Making Large-Scale SVM Learning Practical
面向小样本学习的轻量化知识蒸馏
面向小样本学习的轻量化知识蒸馏面向小样本学习的轻量化知识蒸馏随着深度学习的迅猛发展，人工智能技术在多个领域展现出了巨大的潜力。
然而，深度神经网络需要大量的标记样本进行训练，以获得良好的性能。
这在许多实际应用中，如医疗诊断和工业控制等领域，往往是一个挑战。
因此，如何在小样本学习中取得良好的效果成为了一个研究热点。
在小样本学习中，一个常见的方法是使用迁移学习来利用已有的知识。
知识蒸馏（knowledge distillation）作为一种迁移学习的方法，可以帮助将复杂模型的知识转移到简化的模型上，以便在小样本学习任务中发挥作用。
知识蒸馏的基本思想是将复杂模型（教师模型）的知识转化为简化模型（学生模型）可以理解的形式。
这样，学生模型在学习过程中可以依靠教师模型的知识来辅助训练，从而在小样本学习任务中获得更好的性能。
知识蒸馏方法通常通过两个步骤来实现：首先，使用教师模型对大规模数据进行训练，得到教师模型的预测结果和中间层特征；然后，使用这些预测结果和特征来训练学生模型。
通过这种方式，学生模型可以从教师模型的丰富知识中受益，提高在小样本学习任务中的性能。
然而，传统的知识蒸馏方法存在一个问题，就是学生模型往往比教师模型更大，带来了额外的计算和存储开销。
为了解决这个问题，研究者们提出了一种轻量化知识蒸馏的方法，即将教师模型的复杂信息进行简化和压缩，以满足学生模型的轻量化需求。
轻量化知识蒸馏的主要思想是通过模型的压缩和简化，减少参数和计算量，从而在小样本学习任务中保持高性能。
具体来说，有以下几个步骤：首先，对教师和学生模型进行结构压缩。
通常情况下，教师模型拥有较多的参数和层数，而学生模型需要更小更轻量化。
因此，可以通过剪枝、裁剪或者网络结构优化等方法，减少教师模型的参数和层数，使其适应学生模型的轻量化需求。
其次，对教师模型的知识进行压缩。
教师模型中的知识既包括预测结果，也包括中间层特征。
对于预测结果，可以使用软标签来替代硬标签，软标签是一种概率分布形式，可以提供更加丰富的信息。
基于毫米波雷达的舱内儿童遗留检测系统设计和验证
AUTOMOBILE DESIGN | 汽车设计基于毫米波雷达的舱内儿童遗留检测系统设计和验证祁淼盐城工业职业技术学院 江苏省盐城市 224005摘 要： 为了保护儿童避免被单独遗留在舱内，提出了基于毫米波雷达的传感器的检测方法。
本方法采集毫米波多普勒效应产生的时域和频域信息，在LC-KSVD算法中加入主成分分析和随机森林的降维方法提取特征，对特征最组合。
将组合的特征用SVM做分类，区分出存在和不存在儿童的场景。
实验部分根据用车习惯，收集设计了正样本的采集和负样本的采集。
实验表明，与同类的研究相比，本方法有更好的环境适应性可以避免相机等传统方法的局限性。
关键词：毫米波雷达 LC-KSVD算法 儿童检测 SVM分类1 前言汽车是许多家庭的标配，最近几年车辆设计的趋势之一是大天窗装在越来越多的车型上，2022年销量前十的车型[1]中除了五菱宏光MINIEV外都配有天窗，其中半数配置了全景天窗。
如果车辆暴露在阳光下，更多的热量通过天窗传递到舱内，在密闭环境中热量聚集使舱内温度快速上升。
幼儿被遗留在无人看管的车汽车里几分钟可能导致中暑和死亡。
大多数父母相信自己永远不会忘记坐在后座上的孩子。
现实情况是在过去的15年中美国有1000名儿童在车上因为过热去世，其中超过88%的幼儿小于3个月[2]。
常见的活体检测手段为视觉，文强[3]等人通过图像的几何形态学关系区分成年人和儿童（<6岁）的脸部特征。
公妍苏[4]等人利用树莓派作为计算平台开发基于Adaptive Boosting的儿童车内遗留检测系统。
但是，大多数婴儿座椅会配置遮阳帘，导致婴儿大多数特征无法被摄像头捕捉，造成漏报。
而且舱内过多的布置摄像头也会引起用户的反感。
董启迪[5]等人读取车上压力传感器的数值推测大人和孩子，结合车门开关等信息实现遗留检测。
0-6岁的孩子成长快，体重分布区间规律性不强，存在较大的误报风险。
本文采用基于毫米波雷达的技术方案，利用多普勒效应检测车内的运动情况，通过空间定位过滤车外的和非成员区间的运动，利用人体运动时频过滤出人体的运动。
Large-scale machine learning with stochastic gradient descent
Large-Scale Machine Learningwith Stochastic Gradient DescentL´e on BottouNEC Labs America,Princeton NJ08542,USAleon@Abstract.During the last decade,the data sizes have grown faster than the speed of processors.In this context,the capabilities of statistical machine learning meth-ods is limited by the computing time rather than the sample size.A more pre-cise analysis uncovers qualitatively different tradeoffs for the case of small-scale and large-scale learning problems.The large-scale case involves the computational complexity of the underlying optimization algorithm in non-trivial ways.Unlikely optimization algorithms such as stochastic gradient descent show amazing perfor-mance for large-scale problems.In particular,second order stochastic gradient and averaged stochastic gradient are asymptotically efficient after a single pass on the training set.Keywords:Stochastic gradient descent,Online learning,Efficiency1IntroductionThe computational complexity of learning algorithm becomes the critical limiting factor when one envisions very large datasets.This contribution ad-vocates stochastic gradient algorithms for large scale machine learning prob-lems.Thefirst section describes the stochastic gradient algorithm.The sec-ond section presents an analysis that explains why stochastic gradient algo-rithms are attractive when the data is abundant.The third section discusses the asymptotical efficiency of estimates obtained after a single pass over the training set.The last section presents empirical evidence.2Learning with gradient descentLet usfirst consider a simple supervised learning setup.Each example z is a pair(x,y)composed of an arbitrary input x and a scalar output y.We consider a loss function (ˆy,y)that measures the cost of predictingˆy when the actual answer is y,and we choose a family F of functions f w(x)parametrized by a weight vector w.We seek the function f∈F that minimizes the loss Q(z,w)= (f w(x),y)averaged on the examples.Although we would