Bottom Line: Auto-stopping for stochastic gradient descent training is great in theory, but, in my opinion, having to set the stopping tolerance constant (which depends on how the training data is scaled) and the consecutive-no-improve constant (which is arbitrary), outweigh the benefits of auto-stopping in many (but not all) scenarios.
The goal of a machine learning regression problem is to predict a single numeric value. Many regression techniques create a mathematical prediction equation that has constants called weights. Finding the values of the model weights is called training the model. One common form of training is called stochastic gradient descent (SGD). SGD is iterative where you must specify how mant times to iterate (max epochs) and how much to adjust the weights in each update (learn rate).
If you add auto-stopping to SGD training, the code checks the mean square error (MSE) of the current model after each epoch (one complete pass through the training data). If the current model has not significantly improved (measured using MSE with a small constant called a stop tolerance) for n consecutive epochs, then the training stops early, the idea being that the model is not improving.
Note: The term “early stopping” is sometimes synonymous with “auto-stopping”, but early stopping usually applies to a set of validation data instead of training data.
This sounds great in theory. For some scenarios, SGD auto-stop is a good idea. But in most of the scenarios I work with, the difficulties of specifying the stop tolerance and the number of consecutive no-improves required to exit, outweigh the benefit of an auto-stop.
First, consider determining if there is no improvement in MSE. The standard way to do this is to keep track of the best MSE found. Then the condition (currMSE “gt” bestMSE) indicates extreme no improvement, in fact, the current model is worse. But it’s common to relax this consition a bit to make it a bit easier to have no improvement: (currMSE “gt” bestMSE – a_little_bit). This is surprisingly subtle.
If the a_little_bit is called the stopping tolerance, the problem becomes specifying it.
The smallest possible tolerance is 0, in which case it is realtively easier to improve, which means it is relatively difficult to auto-stop, which leads to a more accurate prediction model, but one that is prone to overfitting.
If the tolerance is relatively large, like 0.01, then it is more diffiuclt to improve, easier to auto-stop, which leads to a model that predicts more poorly on training data but less prone to overfitting.
Unfortunately, MSE depends on how the training data is scaled, which means that specifying a good stopping tolerance dependes on the data. Suppose a predictor variable age has values (33, 51, 27, . . .) and some model MSE is 0.4567. But if the age values are scaled by dividing by 100 to (0.33, 0.51, 0.27 . . .) then the MSE on the same model will be divided by 100: 0.0045.
A second problem is specifying how many consecutive times there is no improvement that are required to indicate no progress and therefore trigger an auto-stop. The default value in the scikit SGDRegressor module is 5, but that is purely arbitrary.
I coded up a demo of auto-stopping for a linear regression model implemented using C#, to illustrate the ideas. The output is:
Begin C# linear regression using SGD training with auto-stopping Loading synthetic train (200) and test (40) data Done First three train X: -0.1660 0.4406 -0.9998 -0.3953 -0.7065 0.0776 -0.1616 0.3704 -0.5911 0.7562 -0.9452 0.3409 -0.1654 0.1174 -0.7192 First three train y: 0.4840 0.1568 0.8054 Setting lrnRate = 0.0010 Setting maxEpochs = 500 Setting stopTol = 1.000000e-004 Creating and training Linear Regression model epoch = 0 MSE = 0.1132 epoch = 25 MSE = 0.0035 epoch = 50 MSE = 0.0026 epoch = 75 MSE = 0.0026 epoch = 100 MSE = 0.0026 Auto stop at epoch 118 Done Weights/coefficients: -0.2655 0.0333 -0.0455 0.0358 -0.1146 Bias/constant: 0.3620 Evaluating model Acc train (within 0.10) = 0.4600 Acc test (within 0.10) = 0.6250 MSE train = 0.0026 MSE test = 0.0020 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 Predicted y = 0.5330 End demo
I used synthetic data. I hard-coded the consecutive-no-improves to 5.
