[jira] [Commented] (SPARK-21919) inconsistent behavior of AFTsurvivalRegression algorithm
[
https://issues.apache.org/jira/browse/SPARK-21919?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=16157027#comment-16157027
]
Yanbo Liang commented on SPARK-21919:
-
[~srowen] You are right, that is caused by line search bug. The error log in
2.2.0 can tell us what happened. Thanks for dig into it.
> inconsistent behavior of AFTsurvivalRegression algorithm
>
>
> Key: SPARK-21919
> URL: https://issues.apache.org/jira/browse/SPARK-21919
> Project: Spark
> Issue Type: Bug
> Components: ML, PySpark
>Affects Versions: 2.2.0
> Environment: Spark Version: 2.2.0
> Cluster setup: Standalone single node
> Python version: 3.5.2
>Reporter: Ashish Chopra
>
> Took the direct example from spark ml documentation.
> {code}
> training = spark.createDataFrame([
> (1.218, 1.0, Vectors.dense(1.560, -0.605)),
> (2.949, 0.0, Vectors.dense(0.346, 2.158)),
> (3.627, 0.0, Vectors.dense(1.380, 0.231)),
> (0.273, 1.0, Vectors.dense(0.520, 1.151)),
> (4.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result is:
> Coefficients: [-0.496304411053,0.198452172529]
> Intercept: 2.6380898963056327
> Scale: 1.5472363533632303
> ||label||censor||features ||prediction || quantiles ||
> |1.218|1.0 |[1.56,-0.605] |5.718985621018951 |
> [1.160322990805951,4.99546058340675]|
> |2.949|0.0 |[0.346,2.158] |18.07678210850554
> |[3.66759199449632,15.789837303662042]|
> |3.627|0.0 |[1.38,0.231] |7.381908879359964
> |[1.4977129086101573,6.4480027195054905]|
> |0.273|1.0 |[0.52,1.151]
> |13.577717814884505|[2.754778414791513,11.859962351993202]|
> |4.199|0.0 |[0.795,-0.226]|9.013087597344805
> |[1.828662187733188,7.8728164067854856]|
> But if we change the value of all labels as label + 20. as:
> {code}
> training = spark.createDataFrame([
> (21.218, 1.0, Vectors.dense(1.560, -0.605)),
> (22.949, 0.0, Vectors.dense(0.346, 2.158)),
> (23.627, 0.0, Vectors.dense(1.380, 0.231)),
> (20.273, 1.0, Vectors.dense(0.520, 1.151)),
> (24.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result changes to:
> Coefficients: [23.9932020748,3.18105314757]
> Intercept: 7.35052273751137
> Scale: 7698609960.724161
> ||label ||censor||features ||prediction ||quantiles||
> |21.218|1.0 |[1.56,-0.605] |4.0912442688237169E18|[0.0,0.0]|
> |22.949|0.0 |[0.346,2.158] |6.011158613411288E9 |[0.0,0.0]|
> |23.627|0.0 |[1.38,0.231] |7.7835948690311181E17|[0.0,0.0]|
> |20.273|1.0 |[0.52,1.151] |1.5880852723124176E10|[0.0,0.0]|
> |24.199|0.0 |[0.795,-0.226]|1.4590190884193677E11|[0.0,0.0]|
> Can someone please explain this exponential blow up in prediction, as per my
> understanding prediction in AFT is a prediction of the time when the failure
> event will occur, not able to understand why it will change exponentially
> against the value of the label.