like to average over the unknown distribution dP(z)that embodies the Laws of2L´e on BottouNature,we must often settle for computing the average on a sample z1...z n.E(f)=(f(x),y)dP(z)E n(f)=1nni=1(f(x i),y i)(1)The empirical risk E n(f)measures the training set performance.The expected risk E(f)measures the generalization performance,that is,the expected performance on future examples.The statistical learning theory(Vapnik and Chervonenkis,1971)justifies minimizing the empirical risk instead of the expected risk when the chosen family F is sufficiently restrictive.2.1Gradient descentIt has often been proposed(e.g.,Rumelhart et al.,1986)to minimize the empirical risk E n(f w)using gradient descent(GD).Each iteration updates the weights w on the basis of the gradient of E n(f w),w t+1=w t−γ1nni=1∇w Q(z i,w t),(2)whereγis an adequately chosen gain.Under sufficient regularity assumptions, when the initial estimate w0is close enough to the optimum,and when the gainγis sufficiently small,this algorithm achieves linear convergence(Dennis and Schnabel,1983),that is,−logρ∼t,whereρrepresents the residual error.Much better optimization algorithms can be designed by replacing the scalar gainγby a positive definite matrixΓt that approaches the inverse of the Hessian of the cost at the optimum:w t+1=w t−Γt 1nni=1∇w Q(z i,w t).(3)This second order gradient descent(2GD)is a variant of the well known Newton algorithm.Under sufficiently optimistic regularity assumptions,and provided that w0is sufficiently close to the optimum,second order gradient descent achieves quadratic convergence.When the cost is quadratic and the scaling matrixΓis exact,the algorithm reaches the optimum after a single iteration.Otherwise,assuming sufficient smoothness,we have−log logρ∼t.2.2Stochastic gradient descentThe stochastic gradient descent(SGD)algorithm is a drastic simplification. Instead of computing the gradient of E n(f w)exactly,each iteration estimates this gradient on the basis of a single randomly picked example z t:w t+1=w t−γt∇w Q(z t,w t).(4)Large-Scale Machine Learning 3The stochastic process {w t ,t =1,...}depends on the examples randomly picked at each iteration.It is hoped that (4)behaves like its expectation (2)despite the noise introduced by this simplified procedure.Since the stochastic algorithm does not need to remember which examples were visited during the previous iterations,it can process examples on the fly in a deployed system.In such a situation,the stochastic gradient descent directly optimizes the expected risk,since the examples are randomly drawn from the ground truth distribution.The convergence of stochastic gradient descent has been studied exten-sively in the stochastic approximation literature.Convergence results usuallyrequire decreasing gains satisfying the conditions t γ2t <∞and t γt =∞.The Robbins-Siegmund theorem (Robbins and Siegmund,1971)provides the means to establish almost sure convergence under mild conditions (Bottou,1998),including cases where the loss function is not everywhere differentiable.The convergence speed of stochastic gradient descent is in fact limited by the noisy approximation of the true gradient.When the gains decrease too slowly,the variance of the parameter estimate w t decreases equally slowly.When the gains decrease too quickly,the expectation of the parameter es-timate w t takes a very long time to approach the optimum.Under suffi-cient regularity conditions (e.g.Murata,1998),the best convergence speed is achieved using gains γt ∼t −1.The expectation of the residual error then decreases with similar speed,that is,E ρ∼t −1.The second order stochastic gradient descent (2SGD)multiplies the gradi-ents by a positive definite matrix Γt approaching the inverse of the Hessian :w t +1=w t −γt Γt ∇w Q (z t ,w t ).