An interesting way to spend some time on a Saturday afternoon.
The concept of auto-stopping for stochastic gradient descent is an interesting topic. Videos of automobile traffic stops can be very interesting too. A minority percentage of the U.S. population think that laws and civilized behavior don’t apply to them. Here are three examples where a simple traffic stop plus failure to follow requests for ID and insurance, turned into felony arrests.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols. (My blog editor chokes on symbols).
using System;
using System.IO;
using System.Collections.Generic;
// SGD training with auto-stopping
namespace LinearRegressionStochasticStopping
{
internal class LinearRegressionProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin C# linear regression " +
"using SGD training with auto-stopping ");
// 1. load data
Console.WriteLine("\nLoading synthetic train" +
" (200) and test (40) data");
string trainFile =
"..\\..\\..\\Data\\synthetic_train_200.txt";
int[] colsX = new int[] { 0, 1, 2, 3, 4 };
double[][] trainX =
MatLoad(trainFile, colsX, ',', "#");
double[] trainY =
MatToVec(MatLoad(trainFile,
new int[] { 5 }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX =
MatLoad(testFile, colsX, ',', "#");
double[] testY =
MatToVec(MatLoad(testFile,
new int[] { 5 }, ',', "#"));
Console.WriteLine("Done ");
Console.WriteLine("\nFirst three train X: ");
for (int i = 0; i "lt" 3; ++i)
VecShow(trainX[i], 4, 8);
Console.WriteLine("\nFirst three train y: ");
for (int i = 0; i "lt" 3; ++i)
Console.WriteLine(trainY[i].ToString("F4").
PadLeft(8));
// 2. create and train model
double lrnRate = 0.001;
int maxEpochs = 500;
double stopTol = 1.0e-4;
// very small stopTol (like 1.0e-8) =
// easy to improve = hard to stop = better acc.
// large stopTol (like 0.0) =
// hard to improve = easy to stop = worse acc.
Console.WriteLine("\nSetting lrnRate = " +
lrnRate.ToString("F4"));
Console.WriteLine("Setting maxEpochs = " +
maxEpochs);
Console.WriteLine("Setting stopTol = " +
stopTol.ToString("e"));
Console.WriteLine("\nCreating and training" +
" Linear Regression model ");
LinearRegressor model =
new LinearRegressor();
model.TrainSGD(trainX, trainY,
lrnRate, maxEpochs, stopTol);
Console.WriteLine("Done ");
// 2b.show model parameters
Console.WriteLine("\nWeights/coefficients: ");
for (int i = 0; i "lt" model.weights.Length; ++i)
Console.Write(model.weights[i].
ToString("F4") + " ");
Console.WriteLine("\nBias/constant: " +
model.bias.ToString("F4"));
// 3. evaluate model
Console.WriteLine("\nEvaluating model ");
double accTrain = model.Accuracy(trainX, trainY,
0.10);
Console.WriteLine("\nAcc train (within 0.10) = " +
accTrain.ToString("F4"));
double accTest = model.Accuracy(testX, testY,
0.10);
Console.WriteLine("Acc test (within 0.10) = " +
accTest.ToString("F4"));
double mseTrain = model.MSE(trainX, trainY);
Console.WriteLine("\nMSE train = " +
mseTrain.ToString("F4"));
double mseTest = model.MSE(testX, testY);
Console.WriteLine("MSE test = " +
mseTest.ToString("F4"));
// 4. use model
double[] x = trainX[0];
Console.WriteLine("\nPredicting for x = ");
VecShow(x, 4, 9);
double predY = model.Predict(x);
Console.WriteLine("\nPredicted y = " +
predY.ToString("F4"));
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main
// ------------------------------------------------------
// helpers for Main()
// ------------------------------------------------------
static double[][] MatLoad(string fn, int[] usecols,
char sep, string comment)
{
List"lt"double[]"gt" result =
new List"lt"double[]"gt"();
string line = "";
FileStream ifs = new FileStream(fn, FileMode.Open);
StreamReader sr = new StreamReader(ifs);
while ((line = sr.ReadLine()) != null)
{
if (line.StartsWith(comment) == true)
continue;
string[] tokens = line.Split(sep);
List"lt"double"gt" lst = new List"lt"double"gt"();
for (int j = 0; j "lt" usecols.Length; ++j)
lst.Add(double.Parse(tokens[usecols[j]]));
double[] row = lst.ToArray();
result.Add(row);
}
sr.Close(); ifs.Close();
return result.ToArray();
}
static double[] MatToVec(double[][] M)
{
int nRows = M.Length;
int nCols = M[0].Length;
double[] result = new double[nRows * nCols];
int k = 0;
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[k++] = M[i][j];
return result;
}
static void VecShow(double[] vec, int dec, int wid)
{
for (int i = 0; i "lt" vec.Length; ++i)
Console.Write(vec[i].ToString("F" + dec).
