# model stability machine learning

1 We will not be discussing the mathematical formulations here, but you should definitely look into it. ) z {\displaystyle V} L z Model monitoring for Machine Learning models. , ∑ z This allows us to see how sensitive it is and what needs to be changed to make it more robust. , z f = {\displaystyle L} {\displaystyle (x,y)} , {\displaystyle \forall S\in Z^{m},\forall i\in \{1,...,m\},|\mathbb {E} _{z}[V(f_{S},z)]-\mathbb {E} _{z}[V(f_{S^{|i}},z)]|\leq \beta }. ( ≤ I y   A stable learning algorithm is one for which the prediction does not change much when the training data is modified slightly. Analysis and Applications, 3(4):397–419, 2005, V.N. Z | {\displaystyle L} , An algorithm Artificial intelligence and machine learning in financial services . . ] Now that we have a model, we need to estimate its performance. {\displaystyle \beta _{EL}^{m}} . . The same machine learning approach could be used for non-cancerous diseases. in , ∀ Y , Some of the simplest machine learning algorithms—for instance, for regression—have hypothesis spaces with unbounded VC-dimension. ∈ (2000), Rifkin, R. Everything Old is New Again: A fresh . This additional randomness gives the model more flexibility when learning, but can make the model less stable (e.g. − X Stability results in learning theory. {\displaystyle \forall i\in \ \{1,...,m\},\mathbb {E} _{S}[|V(f_{S},z_{i})-V(f_{S^{|i}},z_{i})|]\leq \beta .}. i z Two contrasting machine learning techniques were used for deriving the PTFs for predicting the aggregate stability. ( [ { | m ] ( (plus logarithmic factors) from the true error. [ . d z . stability if for each n there exists a ∈ − z f Regardless of how the model is produced, it can be registered in a workspace, where it is represented by a name and a version. ∈ × . . ∈ m r z Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization. ) Wiley, New York, 1998, Poggio, T., Rifkin, R., Mukherjee, S. and Niyogi, P., "Learning Theory: general conditions for predictivity", Nature, Vol. , {\displaystyle \forall i\in \{1,...,m\},\mathbb {P} _{S}\{|V(f_{S},z_{i})-V(f_{S^{|i}},z_{i})|\leq \beta _{CV}\}\geq 1-\delta _{CV}}. ≥ x , − {\displaystyle H} m , m ] That’s the part about putting an upper bound. These keywords were added by machine and not by the authors. ) f } Z . Represents the result of machine learning training. P δ m S In RL you don't collect examples with labels. | . Z m While prediction accuracy may be most desirable, the Businesses do seek out the prominent contributing predictors (i.e. The process of training involved feeding data into this algorithm and building a model. , A model with large weight values is often unstable, meaning that it may suffer from poor performance during learning and sensitivity to input values resulting in higher generalization error. has CVloo stability β with respect to the loss function V if the following holds: ∀ During the training process, an important issue to think about is the stability of the learning algorithm. Model Performance for Test Dataset pre rec spe f1 geo iba sup A 0.87 0.55 0.97 0.67 0.73 0.51 84 D 0.43 0.69 0.66 0.53 0.67 0.45 83 H 0.80 0.69 0.86 0.74 0.77 0.58 139 E i V } This technique was used to obtain generalization bounds for the large class of empirical risk minimization (ERM) algorithms. A central goal in designing a machine learning system is to guarantee that the learning algorithm will generalize, or perform accurately on new examples after being trained on a finite number of them. f | {\displaystyle f} {\displaystyle m} 1. z δ ≥ | | ) 2.3. z , {\displaystyle f_{S}} Jaywing's response to the PRA's findings on ECL calculations. In this article, we point out a new and similar connection between model theory and machine learning, this time developing a correspondence between \emph{stability} and learnability in various settings of \emph{online learning.} We need to make sure that it generalizes well to various training sets. } An algorithm E ) Y S ( { i {\displaystyle Z=X\times Y}. onto a function If we choose a different subset within that training dataset, will the model remain the same? m ) is defined as a mapping from 1 x i } An algorithm i f i This is where stability analysis comes into picture. 1 I m z | Y V 1 Neither condition alone is sufficient for generalization. z However given the dataset changes with time what other factors should I keep in mind: ( m , 04 June 2020. Reinforcement learning differs from other types of machine learning. . , E . m . But it shouldn’t change more than a certain threshold regardless of what subset you choose for training. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. . , {\displaystyle S} ∈ All learning algorithms with Tikhonov regularization satisfies Uniform Stability criteria and are, thus, generalizable. Z . A stable learning algorithm would produce a similar classifier with both the 1000-element and 999-element training sets. An algorithm {\displaystyle m,\rightarrow \infty }. . ) A learning algorithm is said to be stable if the learned model doesn’t change much when the training dataset is modified. . , {\displaystyle L} z Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. ) I am interested in your thoughts on the pros and cons on the different measures of stability such as hypothesis stability vs. cross validation stability. , Theory 25(5) (1979) 601–604. | 1 Testing for stability in a time-series. ∈ , and If it satisfies this condition, it’s said to be “stable”. O S {\displaystyle f} . { f β L The minimum relative entropy algorithm for classification. m z Check out my thoughts: , {\displaystyle S^{|i}=\{z_{1},...,\ z_{i-1},\ z_{i+1},...,\ z_{m}\}}, S Res., 2:499–526, 2002. Based on the morphologies with/without clinical features, machine learning models were constructed and compared to define the morphological determinants and screen the optimal model for predicting aneurysm stability. , o Uniform Stability is a strong condition which is not met by all algorithms but is, surprisingly, met by the large and important class of Regularization algorithms. E β 1 , where z ) ) 23 November 2020. It’s obvious that he has less than 100 million items. − { | f f different results when the same model … z has point-wise hypothesis stability β with respect to the loss function V if the following holds: ∀ f 2008 Feb;26(6):907-15. 1 L {\displaystyle \forall i\in \{1,...,m\},\mathbb {E} _{S,z}[|V(f_{S},z)-V(f_{S^{|i}},z)|]\leq \beta .}. But how can we know? V V { , ≤ (Controlling for Model Stability) Stochastic models, like deep neural networks, add an additional source of randomness. has uniform stability β with respect to the loss function V if the following holds: ∀ X Ask Question Asked 9 years, 5 months ago. As we discussed earlier, the variation comes from how we choose the training dataset. decreases as of a hypothesis H Learn. ( f {\displaystyle d} ] + {\displaystyle f} , maps a training data set, which is a set of labeled examples to i { is {\displaystyle Z_{m}} ) m The notion of stability is centered on putting a bound on the generalization error of the learning algorithm. } i This process is experimental and the keywords may be updated as the learning algorithm improves. | , i i Stability and generalization. S V {\displaystyle I_{S}[f]={\frac {1}{n}}\sum V(f,z_{i})} V ( , ( Log Out /  | V Comput. C ∈ ∞ , ∀ f ( ) An algorithm is said to be stable, when the value of Y , } ) As such, stability analysis is the application of sensitivity analysis to machine learning. 1 November 2017 . ( Mathematically speaking, there are many ways of determining the stability of a learning algorithm. l ( z β S . β , with {\displaystyle Y} P The following years saw a fruitful exchange of ideas between PAC learning and the model theory of NIP structures. {\displaystyle \delta _{EL}^{m}} . [ i In a machine learning code, that computes optimum parameters $\theta _{MLE} ... or not, but if it is, there is already one deliverable in the notebook to fit a regularized linear regression model (through maximizing a posteriori method), ... Browse other questions tagged stability machine-learning inverse-problem or ask your own question. S. Kutin and P. Niyogi, Almost-everywhere algorithmic stability and generalization error, Technical Report TR-2002-03, University of Chicago (2002). Sakiyama Y(1), Yuki H, Moriya T, … V A stable learning algorithm is one for which the prediction does not change much when the training data is modified slightly. , ( is then defined as it does not depend on the order of the elements in the training set. β Change ), Measuring the Stability of Machine Learning Algorithms. H \forall S\in Z^{m},\forall i\in \{1,...,m\},\sup _{z\in Z}|V(f_{S},z)-V(f_{S^{|i}},z)|\leq \beta }. m Put another way, these results could not be applied when the information being learned had a complexity that was too large to measure. The result was later extended to almost-ERM algorithms with function classes that do not have unique minimizers. \delta _{EL}^{m}} S Specifically, the way in which we pick a particular subset of that dataset for training. m As a first step to improving your results, you need to determine the problems with your model. | An artificial intelligence technique—machine learning—is helping accelerate the development of highly tunable materials known as metal-organic frameworks (MOFs) that have important applications in chemical separations, adsorption, catalysis, and sensing. 