(5)Unfortunately,this modification does not reduce the stochastic noise and therefore does not significantly improve the variance of w t .Although con-stants are improved,the expectation of the residual error still decreases like t −1,that is,E ρ∼t −1,(e.g.Bordes et al.,2009,appendix).2.3Stochastic gradient examplesTable 1illustrates stochastic gradient descent algorithms for a number of classic machine learning schemes.The stochastic gradient descent for the Perceptron,for the Adaline,and for k -Means match the algorithms proposed in the original papers.The SVM and the Lasso were first described with traditional optimization techniques.Both Q svm and Q lasso include a regular-ization term controlled by the hyperparameter λ.The K-means algorithm converges to a local minimum because Q kmeans is nonconvex.On the other hand,the proposed update rule uses second order gains that ensure a fast convergence.The proposed Lasso algorithm represents each weight as the difference of two positive variables.Applying the stochastic gradient rule to these variables and enforcing their positivity leads to sparser solutions.4L´e on BottouTable 1.Stochastic gradient algorithms for various learning systems.Loss Stochastic gradient algorithmAdaline (Widrow and Hoff,1960)Q adaline =12 y −w Φ(x ) 2Φ(x )∈R d ,y =±1w ←w +γt y t −w Φ(x t ) Φ(x t )Perceptron (Rosenblatt,1957)Q perceptron =max {0,−y w Φ(x )}Φ(x )∈R d ,y =±1w ←w +γt y t Φ(x t )if y t w Φ(x t )≤00otherwise K-Means (MacQueen,1967)Q kmeans =min k 1(z −w k )2z ∈R d ,w 1...w k ∈R dn 1...n k ∈N ,initially 0k ∗=arg min k (z t −w k )2n k ∗←n k ∗+1w k ∗←w k ∗+1n k∗(z t −w k ∗)SVM (Cortes and Vapnik,1995)Q svm =λw 2+max {0,1−y w Φ(x )}Φ(x )∈R d ,y =±1,λ>0w ←w −γt λw if y t w Φ(x t )>1,λw −y t Φ(x t )sso (Tibshirani,1996)Q lasso =λ|w |1+12 y −w Φ(x ) 2w =(u 1−v 1,...,u d −v d )Φ(x )∈R d ,y ∈R ,λ>0u i ← u i −γt λ−(y t −w Φ(x t ))Φi (x t ) +v i ← v i −γt λ+(y t −w t Φ(x t ))Φi (x t ) +with notation [x ]+=max {0,x }.3Learning with large training setsLet f ∗=arg min f E (f )be the best possible prediction function.Since we seek the prediction function from a parametrized family of functions F ,let f ∗F =arg min f ∈F E (f )be the best function in this family.Since we optimize the empirical risk instead of the expected risk,let f n =arg min f ∈F E n (f )be the empirical optimum.Since this optimization can be costly,let us stop the algorithm when it reaches an solution ˜f n that minimizes the objective function with a predefined accuracy E n (˜f n )<E n (f n )+ρ.3.1The tradeoffs of large scale learningThe excess error E =E E (˜f n)−E (f ∗) can be decomposed in three terms (Bottou and Bousquet,2008):E =E E (f ∗F )−E (f ∗) +E E (f n )−E (f ∗F ) +E E (˜f n )−E (f n ) .(6)•The approximation error E app =E E (f ∗F )−E (f ∗) measures how closelyfunctions in F can approximate the optimal solution f ∗.The approxima-tion error can be reduced by choosing a larger family of functions.•The estimation error E est =E E (f n )−E (f ∗F ) measures the effect of minimizing the empirical risk E n (f )instead of the expected risk E (f ).Large-Scale Machine Learning 5The estimation error can be reduced by choosing a smaller family of functions or by increasing the size of the training set.•The optimization error E opt =E (˜f n )−E (f n )measures the impact of the approximate optimization on the expected risk.The optimization error can be reduced by running the optimizer longer.The additional computing time depends of course on the family of function and on the size of the training set.Given constraints on the maximal computation time T max and the maximal training set size n max ,this decomposition outlines a tradeoffinvolving the size of the family of functions F ,the optimization accuracy ρ,and the number of examples n effectively processed by the optimization algorithm.min F ,ρ,nE =E app +E est +E opt subject to n ≤n max T (F ,ρ,n )≤T max (7)Two cases should be distinguished:•Small-scale learning problems are first constrained by the maximal num-ber of examples.Since the computing time is not an issue,we can reduce the optimization error E opt to insignificant levels by choosing ρarbitrarily small,and we can minimize the estimation error by chosing n =n max .We then recover the approximation-estimation tradeoffthat has been widely studied in statistics and in learning theory.