PadLeft(wid));
Console.WriteLine("");
}
} // class Program
public class LinearRegressor
{
public double[] weights;
public double bias;
private Random rnd;
public LinearRegressor(int seed = 0)
{
this.weights = new double[0]; // keep compiler happy
this.bias = 0;
this.rnd = new Random(seed);
}
public void TrainSGD(double[][] trainX,
double[] trainY, double lrnRate, int maxEpochs,
double stopTol)
{
// auto stop if MSE does not improve over best
// found MSE for 5 consecutive epochs,
// where improve has stopTol
int n = trainX.Length; int dim = trainX[0].Length;
this.weights = new double[dim];
// initialize weights and bias
double low = -0.01; double hi = 0.01;
for (int i = 0; i "lt" dim; ++i)
this.weights[i] = (hi - low) *
this.rnd.NextDouble() + low;
this.bias = (hi - low) *
this.rnd.NextDouble() + low;
int[] indices = new int[n]; // of train data
for (int i = 0; i "lt" n; ++i)
indices[i] = i;
// small tol: easier to improve = harder to stop
// bigger tol: harder to improve = easier to stop
double bestMSE = double.MaxValue;
int consecNoImprove = 0;
for (int epoch = 0; epoch "lt" maxEpochs; ++epoch)
{
// shuffle order pf training data indices
for (int i = 0; i "lt" indices.Length; ++i)
{
int ri = this.rnd.Next(i, indices.Length);
int tmp = indices[i];
indices[i] = indices[ri];
indices[ri] = tmp;
}
for (int i = 0; i "lt" n; ++i) // each train item
{
int ii = indices[i];
double[] x = trainX[ii];
double predY = this.Predict(x);
double actualY = trainY[ii];
for (int j = 0; j "lt" dim; ++j) // each weight
this.weights[j] -= lrnRate *
(predY - actualY) * x[j];
this.bias -= lrnRate * (predY - actualY);
}
//Console.WriteLine("epoch " + epoch);
double currMSE = this.MSE(trainX, trainY);
//Console.WriteLine("curr MSE = " +
// currMSE.ToString("F8"));
if (currMSE "lt" bestMSE)
{
bestMSE = currMSE;
//Console.WriteLine("New best MSE " +
// bestMSE.ToString("F8") + " at epoch " + epoch);
consecNoImprove = 0;
}
// either reset consecutive counter or increment it
// currMSE "gt" bestMSE is extreme no improvement
// but make it a bit easier for no improve:
// currMSE "gt" bestMSE - a_little_bit
// curr_loss "gt" best_loss - tol means no signif improve
if (currMSE "gt" bestMSE - stopTol) // no sig improve
{
//Console.WriteLine("no significant improvement" +
// " at epoch " + epoch);
++consecNoImprove;
}