1 , , , 25 November 2020. | L. Devroye and Wagner, Distribution-free performance bounds for potential function rules, IEEE Trans. Finally, machine learning does enable humans to quantitatively decide, predict, and look beyond the obvious, while sometimes into previously unknown aspects as well. S . . L Another example is language learning algorithms that can produce sentences of arbitrary length. ) So what exactly is stability? z However, these results could not be applied to algorithms with hypothesis spaces of unbounded VC-dimension. ( z f This allows us to understand how a particular model is going to turn out. z z ( z A model changes when you change the training set. δ , − It was shown that for large classes of learning algorithms, notably empirical risk minimization algorithms, certain types of stability ensure good generalization. z ( Log Out / look at historical approaches in machine learning. i X E STABILITY OF MACHINE LEARNING ALGORITHMS A Dissertation Submitted to the Faculty of Purdue University by Wei Sun In Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy May 2015 ... model as a diligent researcher to pursue important and deep topics. z n} 1 f f V Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics. ≤ Therefore, we applied the machine-learning approach based on compressed sensing (a method widely used to compress images) to develop a very accurate and predictive surrogate model," Levchenko notes. x Inf. First, the GLM model was developed using the glm R Package (Guisan et al., 2002, R Core Team, 2018). Things have changed with the adoption of more sophisticated MLOps solutions. n L Imagine you want to teach a machine to play a very basic video game and never lose. and Market developments and financial stability implications . A few years ago, it was extremely uncommon to retrain a machine learning model with new observations systematically. This year the workshop is organized in two tracks 1) machine learning and 2) clinical neuroimaging. x For instance, the team is … The agents | 1 J. Mach. . → V , n If we repeat this experiment with different subsets of the same size, will the model perform its job with the same efficiency? ) f The stability of an algorithm is a property of the learning process, rather than a direct property of the hypothesis space z=(x,y)} The generalization bound is given in the article. ( and Predicting human liver microsomal stability with machine learning techniques. f { , Stability analysis enables us to determine how the input variations are going to impact the output of our system. their relation to generalization performances. Why do we need to analyze “stability”? L going to zero for \forall i\in \{1,...,m\},\mathbb {P} _{S}\{|I[f_{S}]-{\frac {1}{m}}\sum _{i=1}^{m}V(f_{S^{|i}},z_{i})|\leq \beta _{EL}^{m}\}\geq 1-\delta _{EL}^{m}} V Six pointers to prepare collections strategies for the challenges ahead. . L} y S In our case, the system is a learning algorithm that ingests data to learn from it. Utilizing data about the properties of more than 200 existing MOFs, the machine learning … For instance, consider a machine learning algorithm that is being trained to recognize handwritten lettersof the alphabet, using 1000 examples of handwritten letters and their labels ("A" to "Z") as a training set. Vapnik's work, using what became known as VC theory, established a relationship between generalization of a learning algorithm and properties of the hypothesis space , S = | y L V , n The accuracy metric tells us how many samples were classified correctly, but it doesn’t tell us anything about how the training dataset influenced this process. 1 O\left({\sqrt {\frac {d}{n}}}\right)} , . { 1 X} sup e ) ( ∈ ( Log Out / ≤ In this case, the model would have to be re-taught with data related to that disease. H} Your friend, Carl, asks you to buy some cardboard boxes to move all his stuff to his new apartment. , Stability, also known as algorithmic stability, is a notion in computational learning theory of how a machine learning algorithm is perturbed by small changes to its inputs. { The stability of these aneurysms and other clinical characteristics were reviewed from the medical records. . ———————————————————————————————————————————————————————————. ∈ During that call, Carl tells you that he definitely has less than 100 million items. Z Machine learning techniques. V H Is it possible to know which models will work best or to simply see the data? L f } In our case, the system is a learning algorithm that ingests data to learn from it. β S The two possible sources would be: The noise factor is a part of the data collection problem, so we will focus our discussion on the training dataset. In: Analysing Economic Data. , This repeated holdout procedure, sometimes also called Monte Carlo Cross-Validation, provides with a better estimate of how well our model may perform on a random test set, and it can also give us an idea about our model’s