{"action":"created","comment":{"url":"https://api.github.com/repos/apache/spark/pulls/comments/75215584","id":75215584,"diff_hunk":"@@ -933,32 +946,312 @@ class BinaryLogisticRegressionSummary private[classification] (\n }\n \n /**\n- * LogisticAggregator computes the gradient and loss for binary logistic loss function, as used\n- * in binary classification for instances in sparse or dense vector in an online fashion.\n- *\n- * Note that multinomial logistic loss is not supported yet!\n+ * LogisticAggregator computes the gradient and loss for binary or multinomial logistic (softmax)\n+ * loss function, as used in classification for instances in sparse or dense vector in an online\n+ * fashion.\n  *\n- * Two LogisticAggregator can be merged together to have a summary of loss and gradient of\n+ * Two LogisticAggregators can be merged together to have a summary of loss and gradient of\n  * the corresponding joint dataset.\n  *\n+ * For improving the convergence rate during the optimization process and also to prevent against\n+ * features with very large variances exerting an overly large influence during model training,\n+ * packages like R's GLMNET perform the scaling to unit variance and remove the mean in order to\n+ * reduce the condition number. The model is then trained in this scaled space, but returns the\n+ * coefficients in the original scale. See page 9 in\n+ * http://cran.r-project.org/web/packages/glmnet/glmnet.pdf\n+ *\n+ * However, we don't want to apply the [[org.apache.spark.ml.feature.StandardScaler]] on the\n+ * training dataset, and then cache the standardized dataset since it will create a lot of overhead.\n+ * As a result, we perform the scaling implicitly when we compute the objective function (though\n+ * we do not subtract the mean).\n+ *\n+ * Note that there is a difference between multinomial (softmax) and binary loss. The binary case\n+ * uses one outcome class as a \"pivot\" and regresses the other class against the pivot. In the\n+ * multinomial case, the softmax loss function is used to model each class probability\n+ * independently. Using softmax loss produces `K` sets of coefficients, while using a pivot class\n+ * produces `K - 1` sets of coefficients (a single coefficient vector in the binary case). In the\n+ * binary case, we can say that the coefficients are shared between the positive and negative\n+ * classes. When regularization is applied, multinomial (softmax) loss will produce a result\n+ * different from binary loss since the positive and negative don't share the coefficients while the\n+ * binary regression shares the coefficients between positive and negative.\n+ *\n+ * The following is a mathematical derivation for the multinomial (softmax) loss.\n+ *\n+ * The probability of the multinomial outcome $y$ taking on any of the K possible outcomes is:\n+ *\n+ * <p><blockquote>\n+ *    $$\n+ *    P(y_i=0|\\vec{x}_i, \\beta) = \\frac{e^{\\vec{x}_i^T \\vec{\\beta}_0}}{\\sum_{k=0}^{K-1}\n+ *       e^{\\vec{x}_i^T \\vec{\\beta}_k}} \\\\\n+ *    P(y_i=1|\\vec{x}_i, \\beta) = \\frac{e^{\\vec{x}_i^T \\vec{\\beta}_1}}{\\sum_{k=0}^{K-1}\n+ *       e^{\\vec{x}_i^T \\vec{\\beta}_k}}\\\\\n+ *    P(y_i=K-1|\\vec{x}_i, \\beta) = \\frac{e^{\\vec{x}_i^T \\vec{\\beta}_{K-1}}\\,}{\\sum_{k=0}^{K-1}\n+ *       e^{\\vec{x}_i^T \\vec{\\beta}_k}}\n+ *    $$\n+ * </blockquote></p>\n+ *\n+ * The model coefficients $\\beta = (\\beta_1, \\beta_2, ..., \\beta_{K-1})$ become a matrix\n+ * which has dimension of $K \\times (N+1)$ if the intercepts are added. If the intercepts are not\n+ * added, the dimension will be $K \\times N$.\n+ *\n+ * Note that the coefficients in the model above lack identifiability. That is, any constant scalar\n+ * can be added to all of the coefficients and the probabilities remain the same.