else // improved
{
//Console.WriteLine("some improvement at" +
// " epoch " + epoch);
consecNoImprove = 0;
}
if (consecNoImprove == 5)
{
Console.WriteLine("Auto stop at epoch " + epoch);
break;
}
if (epoch % (int)(maxEpochs / 20) == 0) // progress
{
double mse = this.MSE(trainX, trainY);
string s = "";
s += "epoch = " + epoch.ToString().PadLeft(5);
s += " MSE = " + mse.ToString("F4").PadLeft(8);
Console.WriteLine(s);
}
} // each epoch
} // Train
// ------------------------------------------------------
public double Predict(double[] x)
{
double result = 0.0;
for (int j = 0; j "lt" x.Length; ++j)
result += x[j] * this.weights[j];
result += this.bias;
return result;
}
// ------------------------------------------------------
public double Accuracy(double[][] dataX, double[] dataY,
double pctClose)
{
int numCorrect = 0; int numWrong = 0;
for (int i = 0; i "lt" dataX.Length; ++i)
{
double actualY = dataY[i];
double predY = this.Predict(dataX[i]);
if (Math.Abs(predY - actualY) "lt"
(pctClose * actualY))
++numCorrect;
else
++numWrong;
}
return (numCorrect * 1.0) / (numWrong + numCorrect);
}
// ------------------------------------------------------
public double MSE(double[][] dataX, double[] dataY)
{
int n = dataX.Length;
double sum = 0.0;
for (int i = 0; i "lt" n; ++i)
{
double actualY = dataY[i];
double predY = this.Predict(dataX[i]);
sum += (actualY - predY) * (actualY - predY);
}
return sum / n;
}
} // class LinearRegressor
} // ns
Training data:
# synthetic_train_200.txt # -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 0.9365, -0.3732, 0.3846, 0.7528, 0.7892, 0.1345 -0.8299, -0.9219, -0.6603, 0.7563, -0.8033, 0.7955 0.0663, 0.3838, -0.3690, 0.3730, 0.6693, 0.3206 -0.9634, 0.5003, 0.9777, 0.4963, -0.4391, 0.7377 -0.1042, 0.8172, -0.4128, -0.4244, -0.7399, 0.4801 -0.9613, 0.3577, -0.5767, -0.4689, -0.0169, 0.6861 -0.7065, 0.1786, 0.3995, -0.7953, -0.1719, 0.5569 0.3888, -0.1716, -0.9001, 0.0718, 0.3276, 0.2500 0.1731, 0.8068, -0.7251, -0.7214, 0.6148, 0.3297 -0.2046, -0.6693, 0.8550, -0.3045, 0.5016, 0.2129 0.2473, 0.5019, -0.3022, -0.4601, 0.7918, 0.2613 -0.1438, 0.9297, 0.3269, 0.2434, -0.7705, 0.5171 0.1568, -0.1837, -0.5259, 0.8068, 0.1474, 0.3307 -0.9943, 0.2343, -0.3467, 0.0541, 0.7719, 0.5581 0.2467, -0.9684, 0.8589, 0.3818, 0.9946, 0.1092 -0.6553, -0.7257, 0.8652, 