Q ( ω )/ ε 的指数分布时，即 <math id="M26"><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>ε</mi><mo>·</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="bold-italic">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mi>Δ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118728&type= 53.84799576 4.14866638 , <math display="block" id="M27"><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo><mo>∝</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>ε</mi><mo>·</mo><mi>Q</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow/><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></msub></mrow></munder><mi mathvariant="normal">Δ</mi><mi>Q</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118734&type= 46.56666946 11.09133339 则此过程满足指数机制下的 ε ⁃差分隐私（Δ Q 为全局敏感度，证明参见文献[ 7 7 ]）。 依据组合特性（证明参见文献[ 5 5 ]），差分隐私在监督学习的迭代优化中可以提供达到组合强度级别的隐私保护。 性质1 （组合特性） [ 5 5 ] 满足差分隐私的算法 A 1 , A 2 ，…， A n ，其隐私预算分别为 ε 1 ， ε 2 ，…， ε n ，则算法组合满足 <math id="M28"><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>n</mi></mrow></mover><msub><mrow><mi>ε</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118745&type= 6.94266701 8.46666718 ⁃差分隐私。 本文提出的满足差分隐私的监督学习模型拟合框架，以全局敏感度、指数机制、组合特性为理论基础，实现基于粒子群优化的差分隐私模型参数全局迭代优化。 1.3　粒子群优化算法 2 1995年提出的粒子群优化算法（particle swarm optimization, PSO）基于种群内个体间协作和信息资源共享实现 [ 17 17 ] ，在个体和群体最优的导向选择模式中，以粒子的位置、速度、适应度三元组为核心，其位置 x ( t )、速度 v ( t )的迭代更新公式为 v ( t )= v ( t -1)+ c 1 r 1 ( p i x ( t -1))+ c 2 r 2 ( p g - x ( t -1))（9） x ( t +1)= x ( t )+ v ( t )（10） 其中， c 1 、 c 2 、 r 1 、 r 2 为预设参数， p i 和 p g 分别为个体最优和群体最优，适应度值与目标函数值对应。粒子群优化的全局收敛性由吸引子定理保证，证明参见文献[ 18 18 ]，其中 φ 1 = c 1 r 1 ， φ 2 = c 2 r 2 。 定理2 （吸引子定理） [ 18 18 ] 对于简化的PSO模型，粒子位置序列{ x ( t )}满足如下极限 <math display="block" id="M29"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi mathvariant="bold-italic">x</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118750&type= 18.37266731 4.82599974 <math display="block" id="M30"><mfrac><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi mathvariant="bold-italic">g</mi></mrow></msub></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118756&type= 19.64266586 8.55133343 2　 支持差分隐私的拟合框架 1 2.1　差分隐私模型拟合算法框架 2 根据1.1节中的监督学习模型拟合问题定义，借鉴粒子群优化算法中个体和群体最优的导向选择模式，按照差分隐私中指数机制来实现模型学习性能与数据集隐私性保护的平衡，基于粒子群优化的差分隐私模型拟合算法（PSO⁃guided differential privacy fitting，PSO⁃DP），如算法1所示，其关键步骤如 图1 图1 所示。 算法1 PSO⁃DP Input ：数据集 D 、拟合函数 f 、隐私预算 ε ，参数向量候选集 Ω 大小 m ，参数向量选定集 Ω' 大小 m <math id="M31"><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118774&type= 0.76199996 3.21733332 ，迭代次数 t 。 Output ：最优参数 <math id="M32"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118779&type= 3.64066648 3.04799986 。 ① 随机初始化具有 m 个参数向量候选集 Ω ； FOR i =1 TO r DO ② 隐私预算为 ε/r ，以差分隐私选择策略从候选集 Ω 中选择个体最优和群体最优参数向量，并存入参数向量选定集 Ω' ； ③ 根据位置和速度更新公式（9）和（10），在选定集 Ω' 中选择个体最优和群体最优粒子来生成新的参数向量，并更新候选集 Ω ，进而按照差分隐私的方式向全局最优方向进化； END FOR ④ 最后从选定集 Ω' 中按照改进的指数机制输出最优参数向量 <math id="M33"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118786&type= 3.64066648 3.04799986 。 在算法1中，参数向量候选集 Ω 大小 m 为粒子群优化中的种群大小，参数向量选定集为每轮迭代的个体最优和群体最优集合，步骤②中涉及的满足差分隐私的选择策略由算法2中DP⁃Select实现，其中，算法2中步骤③的指数机制将在2.2小节详细介绍。 算法2 DP ⁃ Select Input ：数据集 D 、拟合函数 f 、隐私预算 <math id="M34"><mi>ε</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118792&type= 2.20133328 3.30199981 ，参数向量候选集 Ω ，选定集大小 <math id="M35"><mi>m</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118799&type= 3.55599999 3.30199981 。 Output ：参数向量选定集 Ω' 。 ① 初始化参数向量选定集 Ω' 为空集； ② 计算所有 ω ∈ Ω 的 f ( D , ω )； FOR i =1 TO <math id="M36"><mi>m</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118807&type= 3.55599999 3.30199981 DO ③ 按照参数为 <math id="M37"><mi>ε</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118814&type= 2.20133328 3.30199981 / <math id="M38"><mi>m</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118822&type= 3.55599999 3.30199981 的指数机制来选择 <math id="M39"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118830&type= 3.64066648 3.04799986 ，使得 f ( D , <math id="M40"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118837&type= 3.64066648 3.04799986 )最大化； ④ 把 <math id="M41"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118846&type= 3.64066648 3.04799986 加入选定集 Ω' 并从候选集 Ω 中删除； END FOR ⑤ 输出选定集 Ω' 。 2.2　改进的指数机制 2 针对模型拟合中差分隐私选择策略，利用改进指数机制实现粒子群优化中个体最优和群体最优粒子的选择。