stability — how the model produced by a learning algorithm changes with different training set splits. . m H} A machine learning algorithm has two types of parameters. i I[f]=\mathbb {E} _{z}V(f,z)}. The goal of all these different metrics is to put a bound on the generalization error. S . \beta _{EL}^{m}} Shalev Shwartz, S., Shamir, O., Srebro, N., Sridharan, K., Learnability, Stability and Uniform Convergence, Journal of Machine Learning Research, 11(Oct):2635-2670, 2010. Safe Model-based Reinforcement Learning with Stability Guarantees Felix Berkenkamp Department of Computer Science ETH Zurich befelix@inf.ethz.ch Matteo Turchetta Department of Computer Science, ETH Zurich matteotu@inf.ethz.ch Angela P. Schoellig Institute for Aerospace Studies University of Toronto schoellig@utias.utoronto.ca Andreas Krause A machine learning algorithm, also known as a learning map β i 1. . Am I wrong in looking at Stability in this way? H i m i m Statistical learning theory deals with the problem of finding a predictive function based on data. One way to modify this training set is to leave out an example, so that only 999 examples of handwritten letters and their labels are available. V It’s important to notice the word “much” in this definition. } I can’t find any follow button. m Machine Learning Model Explanation using Shapley Values. , , V } O({\frac {1}{m}})} ) Please explain stable and unstable learning algorithms with examples and then categorize different classifiers into them. For ERM algorithms specifically (say for the square loss), Leave-one-out cross-validation (CVloo) Stability is both necessary and sufficient for consistency and generalization. Credit: Pixabay/CC0 Public Domain. L S ) The machine learning model can be trained to predict other properties as long as a sufficient amount of data exists. Change ), You are commenting using your Twitter account. E a descriptive model or its resulting explainability) as well. Stability analysis was developed in the 2000s for computational learning theory and is an alternative method for obtaining generalization bounds. The technique historically used to prove generalization was to show that an algorithm was consistent, using the uniform convergence properties of empirical quantities to their means. One of the most common forms of pre-processing consists of a simple linear rescaling of the input variables. of UAI 18, 2002, S. Rakhlin, S. Mukherjee, and T. Poggio. 1 , and a It’s actually quite interesting! i { is ) | , i.e. } | ) View at Medium.com. Stability, also known as algorithmic stability, is a notion in computational learning theory of how a machine learning algorithm is perturbed by small changes to its inputs. f 02 September 2020. L ( Ikano Bank partners with Jaywing. f { has − = f This is an important result for the foundations of learning theory, because it shows that two previously unrelated properties of an algorithm, stability and consistency, are equivalent for ERM (and certain loss functions). { ≤ | O In Proc. Here, we consider only deterministic algorithms where Some of the common methods include hypothesis stability, error stability, leave-one-out cross-validation stability, and a few more. ∈ . I have thought a lot about this issue but express it a bit different. H ∑ , The empirical error of L} A lot of research is centered on developing algorithms that are accurate and can predict the outcome with a high degree of confidence. f ( Learning curves require you to verify against a test set as you vary the number of training instances. Market Stability with Machine Learning Agents Christophre Georgesy Javier Pereiraz Department of Economics Hamilton College April 18, 2019 Abstract We consider the e ect of adaptive model selection and regularization by agents on price volatility and market stability in a simple agent-based model of a nancial market. ( − with VC-dimension , They use different approaches to compute it. , ( , is symmetric with respect to x , E O. Bousquet and A. Elisseeff. E Technical } has error stability β with respect to the loss function V if the following holds: ∀ r sup Developing Simple and Stable Machine Learning Models by Meir Maor 29 Apr 2019 A current challenge and debate in artificial intelligence is building simple and stable machine learning models capable of identifying patterns and even objects. i ( − I The Nature of Statistical Learning Theory. ( = such that: ∀ Furthermore, we assume that all functions are measurable and all sets are countable. L i 1 f Do I use a known tagged source (different from the original training dataset) and measure and track its precision and recall at that time? Stability of a learning algorithm refers to the changes in the output of the system when we change the training dataset. m S.Kutin and P.Niyogi.Almost-everywhere algorithmic stability and generalization error. Machine Learning in Healthcare: An Investigation into Model Stability by Shivapratap Gopakumar M.Tech Submitted in fulﬁlment of the requirements for the degree … I am thinking in terms of tracking only Precision and Recall and not Accuracy as many practical domains/business problems tend to have class imbalances. 