•Large-scale learning problems are first constrained by the maximal com-puting time.Approximate optimization can achieve better expected risk because more training examples can be processed during the allowed time.The specifics depend on the computational properties of the chosen op-timization algorithm.3.2Asymptotic analysisSolving (7)in the asymptotic regime amounts to ensuring that the terms of the decomposition (6)decrease at similar rates.Since the asymptotic conver-gence rate of the excess error (6)is the convergence rate of its slowest term,the computational effort required to make a term decrease faster would be wasted.For simplicity,we assume in this section that the Vapnik-Chervonenkis dimensions of the families of functions F are bounded by a common constant.We also assume that the optimization algorithms satisfy all the assumptions required to achieve the convergence rates discussed in section 2.Similar anal-yses can be carried out for specific algorithms under weaker assumptions (e.g.Shalev-Shwartz and Srebro,2008).A simple application of the uniform convergence results of (Vapnik and Chervonenkis,1971)gives then the upper bound E =E app +E est +E opt =E app +O log n n+ρ .6L´e on BottouTable 2.Asymptotic equivalents for various optimization algorithms:gradient descent(GD,eq.2),second order gradient descent(2GD,eq.3),stochastic gradient descent(SGD,eq.4),and second order stochastic gradient descent(2SGD,eq.5). Although they are the worst optimization algorithms,SGD and2SGD achieve the fastest convergence speed on the expected risk.They differ only by constant factors not shown in this table,such as condition numbers and weight vector dimension.GD2GD SGD2SGD Time per iteration:n n11Iterations to accuracyρ:log1ρlog log1ρ1ρ1ρTime to accuracyρ:n log1ρn log log1ρ1ρ1ρTime to excess error E:1E1/αlog21E1E1/αlog1Elog log1E1E1EUnfortunately the convergence rate of this bound is too pessimistic.Faster convergence occurs when the loss function has strong convexity properties (Lee et al.,2006)or when the data distribution satisfies certain assumptions (Tsybakov,2004).The equivalenceE=E app+E est+E opt∼E app+log nnα+ρ,for someα∈1,1,(8)provides a more realistic view of the asymptotic behavior of the excess er-ror(e.g.Massart,2000,Bousquet,2002).Since the three component of the excess error should decrease at the same rate,the solution of the tradeoffproblem(7)must then obey the multiple asymptotic equivalencesE∼E app∼E est∼E opt∼log nnα∼ρ.(9)Table2summarizes the asymptotic behavior of the four gradient algo-rithm described in section2.Thefirst three rows list the computational cost of each iteration,the number of iterations required to reach an optimization accuracyρ,and the corresponding computational cost.The last row provides a more interesting measure for large scale machine learning purposes.Assum-ing we operate at the optimum of the approximation-estimation-optimization tradeoff(7),this line indicates the computational cost necessary to reach a predefined value of the excess error,and therefore of the expected risk.This is computed by applying the equivalences(9)to eliminate n andρfrom the third row results.Although the stochastic gradient algorithms,SGD and2SGD,are clearly the worst optimization algorithms(third row),they need less time than the other algorithms to reach a predefined expected risk(fourth row).Therefore, in the large scale setup,that is,when the limiting factor is the computing time rather