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[jira] [Commented] (SPARK-21919) inconsistent behavior of AFTsurvivalRegression algorithm
[
https://issues.apache.org/jira/browse/SPARK-21919?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=16157013#comment-16157013
]
Sean Owen commented on SPARK-21919:
---
Hm, yeah, I suppose I should have tried it too. On {{master}}, and in Scala, I
get:
{code}
import org.apache.spark.ml.linalg._
import org.apache.spark.ml.regression._
val training = spark.createDataFrame(Seq(
(21.218, 1.0, Vectors.dense(1.560, -0.605)),
(22.949, 0.0, Vectors.dense(0.346, 2.158)),
(23.627, 0.0, Vectors.dense(1.380, 0.231)),
(20.273, 1.0, Vectors.dense(0.520, 1.151)),
(24.199, 0.0, Vectors.dense(0.795, -0.226))
)).toDF("label", "censor", "features")
val aft = new AFTSurvivalRegression().
setQuantileProbabilities(Array(0.3, 0.6)).
setQuantilesCol("quantiles")
val model = aft.fit(training)
println(s"Coefficients: ${model.coefficients}")
println(s"Intercept: ${model.intercept}")
println(s"Scale: ${model.scale}")
model.transform(training).show(truncate=false)
{code}
{code}
17/09/07 15:30:14 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.5
17/09/07 15:30:14 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.25
17/09/07 15:30:14 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.5
17/09/07 15:30:14 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.25
17/09/07 15:30:14 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.125
...
+--+--+--+--+---+
|label |censor|features |prediction|quantiles
|
+--+--+--+--+---+
|21.218|1.0 |[1.56,-0.605] |24.20972861807431
|[21.617443110471118,23.97833624826161] |
|22.949|0.0 |[0.346,2.158]
|26.461225875981285|[23.627858619625105,26.208314087493857]|
|23.627|0.0 |[1.38,0.231]
|24.565240805031497|[21.934888406858644,24.330450511651165]|
|20.273|1.0 |[0.52,1.151]
|26.074003958175602|[23.28209894956245,25.82479316934075] |
|24.199|0.0
|[0.795,-0.226]|25.491396901107077|[22.761875236582238,25.247754569057985]|
+--+--+--+--+---+
{code}
But in 2.2.0, I get
{code}
ERROR optimize.LBFGS: Failure! Resetting history:
breeze.optimize.FirstOrderException: Line search failed
17/09/07 14:32:35 ERROR optimize.LBFGS: Failure again! Giving up and returning.
Maybe the objective is just poorly behaved?
...
+--+--+--+-+-+
|label |censor|features |prediction |quantiles|
+--+--+--+-+-+
|21.218|1.0 |[1.56,-0.605] |4.091244268823746E18 |[0.0,0.0]|
|22.949|0.0 |[0.346,2.158] |6.011158613411288E9 |[0.0,0.0]|
|23.627|0.0 |[1.38,0.231] |7.7835948690311731E17|[0.0,0.0]|
|20.273|1.0 |[0.52,1.151] |1.5880852723124233E10|[0.0,0.0]|
|24.199|0.0 |[0.795,-0.226]|1.459019088419373E11 |[0.0,0.0]|
+--+--+--+-+-+
{code}
So I'm almost sure this is just another symptom of the Breeze / strong Wolfe
line search bug: https://issues.apache.org/jira/browse/SPARK-21523
> inconsistent behavior of AFTsurvivalRegression algorithm
>
>
> Key: SPARK-21919
> URL: https://issues.apache.org/jira/browse/SPARK-21919
> Project: Spark
> Issue Type: Bug
> Components: ML, PySpark
>Affects Versions: 2.2.0
> Environment: Spark Version: 2.2.0
> Cluster setup: Standalone single node
> Python version: 3.5.2
>Reporter: Ashish Chopra
>
> Took the direct example from spark ml documentation.
> {code}
> training = spark.createDataFrame([
> (1.218, 1.0, Vectors.dense(1.560, -0.605)),
> (2.949, 0.0, Vectors.dense(0.346, 2.158)),
> (3.627, 0.0, Vectors.dense(1.380, 0.231)),
> (0.273, 1.0, Vectors.dense(0.520, 1.151)),
> (4.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result is:
> Co
[jira] [Commented] (SPARK-21919) inconsistent behavior of AFTsurvivalRegression algorithm
[
https://issues.apache.org/jira/browse/SPARK-21919?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=16156979#comment-16156979
]
Yanbo Liang commented on SPARK-21919:
-
[~ashishchopra0308] [~srowen] I can't reproduce this issue, I can get correct
result which is consistent with R {{survreg}}.