\n+ *\n+ * <p><blockquote>\n+ *    $$\n+ *    \\begin{align}\n+ *    \\frac{e^{\\vec{x}_i^T \\left(\\vec{\\beta}_0 + \\vec{c}\\right)}}{\\sum_{k=0}^{K-1}\n+ *       e^{\\vec{x}_i^T \\left(\\vec{\\beta}_k + \\vec{c}\\right)}}\n+ *    = \\frac{e^{\\vec{x}_i^T \\vec{\\beta}_0}e^{\\vec{x}_i^T \\vec{c}}\\,}{e^{\\vec{x}_i^T \\vec{c}}\n+ *       \\sum_{k=0}^{K-1} e^{\\vec{x}_i^T \\vec{\\beta}_k}}\n+ *    = \\frac{e^{\\vec{x}_i^T \\vec{\\beta}_0}}{\\sum_{k=0}^{K-1} e^{\\vec{x}_i^T \\vec{\\beta}_k}}\n+ *    \\end{align}\n+ *    $$\n+ * </blockquote></p>\n+ *\n+ * However, when regularization is added to the loss function, the coefficients are indeed\n+ * identifiable because there is only one set of coefficients which minimizes the regularization\n+ * term. When no regularization is applied, we choose the coefficients with the minimum L2\n+ * penalty for consistency and reproducibility. For further discussion see:\n+ *\n+ * Friedman, et al. \"Regularization Paths for Generalized Linear Models via Coordinate Descent\"\n+ *\n+ * The loss of objective function for a single instance of data (we do not include the\n+ * regularization term here for simplicity) can be written as\n+ *\n+ * <p><blockquote>\n+ *    $$\n+ *    \\begin{align}\n+ *    \\ell\\left(\\beta, x_i\\right) &= -log{P\\left(y_i \\middle| \\vec{x}_i, \\beta\\right)} \\\\\n+ *    &= log\\left(\\sum_{k=0}^{K-1}e^{\\vec{x}_i^T \\vec{\\beta}_k}\\right) - \\vec{x}_i^T \\vec{\\beta}_y\\\\\n+ *    &= log\\left(\\sum_{k=0}^{K-1} e^{margins_k}\\right) - margins_y\n+ *    \\end{align}\n+ *    $$\n+ * </blockquote></p>\n+ *\n+ * where ${margins}_k = \\vec{x}_i^T \\vec{\\beta}_k$.\n+ *\n+ * For optimization, we have to calculate the first derivative of the loss function, and a simple\n+ * calculation shows that\n+ *\n+ * <p><blockquote>\n+ *    $$\n+ *    \\begin{align}\n+ *    \\frac{\\partial \\ell(\\beta, \\vec{x}_i, w_i)}{\\partial \\beta_{j, k}}\n+ *    &= x_{i,j} \\cdot w_i \\cdot \\left(\\frac{e^{\\vec{x}_i \\cdot \\vec{\\beta}_k}}{\\sum_{k'=0}^{K-1}\n+ *      e^{\\vec{x}_i \\cdot \\vec{\\beta}_{k'}}\\,} - I_{y=k}\\right) \\\\\n+ *    &= x_{i, j} \\cdot w_i \\cdot multiplier_k\n+ *    \\end{align}\n+ *    $$\n+ * </blockquote></p>\n+ *\n+ * where $w_i$ is the sample weight, $I_{y=k}$ is an indicator function\n+ *\n+ *  <p><blockquote>\n+ *    $$\n+ *    I_{y=k} = \\begin{cases}\n+ *          1 & y = k \\\\\n+ *          0 & else\n+ *       \\end{cases}\n+ *    $$\n+ * </blockquote></p>\n+ *\n+ * and\n+ *\n+ * <p><blockquote>\n+ *    $$\n+ *    multiplier_k = \\left(\\frac{e^{\\vec{x}_i \\cdot \\vec{\\beta}_k}}{\\sum_{k=0}^{K-1}\n+ *       e^{\\vec{x}_i \\cdot \\vec{\\beta}_k}} - I_{y=k}\\right)\n+ *    $$\n+ * </blockquote></p>\n+ *\n+ * If any of margins is larger than 709.78, the numerical computation of multiplier and loss\n+ * function will suffer from arithmetic overflow. This issue occurs when there are outliers in\n+ * data which are far away from the hyperplane, and this will cause the failing of training once\n+ * infinity is introduced. Note that this is only a concern when max(margins) > 0.\n+ *\n+ * Fortunately, when max(margins) = maxMargin > 0, the loss function and the multiplier can easily\n+ * be rewritten into the following equivalent numerically stable formula.\n+ *\n+ * <p><blockquote>\n+ *    $$\n+ *    \\ell\\left(\\beta, x\\right) = log\\left(\\sum_{k=0}^{K-1} e^{margins_k - maxMargin}\\right) -\n+ *       margins_{y} + maxMargin\n+ *    $$\n+ * </blockquote></p>\n+ *\n+ * Note that each term, $(margins_k - maxMargin)$ in the exponential is no greater than zero; as a\n+ * result, overflow will not happen with this formula.\n+ *\n+ * For $multiplier$, a similar trick can be applied as the following,\n+ *\n+ * <p><blockquote>\n+ *    $$\n+ *    multiplier_k = \\left(\\frac{e^{\\vec{x}_i \\cdot \\vec{\\beta}_k - maxMargin}}{\\sum_{k'=0}^{K-1}\n+ *       e^{\\vec{x}_i \\cdot \\vec{\\beta}_{k'} - maxMargin}} - I_{y=k}\\right)\n+ *    $$\n+ * </blockquote></p>\n+ *\n  * @param bcCoefficients The broadcast coefficients corresponding to the features.\n  * @param bcFeaturesStd The broadcast standard deviation values of the features.