than the number of examples,the stochastic learning algorithms performs asymptotically better!Large-Scale Machine Learning 74Efficient learningLet us add an additional example z t to a training set z 1...z t −1.Since the new empirical risk E t (f )remains close to E t −1(f ),the empirical minimum w ∗t +1=arg min w E t (f w )remains close to w ∗t =arg min w E t −1(f w ).With sufficient regularity assumptions,a first order calculation gives the resultw ∗t +1=w ∗t −t −1Ψt ∇w Q (z t ,w ∗t )+O t −2 ,(10)where Ψt is the inverse of the Hessian of E t (f w )in w ∗t .The similarity be-tween this expression and the second order stochastic gradient descent rule(5)has deep consequences.Let w t be the sequence of weights obtained by performing a single second order stochastic gradient pass on the randomly shuffled training set.With adequate regularity and convexity assumptions,we can prove (e.g.Bottou and LeCun,2004)lim t →∞t E (f w t )−E (f ∗F ) =lim t →∞t E (f w ∗t )−E (f ∗F ) =I >0.(11)Therefore,a single pass of second order stochastic gradient provides a pre-diction function f w t that approaches the optimum f ∗Fas efficiently as the empirical optimum f w ∗t .In particular,when the loss function is the log like-lihood,the empirical optimum is the asymptotically efficient maximum like-lihood estimate,and the second order stochastic gradient estimate is also asymptotically efficient.Unfortunately,second order stochastic gradient descent is computation-ally costly because each iteration (5)performs a computation that involves the large dense matrix Γt .Two approaches can work around this problem.•Computationally efficient approximations of the inverse Hessian trade asymptotic optimality for computation speed.For instance,the SGDQN algorithm (Bordes et al.,2009)achieves interesting speeds using a diag-onal approximation.•The averaged stochastic gradient descent (ASGD)algorithm (Polyak and Juditsky,1992)performs the normal stochastic gradient update (4)and recursively computes the average ¯w t =1tt i =1w t :w t +1=w t −γt ∇w Q (z t ,w t ),¯w t +1=t t +1¯w t +1t +1w t +1.(12)When the gains γt decrease slower than t −1,the ¯w t converges with the optimal asymptotic speed (11).Reaching this asymptotic regime can take a very long time in practice.A smart selection of the gains γt helps achieving the promised performance (Xu,2010).8L´e on BottouAlgorithm Time Test ErrorHinge loss SVM,λ=10−4.SVMLight 23,642s. 6.02%SVMPerf 66s. 6.03%SGD 1.4s. 6.02%Log loss SVM,λ=10−5.TRON (-e0.01)30s. 5.68%TRON (-e0.001)44s. 5.70%SGD 2.3s. 5.66%Optimization accuracy (trainingCost−optimalTrainingCost)Fig.1.Results achieved with a linear SVM on the RCV1task.The lower half of the plot shows the time required by SGD and TRON to reach a predefined accuracy ρon the log loss task.The upper half shows that the expected risk stops improving long before the superlinear TRON algorithm overcomes SGD.0.300.320.340.360.380.400 1 2 34 5E x p e c t e d r i s k Number of epochs SGD SGDQN ASGD 21.022.023.024.025.026.027.0 0 1 2 3 4 5T e s t E r r o r (%)Number of epochsSGD SGDQN ASGD paraison of the test set performance of SGD,SGDQN,and ASGD for a linear squared hinge SVM trained on the ALPHA task of the 2008Pascal Large Scale Learning Challenge.ASGD nearly reaches the optimal expected risk after a single pass.44004500460047004800490050005100520053005400epochs epochsparison of the test set performance of SGD,SGDQN,and ASGD on a CRF trained on the CONLL Chunking task.On this task,SGDQN appears more attractive because ASGD does not reach its asymptotic performance.Large-Scale Machine Learning9 5ExperimentsThis section briefly reports experimental results illustrating the actual per-formance of stochastic gradient algorithms on a variety of linear systems. We use gainsγt=γ0(1+λγ0t)−1for SGD and,following(Xu,2010),γt=γ0(1+λγ0t)−0.75for ASGD.The initial gainsγ0were set