{code}
>>> from pyspark.ml.regression import AFTSurvivalRegression
>>> from pyspark.ml.linalg import Vectors
>>> training = spark.createDataFrame([
... (21.218, 1.0, Vectors.dense(1.560, -0.605)),
... (22.949, 0.0, Vectors.dense(0.346, 2.158)),
... (23.627, 0.0, Vectors.dense(1.380, 0.231)),
... (20.273, 1.0, Vectors.dense(0.520, 1.151)),
... (24.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
... "features"])
>>> quantileProbabilities = [0.3, 0.6]
>>> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
... quantilesCol="quantiles")
>>> model = aft.fit(training)
17/09/07 21:54:31 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.5
17/09/07 21:54:31 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.25
17/09/07 21:54:31 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.5
17/09/07 21:54:31 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.25
17/09/07 21:54:31 ERROR StrongWolfeLineSearch: Encountered bad values in
function evaluation. Decreasing step size to 0.125
>>> print("Coefficients: " + str(model.coefficients))
Coefficients: [-0.065814695216,0.00326705958509]
>>> print("Intercept: " + str(model.intercept))
Intercept: 3.29140205698
>>> print("Scale: " + str(model.scale))
Scale: 0.109856123692
>>> model.transform(training).show(truncate=False)
17/09/07 21:55:05 WARN BLAS: Failed to load implementation from:
com.github.fommil.netlib.NativeSystemBLAS
17/09/07 21:55:05 WARN BLAS: Failed to load implementation from:
com.github.fommil.netlib.NativeRefBLAS
+--+--+--+--+---+
|label |censor|features |prediction|quantiles
|
+--+--+--+--+---+
|21.218|1.0 |[1.56,-0.605] |24.20972861807431
|[21.617443110471118,23.97833624826161] |
|22.949|0.0 |[0.346,2.158]
|26.461225875981285|[23.627858619625105,26.208314087493857]|
|23.627|0.0 |[1.38,0.231]
|24.565240805031497|[21.934888406858644,24.330450511651165]|
|20.273|1.0 |[0.52,1.151]
|26.074003958175602|[23.28209894956245,25.82479316934075] |
|24.199|0.0
|[0.795,-0.226]|25.491396901107077|[22.761875236582238,25.247754569057985]|
+--+--+--+--+---+
{code}
> inconsistent behavior of AFTsurvivalRegression algorithm
>
>
> Key: SPARK-21919
> URL: https://issues.apache.org/jira/browse/SPARK-21919
> Project: Spark
> Issue Type: Bug
> Components: ML, PySpark
>Affects Versions: 2.2.0
> Environment: Spark Version: 2.2.0
> Cluster setup: Standalone single node
> Python version: 3.5.2
>Reporter: Ashish Chopra
>
> Took the direct example from spark ml documentation.
> {code}
> training = spark.createDataFrame([
> (1.218, 1.0, Vectors.dense(1.560, -0.605)),
> (2.949, 0.0, Vectors.dense(0.346, 2.158)),
> (3.627, 0.0, Vectors.dense(1.380, 0.231)),
> (0.273, 1.0, Vectors.dense(0.520, 1.151)),
> (4.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result is:
> Coefficients: [-0.496304411053,0.198452172529]
> Intercept: 2.6380898963056327
> Scale: 1.5472363533632303
> ||label||censor||features ||prediction || quantiles ||
> |1.218|1.0 |[1.56,-0.605] |5.718985621018951 |
> [1.160322990805951,4.99546058340675]|
> |2.949|0.0 |[0.346,2.158] |18.07678210850554
> |[3.66759199449632,15.789837303662042]|
> |3.627|0.0 |[1.38,0.231] |7.381908879359964
> |[1.4977129086101573,6.4480027195054905]|
> |0.273|1.0 |[0.