0.3936, -0.8680, 0.7018 0.8460, 0.4230, -0.7515, -0.9602, -0.9476, 0.1996 -0.9434, -0.5076, 0.7201, 0.0777, 0.1056, 0.5664 0.9392, 0.1221, -0.9627, 0.6013, -0.5341, 0.1533 0.6142, -0.2243, 0.7271, 0.4942, 0.1125, 0.1661 0.4260, 0.1194, -0.9749, -0.8561, 0.9346, 0.2230 0.1362, -0.5934, -0.4953, 0.4877, -0.6091, 0.3810 0.6937, -0.5203, -0.0125, 0.2399, 0.6580, 0.1460 -0.6864, -0.9628, -0.8600, -0.0273, 0.2127, 0.5387 0.9772, 0.1595, -0.2397, 0.1019, 0.4907, 0.1611 0.3385, -0.4702, -0.8673, -0.2598, 0.2594, 0.2270 -0.8669, -0.4794, 0.6095, -0.6131, 0.2789, 0.4700 0.0493, 0.8496, -0.4734, -0.8681, 0.4701, 0.3516 0.8639, -0.9721, -0.5313, 0.2336, 0.8980, 0.1412 0.9004, 0.1133, 0.8312, 0.2831, -0.2200, 0.1782 0.0991, 0.8524, 0.8375, -0.2102, 0.9265, 0.2150 -0.6521, -0.7473, -0.7298, 0.0113, -0.9570, 0.7422 0.6190, -0.3105, 0.8802, 0.1640, 0.7577, 0.1056 0.6895, 0.8108, -0.0802, 0.0927, 0.5972, 0.2214 0.1982, -0.9689, 0.1870, -0.1326, 0.6147, 0.1310 -0.3695, 0.7858, 0.1557, -0.6320, 0.5759, 0.3773 -0.1596, 0.3581, 0.8372, -0.9992, 0.9535, 0.2071 -0.2468, 0.9476, 0.2094, 0.6577, 0.1494, 0.4132 0.1737, 0.5000, 0.7166, 0.5102, 0.3961, 0.2611 0.7290, -0.3546, 0.3416, -0.0983, -0.2358, 0.1332 -0.3652, 0.2438, -0.1395, 0.9476, 0.3556, 0.4170 -0.6029, -0.1466, -0.3133, 0.5953, 0.7600, 0.4334 -0.4596, -0.4953, 0.7098, 0.0554, 0.6043, 0.2775 0.1450, 0.4663, 0.0380, 0.5418, 0.1377, 0.2931 -0.8636, -0.2442, -0.8407, 0.9656, -0.6368, 0.7429 0.6237, 0.7499, 0.3768, 0.1390, -0.6781, 0.2185 -0.5499, 0.1850, -0.3755, 0.8326, 0.8193, 0.4399 -0.4858, -0.7782, -0.6141, -0.0008, 0.4572, 0.4197 0.7033, -0.1683, 0.2334, -0.5327, -0.7961, 0.1776 0.0317, -0.0457, -0.6947, 0.2436, 0.0880, 0.3345 0.5031, -0.5559, 0.0387, 0.5706, -0.9553, 0.3107 -0.3513, 0.7458, 0.6894, 0.0769, 0.7332, 0.3170 0.2205, 0.5992, -0.9309, 0.5405, 0.4635, 0.3532 -0.4806, -0.4859, 0.2646, -0.3094, 0.5932, 0.3202 0.9809, -0.3995, -0.7140, 0.8026, 0.0831, 0.1600 0.9495, 0.2732, 0.9878, 0.0921, 0.0529, 0.1289 -0.9476, -0.6792, 0.4913, -0.9392, -0.2669, 0.5966 0.7247, 0.3854, 0.3819, -0.6227, -0.1162, 0.1550 -0.5922, -0.5045, -0.4757, 0.5003, -0.0860, 0.5863 -0.8861, 0.0170, -0.5761, 0.5972, -0.4053, 0.7301 0.6877, -0.2380, 0.4997, 0.0223, 0.0819, 0.1404 0.9189, 0.6079, -0.9354, 0.4188, -0.0700, 0.1907 -0.1428, -0.7820, 0.2676, 0.6059, 0.3936, 0.2790 