manually by observing the performance of each algorithm running on a subset of the training examples.Figure1reports results achieved using SGD for a linear SVM trained for the recognition of the CCAT category in the RCV1dataset(Lewis et al.,2004)using both the hinge loss(Q svm in table1),and the log loss, (Q logsvm=λw2+log(1+exp(−y w Φ(x)))).The training set contains781,265 documents represented by47,152relatively sparse TF/IDF features.SGD runs considerably faster than either the standard SVM solvers SVMLight and SVMPerf(Joachims,2006)or the superlinear optimization algorithm TRON(Lin et al.,2007).Figure2reports results achieved using SGD,SGDQN,and ASGD for a linear SVM trained on the ALPHA task of the2008Pascal Large Scale Learning Challenge(see Bordes et al.,2009)using the squared hinge loss (Q sqsvm=λw2+max{0,1−y w Φ(x)}2).The training set contains100,000 patterns represented by500centered and normalized variables.Performances measured on a separate testing set are plotted against the number of passes over the training set.ASGD achieves near optimal results after one pass.Figure3reports results achieved using SGD,SGDQN,and ASGD for a CRF(Lafferty et al.,2001)trained on the CONLL2000Chunking task (Tjong Kim Sang and Buchholz,2000).The training set contains8936sen-tences for a1.68×106dimensional parameter space.Performances measured on a separate testing set are plotted against the number of passes over the training set.SGDQN appears more attractive because ASGD does not reach its asymptotic performance.All three algorithms reach the best test set per-formance in a couple minutes.The standard CRF L-BFGS optimizer takes 72minutes to compute an equivalent solution.ReferencesBORDES.A.,BOTTOU,L.,and GALLINARI,P.(2009):SGD-QN:Careful Quasi-Newton Stochastic Gradient Descent.Journal of Machine Learning Research, 10:1737-1754.With Erratum(to appear).BOTTOU,L.and BOUSQUET,O.(2008):The Tradeoffs of Large Scale Learning, In Advances in Neural Information Processing Systems,vol.20,161-168. 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MASSART,P.(2000):Some applications of concentration inequalities to Statistics, Annales de la Facult´e des Sciences de Toulouse,series6,9,(2):245-303. MURATA,N.(1998):A Statistical Study of On-line Learning.In Online Learning and Neural Networks,Cambridge University Press.POLYAK,B.T.and JUDITSKY,A.B.(1992):Acceleration of stochastic approx-imation by averaging.SIAM J.Control and Optimization,30(4):838-855. ROSENBLATT,F.(1957):The Perceptron:A perceiving and recognizing automa-ton.Technical Report85-460-1,Project PARA,Cornell Aeronautical Lab. RUMELHART,D.E.,HINTON,G.E.,and WILLIAMS,R.J.(1986):Learning internal representations by error propagation.In Parallel distributed processing: Explorations in the microstructure of cognition,vol.I,318-362,Bradford Books. SHALEV-SHWARTZ,S.and SREBRO,N.(2008):SVM optimization:inverse de-pendence on training set size.In Proceedings of the ICML2008,928-935,ACM. TIBSHIRANI,R.(1996):Regression shrinkage and selection via the Lasso.Journal of the Royal Statistical Society,Series B,58(1):267-288.TJONG KIM SANG E. F.,and BUCHHOLZ,S.(2000):Introduction to the CoNLL-2000Shared Task:Chunking.In Proceedings of CoNLL-2000,127-132. TSYBAKOV,A.B.(2004):Optimal aggregation of classifiers in statistical learning, Annals of Statististics,32(1).VAPNIK,V.N.and CHERVONENKIS,A.YA.(1971):On the Uniform Con-vergence of Relative Frequencies of Events to Their Probabilities.Theory of Probability and its Applications,16(2):264-280.WIDROW, B.and HOFF,M. E.(1960):Adaptive switching circuits.IRE WESCON Conv.Record,Part4.,96-104.XU,W.(2010):Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent.Journal of Machine Learning Research(to ap-pear).。
支持向量机(SVM)简介
D(x, y) = K( x, x) + K( y, y) − 2K( x, y)
核函数构造