[jira] [Commented] (SPARK-21919) inconsistent behavior of AFTsurvivalRegression algorithm
[
https://issues.apache.org/jira/browse/SPARK-21919?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=16156579#comment-16156579
]
Yanbo Liang commented on SPARK-21919:
-
[~srowen] I will take a look at it. Thanks.
> inconsistent behavior of AFTsurvivalRegression algorithm
>
>
> Key: SPARK-21919
> URL: https://issues.apache.org/jira/browse/SPARK-21919
> Project: Spark
> Issue Type: Bug
> Components: ML, PySpark
>Affects Versions: 2.2.0
> Environment: Spark Version: 2.2.0
> Cluster setup: Standalone single node
> Python version: 3.5.2
>Reporter: Ashish Chopra
>
> Took the direct example from spark ml documentation.
> {code}
> training = spark.createDataFrame([
> (1.218, 1.0, Vectors.dense(1.560, -0.605)),
> (2.949, 0.0, Vectors.dense(0.346, 2.158)),
> (3.627, 0.0, Vectors.dense(1.380, 0.231)),
> (0.273, 1.0, Vectors.dense(0.520, 1.151)),
> (4.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result is:
> Coefficients: [-0.496304411053,0.198452172529]
> Intercept: 2.6380898963056327
> Scale: 1.5472363533632303
> ||label||censor||features ||prediction || quantiles ||
> |1.218|1.0 |[1.56,-0.605] |5.718985621018951 |
> [1.160322990805951,4.99546058340675]|
> |2.949|0.0 |[0.346,2.158] |18.07678210850554
> |[3.66759199449632,15.789837303662042]|
> |3.627|0.0 |[1.38,0.231] |7.381908879359964
> |[1.4977129086101573,6.4480027195054905]|
> |0.273|1.0 |[0.52,1.151]
> |13.577717814884505|[2.754778414791513,11.859962351993202]|
> |4.199|0.0 |[0.795,-0.226]|9.013087597344805
> |[1.828662187733188,7.8728164067854856]|
> But if we change the value of all labels as label + 20. as:
> {code}
> training = spark.createDataFrame([
> (21.218, 1.0, Vectors.dense(1.560, -0.605)),
> (22.949, 0.0, Vectors.dense(0.346, 2.158)),
> (23.627, 0.0, Vectors.dense(1.380, 0.231)),
> (20.273, 1.0, Vectors.dense(0.520, 1.151)),
> (24.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result changes to:
> Coefficients: [23.9932020748,3.18105314757]
> Intercept: 7.35052273751137
> Scale: 7698609960.724161
> ||label ||censor||features ||prediction ||quantiles||
> |21.218|1.0 |[1.56,-0.605] |4.0912442688237169E18|[0.0,0.0]|
> |22.949|0.0 |[0.346,2.158] |6.011158613411288E9 |[0.0,0.0]|
> |23.627|0.0 |[1.38,0.231] |7.7835948690311181E17|[0.0,0.0]|
> |20.273|1.0 |[0.52,1.151] |1.5880852723124176E10|[0.0,0.0]|
> |24.199|0.0 |[0.795,-0.226]|1.4590190884193677E11|[0.0,0.0]|
> Can someone please explain this exponential blow up in prediction, as per my
> understanding prediction in AFT is a prediction of the time when the failure
> event will occur, not able to understand why it will change exponentially
> against the value of the label.
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[jira] [Commented] (SPARK-21919) inconsistent behavior of AFTsurvivalRegression algorithm
[
https://issues.apache.org/jira/browse/SPARK-21919?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=16153367#comment-16153367
]
Sean Owen commented on SPARK-21919:
---
It does look like a problem. From R's survreg I get:
{code}
survreg(formula = Surv(data$label, data$censor) ~ data$feature1 +
data$feature2, dist = "weibull")
Value Std. Error zp
(Intercept)3.29140 0.295 11.1737 5.49e-29
data$feature1 -0.06581 0.245 -0.2688 7.88e-01
data$feature2 0.00327 0.123 0.0265 9.79e-01
Log(scale)-2.20858 0.642 -3.4390 5.84e-04
Scale= 0.11
{code}
[~yanboliang] I think you originally created this; does it ring any bells?