0.5324, -0.3151, 0.6917, -0.1425, 0.6480, 0.1071 -0.8432, -0.9633, -0.8666, -0.0828, -0.7733, 0.7784 -0.9444, 0.5097, -0.2103, 0.4939, -0.0952, 0.6787 -0.0520, 0.6063, -0.1952, 0.8094, -0.9259, 0.4836 0.5477, -0.7487, 0.2370, -0.9793, 0.0773, 0.1241 0.2450, 0.8116, 0.9799, 0.4222, 0.4636, 0.2355 0.8186, -0.1983, -0.5003, -0.6531, -0.7611, 0.1511 -0.4714, 0.6382, -0.3788, 0.9648, -0.4667, 0.5950 0.0673, -0.3711, 0.8215, -0.2669, -0.1328, 0.2677 -0.9381, 0.4338, 0.7820, -0.9454, 0.0441, 0.5518 -0.3480, 0.7190, 0.1170, 0.3805, -0.0943, 0.4724 -0.9813, 0.1535, -0.3771, 0.0345, 0.8328, 0.5438 -0.1471, -0.5052, -0.2574, 0.8637, 0.8737, 0.3042 -0.5454, -0.3712, -0.6505, 0.2142, -0.1728, 0.5783 0.6327, -0.6297, 0.4038, -0.5193, 0.1484, 0.1153 -0.5424, 0.3282, -0.0055, 0.0380, -0.6506, 0.6613 0.1414, 0.9935, 0.6337, 0.1887, 0.9520, 0.2540 -0.9351, -0.8128, -0.8693, -0.0965, -0.2491, 0.7353 0.9507, -0.6640, 0.9456, 0.5349, 0.6485, 0.1059 -0.0462, -0.9737, -0.2940, -0.0159, 0.4602, 0.2606 -0.0627, -0.0852, -0.7247, -0.9782, 0.5166, 0.2977 0.0478, 0.5098, -0.0723, -0.7504, -0.3750, 0.3335 0.0090, 0.3477, 0.5403, -0.7393, -0.9542, 0.4415 -0.9748, 0.3449, 0.3736, -0.1015, 0.8296, 0.4358 0.2887, -0.9895, -0.0311, 0.7186, 0.6608, 0.2057 0.1570, -0.4518, 0.1211, 0.3435, -0.2951, 0.3244 0.7117, -0.6099, 0.4946, -0.4208, 0.5476, 0.1096 -0.2929, -0.5726, 0.5346, -0.3827, 0.4665, 0.2465 0.4889, -0.5572, -0.5718, -0.6021, -0.7150, 0.2163 -0.7782, 0.3491, 0.5996, -0.8389, -0.5366, 0.6516 -0.5847, 0.8347, 0.4226, 0.1078, -0.3910, 0.6134 0.8469, 0.4121, -0.0439, -0.7476, 0.9521, 0.1571 -0.6803, -0.5948, -0.1376, -0.1916, -0.7065, 0.7156 0.2878, 0.5086, -0.5785, 0.2019, 0.4979, 0.2980 0.2764, 0.1943, -0.4090, 0.4632, 0.8906, 0.2960 -0.8877, 0.6705, -0.6155, -0.2098, -0.3998, 0.7107 -0.8398, 0.8093, -0.2597, 0.0614, -0.0118, 0.6502 -0.8476, 0.0158, -0.4769, -0.2859, -0.7839, 0.7715 0.5751, -0.7868, 0.9714, -0.6457, 0.1448, 0.1175 0.4802, -0.7001, 0.1022, -0.5668, 0.5184, 0.1090 0.4458, -0.6469, 0.7239, -0.9604, 0.7205, 0.0779 0.5175, 0.4339, 0.9747, -0.4438, -0.9924, 0.2879 0.8678, 0.7158, 0.4577, 0.0334, 0.4139, 0.1678 0.5406, 0.5012, 0.2264, -0.1963, 0.3946, 0.2088 -0.9938, 0.5498, 0.7928, -0.5214, -0.7585, 0.7687 0.7661, 0.0863, -0.4266, -0.7233, -0.4197, 0.1466 0.2277, -0.3517, -0.0853, -0.1118, 0.6563, 0.1767 