机器学习和模式识别中的很多算法要求输入模式是向 量空间中的元素。 但是，输入模式可能是非向量的形式，可能是任何对 象——串、树，图、蛋白质结构、人… 一种做法：把对象表示成向量的形式，传统算法得以 应用。 问题：在有些情况下，很难把关于事物的直观认识抽 象成向量形式。比如，文本分类问题。或者构造的向 量维度非常高，以至于无法进行运算。
学习问题
学习问题就是从给定的函数集f(x,w),w W中选择出 ∈ 能够最好的近训练器响应的函数。而这种选择是 基于训练集的，训练集由根据联合分布 F(x,y)=F(x)F(y|x)抽取的n个独立同分布样本 (xi,yi)， i=1,2,…,n 组成 。
学习问题的表示
学习的目的就是，在联合概率分布函数F(x,y)未知、 所有可用的信息都包含在训练集中的情况下，寻找 函数f(x,w0)，使它（在函数类f(x,w)，(w W）上 最小化风险泛函
支持向量机(SVM)简介
付岩
2007年6月12日
提纲
统计学习理论基本思想 标准形式的分类SVM 核函数技术 SVM快速实现算法 SVM的一些扩展形式
学习问题
x G S LM y _ y
x∈ Rn，它带有一定 产生器（G），随机产生向量
但未知的概率分布函数F(x) 训练器（S）,条件概率分布函数F(y|x) ，期望响应y 和输入向量x关系为y=f(x,v) 学习机器（LM）,输入-输出映射函数集y=f(x,w)， ∈ w W，W是参数集合。
核函数构造
String matching kernel
定义：
K( x, x′) =
svm回归算法
支持向量机回归算法（Support Vector Machine Regression，简称SVM Regression）是一种监督学习算法，用于解决回归问题。
它通过构建超平面来分割数据集，并使用特定的误差函数来评估模型的预测性能。
在SVM回归算法中，采用了一种称为ε-不敏感误差函数的方法。
该误差函数定义为，如果预测值与真实值之间的差值小于一个阈值ε，则不对此样本点做惩罚。
如果差值超过阈值，则惩罚量为
|yn−tn|−ε，其中yn是预测值，tn是真实值。
这种误差函数实际上形成了一个管道，在管道中样本点不做惩罚被称为
ε-tube。
SVM回归算法的目标是找到一个超平面，使得管道内的样本点数量最大化。
为了获得稀疏解，即计算超平面参数不依靠所有样本数据，而是部分数据，采用了这种误差函数来定义最小化误差函数作为优化目标。
由于上述目标函数含有绝对值项不可微，因此在实际应用中可能会遇到一些问题。
在训练SVM回归模型时，需要提前指定管道的宽度（即ε
的大小），并且算法引入了超参数C来控制对误差的惩罚程度。
在具体训练过程中，通过优化目标函数来找到最优的超平面和参数。
SVM回归算法可以应用于各种回归问题，如房价预测、股票价格预测等。
它的优点包括能够处理非线性问题、对异常值和噪声具有鲁棒性等。
然而，SVM回归算法也有一些局限性，例如在高维空间中可能会遇到维数灾难等问
题。
因此，在使用SVM回归算法时需要根据具体问题来选择合适的算法参数和核函数，并进行充分的实验验证和模型评估。
llm大语言模型 参数的作用
一、介绍LLM大语言模型LLM大语言模型（Large Language Model）是一种利用深度学习技术训练的语言模型，它可以自动学习和处理人类语言的规律和特点，从而实现自然语言理解、生成和处理的功能。
LLM大语言模型在自然语言处理领域具有重要的应用价值，广泛应用于机器翻译、问答系统、智能对话等方面，成为推动人工智能技术发展的重要手段之一。
二、LLM大语言模型参数的作用1. 参数对模型性能的影响LLM大语言模型的参数数量是对模型容量的一种度量，参数的数量越多，模型的容量越大，能够表示和学习的语言知识和规律也就越多。
在训练LLM大语言模型时，合理设置参数能够显著提升模型的性能，包括语言生成的准确性、语言理解的效果等。
2. 参数调节和优化在训练LLM大语言模型时，参数的调节和优化是一个重要的过程。
不同的参数设置会导致模型性能的巨大差异，因此需要通过对参数的调节和优化来获取最优的模型表现。
这涉及到参数的初始化、学习率的选择、正则化项的设置等方面，需要结合具体的任务和数据特点进行调节。
3. 参数对模型复杂度的影响LLM大语言模型的参数数量直接影响了模型的复杂度。
复杂度越高的模型能够更准确地捕捉和表达语言中的复杂规律和特点，但同时也容易导致过拟合的问题。
参数的作用还涉及到了在模型复杂度和泛化能力之间进行合理的权衡。
4. 参数的调节和调整方法针对LLM大语言模型的参数，研究人员提出了多种参数的调节和调整方法，包括网格搜索、随机搜索、贝叶斯优化等。
这些方法可以帮助研究人员在大规模的参数空间中找到最优的参数配置，从而提升LLM大语言模型的性能和效果。
5. 参数对模型性能的稳定性影响在训练LLM大语言模型时，参数的设置会影响模型的稳定性。
合理的参数设置能够提高模型的稳定性，避免模型出现梯度爆炸、梯度消失等问题，从而保证模型能够有效地学习和表达语言知识。
6. 参数对模型训练时间和资源的消耗影响LLM大语言模型的参数数量直接影响了模型的训练时间和资源的消耗。
基于支持向量机的人脸识别
3
崔国勤等
基于支持向量机的人脸识别方法
y
α1 y1
K ( x1 , x) K ( x2 , x)
输出结果（决策规则）
α 2 y2
1.引言
人脸是人类视觉中的常见模式,人脸识别在安全验证系统、公安(犯罪识别等)、医学、视频 会议、交通量控制等方面有着广阔的应用前景[28]。现有的基于生物特征的识别技术, 包括语音 识别，虹膜识别，指纹识别等，都已用于商业应用。然而最吸引人的还是人脸识别，因为从人机 交互的方式来看，人脸识别更符合人们的理想。虽然人能毫不费力地识别出人脸及其表情，但人 脸的机器自动识别仍然是一个具挑战性的研究领域。由于人脸结构的复杂性以及人脸表情的多样 性、成像过程的光照、图像的尺寸、旋转及姿势的变化等,即使同一个人，在不同的环境下拍摄 得到的人脸图像也可能不同, 所以, 虽然人脸识别的研究已有２０多年的历史, 至今还没有通用 成熟的人脸自动识别系统出现。 从算法的实践看,人脸识别不同于很多经典的识别问题，经典的模式识别,譬如文字识别等要 处理的是相对较少的类同时每个类有大量的训练样本,人脸识别中通常处理的是有相当多的类， 对于每个类则存在很少的样本[27] (譬如身份照)，识别算法必须在很少的样本中提取特征，通过 训练进行人脸图像的匹配。 统计学习理论(Statistical Learning Theory-SLT)是一种专门研究小样本情况下机器学习 规律的理论, 该理论针对小样本统计问题建立了一套新的理论体系[1][2][7][12]。支持向量机是建 立在统计学习理论基础上的解决两类问题的学习方法，由于其快速性和有效性，近年来得到了广 泛的研究和应用[13][20][21]。我们的人脸识别系统采用 Eigenface 技术[3]得到的人脸特征向量表
请简述 SVM(支持向量机)的原理以及如何处理非线性问题。
请简述 SVM(支持向量机)的原理以及如何处理非线性问题。
支持向量机（Support Vector Machine，SVM）是一种常用的机器学习算法，常用于分类和回归问题。
它的原理是基于统计学习理论和结构风险最小化原则，通过寻找最优超平面来实现分类。
SVM在处理非线性问题时，可以通过核函数的引入来将数据映射到高维空间，从而实现非线性分类。
一、SVM原理支持向量机是一种二分类模型，它的基本思想是在特征空间中找到一个超平面来将不同类别的样本分开。
具体而言，SVM通过寻找一个最优超平面来最大化样本间的间隔，并将样本分为两个不同类别。
1.1 线性可分情况在特征空间中，假设有两个不同类别的样本点，并且这两个类别可以被一个超平面完全分开。
这时候我们可以找到无数个满足条件的超平面，但我们要寻找具有最大间隔（Margin）的超平面。
Margin是指离超平面最近的训练样本点到该超平面之间距离之和。
我们要选择具有最大Margin值（即支持向量）对应的决策函数作为我们模型中使用。
1.2 线性不可分情况在实际问题中，很多情况下样本不是线性可分的，这时候我们需要引入松弛变量（Slack Variable）来处理这种情况。
松弛变量允许样本点处于超平面错误的一侧，通过引入惩罚项来平衡Margin和错误分类的数量。
通过引入松弛变量，我们可以将线性不可分问题转化为线性可分问题。
同时，为了防止过拟合现象的发生，我们可以在目标函数中加入正则化项。
1.3 目标函数在SVM中，目标函数是一个凸二次规划问题。
我们需要最小化目标函数，并找到最优解。
二、处理非线性问题SVM最初是用于处理线性可分或近似线性可分的数据集。
然而，在实际应用中，很多数据集是非线性的。
为了解决这个问题，SVM引入了核函数（Kernel Function）。
核函数可以将数据从低维空间映射到高维空间，在高维空间中找到一个超平面来实现非线性分类。
通过核技巧（Kernel Trick），SVM 可以在低维空间中计算高维空间中样本点之间的内积。
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(OP1) minimize:
subject to:
W( ) = ?