\n+ * @param numFeatures The number of features for the input data.\n  * @param numClasses the number of possible outcomes for k classes classification problem in\n  *                   Multinomial Logistic Regression.\n  * @param fitIntercept Whether to fit an intercept term.\n+ * @param multinomial Whether to use multinomial or binary loss","path":"mllib/src/main/scala/org/apache/spark/ml/classification/LogisticRegression.scala","position":285,"original_position":285,"commit_id":"61521e6fc9fef06fd0688634024a2c377855f089","original_commit_id":"61521e6fc9fef06fd0688634024a2c377855f089","user":{"login":"dbtsai","id":1134574,"avatar_url":"https://avatars.githubusercontent.com/u/1134574?v=3","gravatar_id":"","url":"https://api.github.com/users/dbtsai","html_url":"https://github.com/dbtsai","followers_url":"https://api.github.com/users/dbtsai/followers","following_url":"https://api.github.com/users/dbtsai/following{/other_user}","gists_url":"https://api.github.com/users/dbtsai/gists{/gist_id}","starred_url":"https://api.github.com/users/dbtsai/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/dbtsai/subscriptions","organizations_url":"https://api.github.com/users/dbtsai/orgs","repos_url":"https://api.github.com/users/dbtsai/repos","events_url":"https://api.github.com/users/dbtsai/events{/privacy}","received_events_url":"https://api.github.com/users/dbtsai/received_events","type":"User","site_admin":false},"body":"multinomial (softmax) or binary loss","created_at":"2016-08-17T22:00:08Z","updated_at":"2016-08-17T22:00:08Z","html_url":"https://github.com/apache/spark/pull/13796#discussion_r75215584","pull_request_url":"https://api.github.com/repos/apache/spark/pulls/13796","_links":{"self":{"href":"https://api.github.com/repos/apache/spark/pulls/comments/75215584"},"html":{"href":"https://github.com/apache/spark/pull/13796#discussion_r75215584"},"pull_request":{"href":"https://api.github.com/repos/apache/spark/pulls/13796"}}},"pull_request":{"url":"https://api.github.com/repos/apache/spark/pulls/13796","id":74546117,"html_url":"https://github.com/apache/spark/pull/13796","diff_url":"https://github.com/apache/spark/pull/13796.diff","patch_url":"https://github.com/apache/spark/pull/13796.patch","issue_url":"https://api.github.com/repos/apache/spark/issues/13796","number":13796,"state":"open","locked":false,"title":"[SPARK-7159][ML] Add multiclass logistic regression to Spark ML","user":{"login":"sethah","id":7275795,"avatar_url":"https://avatars.githubusercontent.com/u/7275795?v=3","gravatar_id":"","url":"https://api.github.com/users/sethah","html_url":"https://github.com/sethah","followers_url":"https://api.github.com/users/sethah/followers","following_url":"https://api.github.com/users/sethah/following{/other_user}","gists_url":"https://api.github.com/users/sethah/gists{/gist_id}","starred_url":"https://api.github.com/users/sethah/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/sethah/subscriptions","organizations_url":"https://api.github.com/users/sethah/orgs","repos_url":"https://api.github.com/users/sethah/repos","events_url":"https://api.github.com/users/sethah/events{/privacy}","received_events_url":"https://api.github.com/users/sethah/received_events","type":"User","site_admin":false},"body":"## What changes were proposed in this pull request?\r\n\r\nThis patch adds a new estimator/transformer `MultinomialLogisticRegression` to spark ML. \r\n\r\nJIRA: [SPARK-7159](https://issues.apache.org/jira/browse/SPARK-7159)\r\n\r\n## How was this patch tested?\r\n\r\nAdded new test suite `MultinomialLogisticRegressionSuite`.