1 Leave-one-out cross-validation (CVloo) Stability. For instance, consider a machine learning algorithm that is being trained to recognize handwritten letters of the alphabet, using 1000 examples of handwritten letters and their labels ("A" to "Z") as a training set. [ Prateek, keep thinking of tracking the Stability of a model in terms of Precision and Recall over time. Testing for Stability in Regression Models. f ] The machine learning track seeks novel contributions that address current methodological gaps in analyzing… , For symmetric learning algorithms with bounded loss, if the algorithm has Uniform Stability with the probabilistic definition above, then the algorithm generalizes. 1 The true error of { z C , m , , − ≤ X} A general result, proved by Vladimir Vapnik for an ERM binary classification algorithms, is that for any target function and input distribution, any hypothesis space i So putting a tight upper bound is very important. = S In order to estimate it, we will consider the stability factor with respect to the changes made to the training set. i i . Math., 25(1-3):161–193, 2006. Estimating the stability becomes crucial in these situations. H How do we estimate it? , mapping a training set S. Mukherjee, P. Niyogi, T. Poggio, and R. M. Rifkin. ... by different I mean either differences in model parameters ... Browse other questions tagged time-series machine-learning or ask your own question. ] S^{i}=\{z_{1},...,\ z_{i-1},\ z_{i}^{'},\ z_{i+1},...,\ z_{m}\}}. i S} . from If we create a set of learning models based on different subset and measure the error for each one, what will it look like? to \forall S\in Z^{m},\forall i\in \{1,...,m\},\mathbb {P} _{S}\{\sup _{z\in Z}|V(f_{S},z)-V(f_{S^{|i}},z)|\leq \beta \}\geq 1-\delta }. The study of stability gained importance in computational learning theory in the 2000s when it was shown to have a connection with generalization[citation needed]. β A stable learning algorithm is one for which the learned function does not change much when the training set is slightly modified, for instance by leaving out an example. \beta } report. Stability can be studied for many types of learning problems, from language learning to inverse problems in physics and engineering, as it is a property of the learning process rather than the type of information being learned. i S L i z When you think of a machine learning algorithm, the first metric that comes to mind is its accuracy. , S 1 L S ) 1 Given a training set S of size m, we will build, for all i = 1....,m, modified training sets as follows: S Elisseeff, A. 7.2 Tunning The Model’s Hyperparameters. ( A probabilistic version of uniform stability β is: ∀ ≤ β ≥ , Springer, 1995, Vapnik, V., Statistical Learning Theory. A study about algorithmic stability and . ) Eloo_{err}} − So far, so good! ( Log Out / S i Even though it’s factually correctly, it’s not very helpful. ) S , Y} That’s just how it is! Introduction. Let’s take the example of supervised learning. 428, 419-422, 2004, Andre Elisseeff, Theodoros Evgeniou, Massimiliano Pontil, Stability of Randomized Learning Algorithms, Journal of Machine Learning Research 6, 55–79, 2010, Elisseeff, A. Pontil, M., Leave-one-out Error and Stability of Learning Algorithms with Applications, NATO SCIENCE SERIES SUB SERIES III COMPUTER AND SYSTEMS SCIENCES, 2003, VOL 190, pages 111-130, Shalev Shwartz, S., Shamir, O., Srebro, N., Sridharan, K., Learnability, Stability and Uniform Convergence, Journal of Machine Learning Research, 11(Oct):2635-2670, 2010, This page was last edited on 5 August 2020, at 20:20. S} , S=\{z_{1}=(x_{1},\ y_{1})\ ,..,\ z_{m}=(x_{m},\ y_{m})\}}, and is of size m f V drawn i.i.d. Stability analysis enables us to determine how the input variations are going to impact the output of our system. Now what are the sources of these changes? A supervised learning algorithm takes a labeled dataset that contains data points and the corresponding labels. z Palgrave Texts in Econometrics. , We want this bound to be as tight as possible. E The training set from which an algorithm learns is defined as, S z Adv. E The loss ∈ ′ ] = S V , Many thanks! + The goal of stability analysis is to come up with a upper bound for this error. z An algorithm m ( L} A learning algorithm is said to be stable if the learned model doesn’t change much when the training dataset is modified. We define several terms related to learning algorithms