主要改进体现在对指数机制的打分函数上，拟合函数在数据集 D 上的打分函数定义为 <math display="block" id="M42"><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="bold-italic">D</mi></mrow></munder><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118855&type= 39.87800217 6.77333355 其中， c 为调节常数， q ( t , ω )用来度量机器学习模型与 D 中数据 t 的拟合程度。由定理1可知，针对 D ，若输出参数向量 ω 服从exp( ε <math id="M43"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118863&type= 1.10066664 2.79399991 f ( D , ω )/ Δ )，则可以得到 <math display="block" id="M44"><mi>Δ</mi><mo>≥</mo><mn mathvariant="normal">2</mn><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118872&type= 54.77933502 5.16466665 Δ 与全局敏感度密切相关，为全局敏感度2倍的上界。若全局敏感度远大于 ε <math id="M45"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118877&type= 1.10066664 2.79399991 f ( D , <math id="M46"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118884&type= 3.64066648 3.04799986 )，则输出的 <math id="M47"><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118890&type= 3.64066648 3.04799986 概率几乎相同。为此，本文改进的指数机制采用的参数 Δ 具有更小下界，满足下式 <math display="block" id="M48"><mi>Δ</mi><mo>≥</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>{</mo><mn mathvariant="normal">2</mn><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118896&type= 55.71066666 5.16466665 , <math display="block" id="M49"><mn mathvariant="normal">2</mn><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118905&type= 41.99466705 5.16466665 } 引理1 当 <math id="M50"><mi>Δ</mi><mo>≥</mo><mn mathvariant="normal">2</mn><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118912&type= 47.32866669 5.16466665 )时，改进的指数机制满足差分隐私。 证 给定邻接数据集 D 、 <math id="M51"><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118919&type= 3.64066648 3.30199981 中任意记录 t 、 <math id="M52"><mi>t</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118926&type= 1.86266661 3.30199981 ，对任意拟合参数向量 <math id="M53"><mi mathvariant="bold-italic">ω</mi><mo>∈</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118933&type= 6.26533318 3.13266659 Ω ，可以得到 <math display="block" id="M54"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi>D</mi><mi mathvariant="normal">'</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118938&type= 43.43400192 5.58799982 = <math display="block" id="M55"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi>D</mi><mi mathvariant="normal">'</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118947&type= 39.28533173 5.58799982 = <math display="block" id="M56"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">t</mi><mo>,</mo><mi>t</mi><mi>'</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118955&type= 39.28533173 5.16466665 引理2 当 <math id="M57"><mi>Δ</mi><mo>≥</mo><mn mathvariant="normal">2</mn><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118965&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118962&type= 50.12266541 5.16466665 时，改进的指数机制满足差分隐私。 证 给定邻接数据集 D 、 <math id="M58"><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118969&type= 3.64066648 3.30199981 中任意记录 t 、 <math id="M59"><mi>t</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118978&type= 1.86266661 3.30199981 ，对于改进指数机制的任意输出 <math id="M60"><mi mathvariant="bold-italic">ω</mi><mo>∈</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118985&type= 6.26533318 3.13266659 Ω ，可以得 <math display="block" id="M61"><mfrac><mrow><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118993&type= 29.97199821 8.80533314 <math display="block" id="M62"><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>ε</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="bold-italic">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mi>Δ</mi><mo stretchy="false">)</mo></mrow><mrow><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><msub><mrow/><mrow><mi>ω</mi><mi>'</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></msub></mrow></munder><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>ε</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mi mathvariant="bold-italic">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mi>Δ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119001&type= 64.09266663 11.51466751 结合公式（7），可以得到 <math