\r\n\r\n## Approach\r\n\r\n### Do not use a \"pivot\" class in the algorithm formulation\r\n\r\nMany implementations of multinomial logistic regression treat the problem as K - 1 independent binary logistic regression models where K is the number of possible outcomes in the output variable. In this case, one outcome is chosen as a \"pivot\" and the other K - 1 outcomes are regressed against the pivot. This is somewhat undesirable since the coefficients returned will be different for different choices of pivot variables. An alternative approach to the problem models class conditional probabilites using the softmax function and will return uniquely identifiable coefficients (assuming regularization is applied). This second approach is used in R's glmnet and was also recommended by @dbtsai. \r\n\r\n### Separate multinomial logistic regression and binary logistic regression\r\n\r\nThe initial design makes multinomial logistic regression a separate estimator/transformer than the existing LogisticRegression estimator/transformer. An alternative design would be to merge them into one. \r\n\r\n**Arguments for:**\r\n\r\n* The multinomial case without pivot is distinctly different than the current binary case since the binary case uses a pivot class. \r\n* The current logistic regression model in ML uses a vector of coefficients and a scalar intercept. In the multinomial case, we require a matrix of coefficients and a vector of intercepts. There are potential workarounds for this issue if we were to merge the two estimators, but none are particularly elegant. \r\n\r\n**Arguments against:**\r\n\r\n* It may be inconvenient for users to have to switch the estimator class when transitioning between binary and multiclass (although the new multinomial estimator can be used for two class outcomes).\r\n* Some portions of the code are repeated.\r\n\r\nThis is a major design point and warrants more discussion.\r\n\r\n### Mean centering\r\n\r\nWhen no regularization is applied, the coefficients will not be uniquely identifiable. This is not hard to show and is discussed in further detail [here](https://core.ac.uk/download/files/153/6287975.pdf). R's glmnet deals with this by choosing the minimum l2 regularized solution (i.e. mean centering). Additionally, the intercepts are never regularized so they are always mean centered. This is the approach taken in this PR as well.\r\n\r\n### Feature scaling\r\n\r\nIn current ML logistic regression, the features are always standardized when running the optimization algorithm. They are always returned to the user in the original feature space, however. This same approach is maintained in this patch as well, but the implementation details are different. In ML logistic regression, the unregularized feature values are divided by the column standard deviation in every gradient update iteration. In contrast, MLlib transforms the entire input dataset to the scaled space _before_ optimizaton. In ML, this means that `numFeatures * numClasses` extra scalar divisions are required in every iteration. Performance testing shows that this has significant (4x in some cases) slow downs in each iteration. This can be avoided by transforming the input to the scaled space ala MLlib once, before iteration begins. This does add some overhead initially, but can make significant time savings in some cases. \r\n\r\nOne issue with this approach is that if the input data is already cached, there may not be enough memory to cache the transformed data, which would make the algorithm _much_ slower. The tradeoffs here merit more discussion. \r\n\r\n### Specifying and inferring the number of outcome classes\r\n\r\nThe estimator checks the dataframe label column for metadata which specifies the number of values. If they are not specified, the length of the `histogram` variable is used, which is essentially the maximum value found in the column. The assumption then, is that the labels are zero-indexed when they are provided to the algorithm. \r\n\r\n## Performance\r\n\r\nBelow are some performance tests I have run so far. I am happy to add more cases or trials if we deem them necessary.