training sets, so that we can then define stability in multiple ways and present theorems from the field. Epub 2007 Jun 27. H} Ideally, we want the model to remain the same and perform its job with the same accuracy. | | δ m = o m 1 f} the first type are the parameters that are learned through the training phase and the second type are the hyperparameters that we pass to the machine learning model. of functions being learned. X Stability of a learning algorithm refers to the changes in the output of the system when we change the training dataset. ) Z = V [ Conceptually, it refers to the inherent instability machine learning models experience in their decision-making in response to variations in the training data. . , [ H} , Improve your training time, model stability and accuracy on Amazon Forecast by leveraging new hyperparameters now supported on DeepAR+ Posted On: Feb 27, 2020 Amazon Forecast is a fully managed service that uses machine learning (ML) to generate accurate forecasts, without requiring any prior ML experience. = S Hi, how can I follow your blog? The functions i The NHS has invested £250m$323m; €275m) to embed machine learning in healthcare, but researchers say the level of consistency (stability) … This is a list of algorithms that have been shown to be stable, and the article where the associated generalization bounds are provided. k-NN classifier with a {0-1} loss function. The generalization bound is given in the article. from an unknown distribution D. Thus, the learning map In the 1990s, milestones were reached in obtaining generalization bounds for supervised learning algorithms. [   A measure of Leave one out error is used in a Cross Validation Leave One Out (CVloo) algorithm to evaluate a learning algorithm's stability with respect to the loss function. . What factors do we consider or keep track in terms of the new dataset used to measure this – size, statistical significance of the sample, feature diversity in the dataset? 1 S , {\displaystyle L} ... Superplasticizers (C5) are water-soluble organic substances that reduce the amount of water require to achieve certain stability of concrete, reduce the water-cement ratio, reduce cement content and increase slump. {\displaystyle f} P , This was mostly because the model retraining tasks were laborious and cumbersome, but machine learning has come a long way in a short time. training examples, the algorithm is consistent and will produce a training error that is at most The definition of (CVloo) Stability is equivalent to Pointwise-hypothesis stability seen earlier. f Ph.D. Thesis, MIT, 2002, http://www.mit.edu/~9.520/spring09/Classes/class10_stability.pdf, https://en.wikipedia.org/w/index.php?title=Stability_(learning_theory)&oldid=971385999, Articles with unsourced statements from September 2019, Creative Commons Attribution-ShareAlike License, For symmetric learning algorithms with bounded loss, if the algorithm has. {\displaystyle Y} with respect to an example . You don’t know how many items he has, so you call him to get that information. Let’s take an example. are in the same space of the training examples. 1 . ∀   {\displaystyle V(f,z)=V(f(x),y)} ( in such a way to minimize the empirical error on a training set Z , and it can be assessed in algorithms that have hypothesis spaces with unbounded or undefined VC-dimension such as nearest neighbor. z .   A model is the result of a Azure Machine learning training Run or some other model training process outside of Azure. from , | The 3rd international workshop on machine learning in clinical neuroimaging (MLCN2020) aims to bring together the top researchers in both machine learning and clinical neuroimaging. View at Medium.com y m S z S An ERM algorithm is one that selects a solution from a hypothesis space J Mol Graph Model. , are selected from a hypothesis space of functions called Change ), You are commenting using your Google account. However, both together ensure generalization (while the converse is not true). S You set up the model (often called an agent in RL) with the game, and you tell the model not to get a "game over" screen. f β i S . m . It’s important to notice the word “much” in this definition. has hypothesis stability β with respect to the loss function V if the following holds: ∀ 1 {\displaystyle L} − m z L , onto a function . , {\displaystyle H} L You’ll immediately notice whether you find much difference between your in-sample and out-of-sample errors. | − {\displaystyle L} One way to modify thi… Vapnik. S Change ), You are commenting using your Facebook account. 1   S } V y δ An algorithm d into V As a friend, he z = E . ∈ E 2. z . } {\displaystyle X} = We need a criterion that’s easy to check so that we can estimate the stability with a certain degree of confidence.

Posted in 게시판.