display="block" id="M63"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>ω</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi>ω</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mi>Δ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119009&type= 70.27333069 14.73199940 由引理1得到 <math display="block" id="M64"><mtable columnalign="left"><mtr><mtd><mi>Δ</mi><mo>≥</mo><mn mathvariant="normal">2</mn><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≥</mo></mtd></mtr><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>-</mo><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mi mathvariant="normal">'</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119016&type= 54.18666840 19.04999924 可知 <math id="M65"><mfrac><mrow><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="bold-italic">P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">D</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>≤</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>ε</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119025&type= 34.03599930 8.80533314 结合引理1和引理2，可证改进的指数机制满足差分隐私。结合定理2及差分隐私组合特性，可证基于粒子群优化的差分隐私拟合框架可以依概率收敛，并满足 ε ⁃差分隐私。 3　 应 用 1 在回归、支持向量机模型中，基于粒子群优化的差分隐私拟合框架涉及的打分函数和与敏感度函数密切相关的参数 Δ 具体应用如下： 在线性回归模型拟合模型和logistic回归模型中，拟合函数分别为（1）式和（4）式，根据2.2节中改进的指数机制定义的打分函数，可以作如下设置， <math id="M66"><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119033&type= 36.49133301 4.06400013 <math id="M67"><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119041&type= 42.24866486 3.97933340 ，同时根据（14）式，可以得到 <math display="block" id="M68"><mtable columnalign="left"><mtr><mtd><mi>Δ</mi><mo>≥</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>{</mo><munder><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>d</mi></mrow></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd><munder><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>d</mi></mrow></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mi mathvariant="normal">'</mi></mrow></mfenced><mo stretchy="false">)</mo><mo>}</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119048&type= 50.63066864 19.21933365 在支持向量机模型拟合中，拟合函数为1.1节中的（5）式，根据改进的指数机制中定义的打分函数，可以设置 <math display="block" id="M69"><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mi>α</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo>{</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>y</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119054&type= 57.99666595 4.06400013 同时根据公式（14），可以得到 <math display="block" id="M70"><mi>Δ</mi><mo>≥</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119062&type= 7.11199999 3.30199981 <math display="block" id="M71"><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="{" close="" separators="|"><mrow><munder><mrow><mn mathvariant="normal">2</mn><mi>α</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>d</mi></mrow></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119071&type= 40.63999939 9.48266697 , <math display="block" id="M72"><mfenced open="" close="}" separators="|"><mrow><munder><mrow><mn mathvariant="normal">2</mn><mi>α</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>∈</mo><mi mathvariant="bold-italic">Ω</mi></mrow></munder><mo stretchy="false">(</mo><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>d</mi></mrow></mover><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mi mathvariant="normal">'</mi></mrow></mfenced><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119078&type= 37.42266846 10.58333302 此外，本文的算法框架及机制均可尝试给出机器学习中的神经网络、深度学习中的随机梯度下降算法的隐私保护解决方案。 4　 结 论 1 本文针对监督学习模型拟合中涉及的训练数据集隐私保护问题，引入满足差分隐私的粒子群优化算法，按照改进的指数机制选择个体最优和群体最优粒，引导以参数向量形式编码的粒子进行全局寻优。理论证明，本文算法框架满足差分隐私，可以实现监督学习中训练数据集的敏感信息保护。最后给出了框架在回归、支持向量机模型参数优化过程中打分函数的设置方式。下一步工作将深入探索本文框架在多种深度神经网络中的应用途径。 参考文献 NOZARI E , TALLAPRAGADA P , CORTES J . 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GAO Zhiqiang,CUI Xiaolong,YANG Weifeng, et al .Framework of Model Fitting under PSO⁃Guided Differential Privacy [J]. J Wuhan Univ （ Nat Sci Ed ）,2019,65(2):213⁃217. DOI:10.14188/j.1671⁃8836.2019.02.011(Ch). GAO Zhiqiang,CUI Xiaolong,YANG Weifeng,et al.Framework of Model Fitting under PSO⁃Guided Differential Privacy [J].J Wuhan Univ（Nat Sci Ed）,2019,65(2):213⁃217. DOI:10.14188/j.1671⁃8836.2019.02.011(Ch). 国家自然科学基金（U1603261）；新疆维吾尔自治区自然科学基金（2016D01A080） 2018-09-08 2019-04-24 2020-11-25 武汉大学学报（理学版） 2 65