> inconsistent behavior of AFTsurvivalRegression algorithm
>
>
> Key: SPARK-21919
> URL: https://issues.apache.org/jira/browse/SPARK-21919
> Project: Spark
> Issue Type: Bug
> Components: ML, PySpark
>Affects Versions: 2.2.0
> Environment: Spark Version: 2.2.0
> Cluster setup: Standalone single node
> Python version: 3.5.2
>Reporter: Ashish Chopra
>
> Took the direct example from spark ml documentation.
> {code}
> training = spark.createDataFrame([
> (1.218, 1.0, Vectors.dense(1.560, -0.605)),
> (2.949, 0.0, Vectors.dense(0.346, 2.158)),
> (3.627, 0.0, Vectors.dense(1.380, 0.231)),
> (0.273, 1.0, Vectors.dense(0.520, 1.151)),
> (4.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result is:
> Coefficients: [-0.496304411053,0.198452172529]
> Intercept: 2.6380898963056327
> Scale: 1.5472363533632303
> ||label||censor||features ||prediction || quantiles ||
> |1.218|1.0 |[1.56,-0.605] |5.718985621018951 |
> [1.160322990805951,4.99546058340675]|
> |2.949|0.0 |[0.346,2.158] |18.07678210850554
> |[3.66759199449632,15.789837303662042]|
> |3.627|0.0 |[1.38,0.231] |7.381908879359964
> |[1.4977129086101573,6.4480027195054905]|
> |0.273|1.0 |[0.52,1.151]
> |13.577717814884505|[2.754778414791513,11.859962351993202]|
> |4.199|0.0 |[0.795,-0.226]|9.013087597344805
> |[1.828662187733188,7.8728164067854856]|
> But if we change the value of all labels as label + 20. as:
> {code}
> training = spark.createDataFrame([
> (21.218, 1.0, Vectors.dense(1.560, -0.605)),
> (22.949, 0.0, Vectors.dense(0.346, 2.158)),
> (23.627, 0.0, Vectors.dense(1.380, 0.231)),
> (20.273, 1.0, Vectors.dense(0.520, 1.151)),
> (24.199, 0.0, Vectors.dense(0.795, -0.226))], ["label", "censor",
> "features"])
> quantileProbabilities = [0.3, 0.6]
> aft = AFTSurvivalRegression(quantileProbabilities=quantileProbabilities,
> quantilesCol="quantiles")
> #aft = AFTSurvivalRegression()
> model = aft.fit(training)
>
> # Print the coefficients, intercept and scale parameter for AFT survival
> regression
> print("Coefficients: " + str(model.coefficients))
> print("Intercept: " + str(model.intercept))
> print("Scale: " + str(model.scale))
> model.transform(training).show(truncate=False)
> {code}
> result changes to:
> Coefficients: [23.9932020748,3.18105314757]
> Intercept: 7.35052273751137
> Scale: 7698609960.724161
> ||label ||censor||features ||prediction ||quantiles||
> |21.218|1.0 |[1.56,-0.605] |4.0912442688237169E18|[0.0,0.0]|
> |22.949|0.0 |[0.346,2.158] |6.011158613411288E9 |[0.0,0.0]|
> |23.627|0.0 |[1.38,0.231] |7.7835948690311181E17|[0.0,0.0]|
> |20.273|1.0 |[0.52,1.151] |1.5880852723124176E10|[0.0,0.0]|
> |24.199|0.0 |[0.795,-0.226]|1.4590190884193677E11|[0.0,0.0]|
> Can someone please explain this exponential blow up in prediction, as per my
> understanding prediction in AFT is a prediction of the time when the failure
> event will occur, not able to understand why it will change exponentially
> against the value of the label.
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