` X i=1
` X i=1
1 i+2
` ` XX i=1 j =1
yi yj
i j k(xi; xj )
(1) (2)
yi i = 0
i
8i : 0
C
(3)
The number of training examples is denoted by `. is a vector of ` variables, where each component i corresponds to a training example (xi ; yi). The solution of OP1 is the vector for which (1) is minimized and the constraints (2) and (3) are ful lled. De ning the matrix Q as (Q)ij = yi yj k(xi; xj ), this can equivalently be written as minimize: subject to:
2 General Decomposition Algorithm
This section presents a generalized version of the decomposition strategy proposed by Osuna et al. 1997a]. This strategy uses a decomposition similar to those used in active set strategies (see Gill et al. 1981]) for the case that all inequality constraints are simple bounds. In each iteration the variables i of OP1 are split into two categories. the set B of free variables the set N of xed variables Free variables are those which can be updated in the current iteration, whereas xed variables are temporarily xed at a particular value. The set of free variables will also be referred to as the working set. The working set has a constant size q much smaller than `. The algorithm works as follows:
{ much less support vectors (SVs) than training examples. { many SVs which have an i at the upper bound C .
Computational improvements like caching and incremental updates of the gradient and the termination criteria.
2
2 GENERAL DECOMPOSITION ALGORITHM
This chapter is structured as follows. First, a generalized version of the decompositon algorithm of Osuna et al. 1997a] is introduced. This identi es the problem of selecting the working set, which is addressed in the following section. In section 4 a method for \shrinking" OP1 is presented and section 5 describes the computational and implementational approach of SV M light . Finally, experimental results on two benchmark tasks, a text classi cation task, and an image recognition task are discussed to evaluate the approach.
Dortmund, 15. June, 1998
Universitat Dortmund Fachbereich Informatik
University of Dortmund Computer Science Department
Forschungsberichte des Lehrstuhls VIII (KI) Research Reports of the unit no. VIII (AI) Fachbereich Informatik Computer Science Department der Universitat Dortmund of the University of Dortmund ISSN 0943-4135 Anforderungen an: Universitat Dortmund Fachbereich Informatik Lehrstuhl VIII D-44221 Dortmund ISSN 0943-4135 Requests to: University of Dortmund Fachbereich Informatik Lehrstuhl VIII D-44221 Dortmund
Making Large-Scale SVM Learning Practical
LS{8 Report 24
Thorsten Joachims
Dortmund, 15. June, 1998
Universitat Dortmund Fachbereich Informatik
Abstract
Training a support vector machine (SVM) leads to a quadratic optimization problem with bound constraints and one linear equality constraint. Despite the fact that this type of problem is well understood, there are many issues to be considered in designing an SVM learner. In particular, for large learning tasks with many training examples, o -the-shelf optimization techniques for general quadratic programs quickly become intractable in their memory and time requirements. SV M light 1 is an implementation of an SVM learner which addresses the problem of large tasks. This chapter presents algorithmic and computational results developed for SV M light V2.0, which make large-scale SVM training more practical. The results give guidelines for the application of SVMs to large domains.
UNIVERSITAT DORTMUND
Fachbereich Informatik Lehrstuhl VIII Kunstliche Intelligenz
Making Large-Scale SVM Learning Practical
LS{8 Report 24
Thorsten Joachims
1 W ( ) = ? T 1 + 2 TQ
Ty = 0
0
C1
(4) (5) (6)
The size of the optimization problem depends on the number of training examples `. Since the size of the matrix Q is `2, for learning tasks with 10000 training examples and more it becomes impossible to keep Q in memory. Many standard implementations of QP solvers require explicit storage of Q which prohibits their application. An alternative would be to recompute Q every time it is needed. But this becomes prohibitively expensive, if Q is needed often. One approach to making the training of SVMs on problems with many training examples tractable is to decompose the problem into a series of smaller tasks. SV M light uses the decomposition idea of Osuna et al. 1997b]. This decomposition splits OP1 in an inactive and an active part - the so call \working set". The main advantage of this decomposition is that it suggests algorithms with memory requirements linear in the number of training examples and linear in the number of SVs. One potential disadvantage is that these algorithms may need a long training time. To tackle this problem, this chapter proposes an algorithm which incorporates the following ideas: An e cient and e ective method for selecting the working set. Successive \shrinking" of the optimization problem. This exploits the property that many SVM learning problems have