\r\n\r\nTest cluster: 4 bare metal nodes, 128 GB RAM each, 48 cores each\r\n\r\nNotes:\r\n\r\n* Time in units of seconds\r\n* Metric is classification accuracy\r\n\r\n| algo   |   elasticNetParam | fitIntercept   |   metric |   maxIter |   numPoints |   numClasses |   numFeatures |    time | standardization   |   regParam |\r\n|--------|-------------------|----------------|----------|-----------|-------------|--------------|---------------|---------|-------------------|------------|\r\n| ml     |                 0 | true           | 0.746415 |        30 |      100000 |            3 |        100000 | 327.923 | true              |          0 |\r\n| mllib  |                 0 | true           | 0.743785 |        30 |      100000 |            3 |        100000 | 390.217 | true              |          0 |\r\n\r\n| algo   |   elasticNetParam | fitIntercept   |   metric |   maxIter |   numPoints |   numClasses |   numFeatures |    time | standardization   |   regParam |\r\n|--------|-------------------|----------------|----------|-----------|-------------|--------------|---------------|---------|-------------------|------------|\r\n| ml     |                 0 | true           | 0.973238 |        30 |     2000000 |            3 |         10000 | 385.476 | true              |          0 |\r\n| mllib  |                 0 | true           | 0.949828 |        30 |     2000000 |            3 |         10000 | 550.403 | true              |          0 |\r\n\r\n| algo   |   elasticNetParam | fitIntercept   |   metric |   maxIter |   numPoints |   numClasses |   numFeatures |    time | standardization   |   regParam |\r\n|--------|-------------------|----------------|----------|-----------|-------------|--------------|---------------|---------|-------------------|------------|\r\n| mllib  |                 0 | true           | 0.864358 |        30 |     2000000 |            3 |         10000 | 543.359 | true              |        0.1 |\r\n| ml     |                 0 | true           | 0.867418 |        30 |     2000000 |            3 |         10000 | 401.955 | true              |        0.1 |\r\n\r\n| algo   |   elasticNetParam | fitIntercept   |   metric |   maxIter |   numPoints |   numClasses |   numFeatures |    time | standardization   |   regParam |\r\n|--------|-------------------|----------------|----------|-----------|-------------|--------------|---------------|---------|-------------------|------------|\r\n| ml     |                 1 | true           | 0.807449 |        30 |     2000000 |            3 |         10000 | 334.892 | true              |       0.05 |\r\n\r\n| algo   |   elasticNetParam | fitIntercept   |   metric |   maxIter |   numPoints |   numClasses |   numFeatures |    time | standardization   |   regParam |\r\n|--------|-------------------|----------------|----------|-----------|-------------|--------------|---------------|---------|-------------------|------------|\r\n| ml     |                 0 | true           | 0.602006 |        30 |     2000000 |          500 |           100 | 112.319 | true              |          0 |\r\n| mllib  |                 0 | true           | 0.567226 |        30 |     2000000 |          500 |           100 | 263.768 | true              |          0 |e           | 0.567226 |        30 |     2000000 |          500 |           100 | 263.768 | true              |          0 |\r\n\r\n## References\r\n\r\nFriedman, et al. [\"Regularization Paths for Generalized Linear Models via Coordinate Descent\"](https://core.ac.uk/download/files/153/6287975.pdf)\r\n[http://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html](http://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html)\r\n\r\n## Follow up items\r\n* Consider using level 2 BLAS routines in the gradient computations\r\n* Add model summary for MLOR\r\n* Add initial model to MLOR and add test for intercept priors\r\n* Consider changing the tree aggregation level for MLOR/BLOR or making it user configurable to avoid memory problems with high dimensional data - [SPARK-17090](https://issues.apache.org/jira/browse/SPARK-17090)\r\n* Refactor helper classes out of 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