0.3499, -0.5570, -0.0655, -0.3705, 0.2537, 0.1632 0.7547, -0.1046, 0.5689, -0.0861, 0.3125, 0.1257 0.8186, 0.2110, 0.5335, 0.0094, -0.0039, 0.1391 0.6858, -0.8644, 0.1465, 0.8855, 0.0357, 0.1845 -0.4967, 0.4015, 0.0805, 0.8977, 0.2487, 0.4663 0.6760, -0.9841, 0.9787, -0.8446, -0.3557, 0.1509 -0.1203, -0.4885, 0.6054, -0.0443, -0.7313, 0.4854 0.8557, 0.7919, -0.0169, 0.7134, -0.1628, 0.2002 0.0115, -0.6209, 0.9300, -0.4116, -0.7931, 0.4052 -0.7114, -0.9718, 0.4319, 0.1290, 0.5892, 0.3661 0.3915, 0.5557, -0.1870, 0.2955, -0.6404, 0.2954 -0.3564, -0.6548, -0.1827, -0.5172, -0.1862, 0.4622 0.2392, -0.4959, 0.5857, -0.1341, -0.2850, 0.2470 -0.3394, 0.3947, -0.4627, 0.6166, -0.4094, 0.5325 0.7107, 0.7768, -0.6312, 0.1707, 0.7964, 0.2757 -0.1078, 0.8437, -0.4420, 0.2177, 0.3649, 0.4028 -0.3139, 0.5595, -0.6505, -0.3161, -0.7108, 0.5546 0.4335, 0.3986, 0.3770, -0.4932, 0.3847, 0.1810 -0.2562, -0.2894, -0.8847, 0.2633, 0.4146, 0.4036 0.2272, 0.2966, -0.6601, -0.7011, 0.0284, 0.2778 -0.0743, -0.1421, -0.0054, -0.6770, -0.3151, 0.3597 -0.4762, 0.6891, 0.6007, -0.1467, 0.2140, 0.4266 -0.4061, 0.7193, 0.3432, 0.2669, -0.7505, 0.6147 -0.0588, 0.9731, 0.8966, 0.2902, -0.6966, 0.4955 -0.0627, -0.1439, 0.1985, 0.6999, 0.5022, 0.3077 0.1587, 0.8494, -0.8705, 0.9827, -0.8940, 0.4263 -0.7850, 0.2473, -0.9040, -0.4308, -0.8779, 0.7199 0.4070, 0.3369, -0.2428, -0.6236, 0.4940, 0.2215 -0.0242, 0.0513, -0.9430, 0.2885, -0.2987, 0.3947 -0.5416, -0.1322, -0.2351, -0.0604, 0.9590, 0.3683 0.1055, 0.7783, -0.2901, -0.5090, 0.8220, 0.2984 -0.9129, 0.9015, 0.1128, -0.2473, 0.9901, 0.4776 -0.9378, 0.1424, -0.6391, 0.2619, 0.9618, 0.5368 0.7498, -0.0963, 0.4169, 0.5549, -0.0103, 0.1614 -0.2612, -0.7156, 0.4538, -0.0460, -0.1022, 0.3717 0.7720, 0.0552, -0.1818, -0.4622, -0.8560, 0.1685 -0.4177, 0.0070, 0.9319, -0.7812, 0.3461, 0.3052 -0.0001, 0.5542, -0.7128, -0.8336, -0.2016, 0.3803 0.5356, -0.4194, -0.5662, -0.9666, -0.2027, 0.1776 -0.2378, 0.3187, -0.8582, -0.6948, -0.9668, 0.5474 -0.1947, -0.3579, 0.1158, 0.9869, 0.6690, 0.2992 0.3992, 0.8365, -0.9205, -0.8593, -0.0520, 0.3154 -0.0209, 0.0793, 0.7905, -0.1067, 0.7541, 0.1864 -0.4928, -0.4524, -0.3433, 0.0951, -0.5597, 0.6261 -0.8118, 0.7404, -0.5263, -0.2280, 0.1431, 0.6349 0.0516, -0.8480, 0.7483, 0.9023, 0.6250, 0.1959 -0.3212, 0.1093, 0.9488, -0.3766, 0.3376, 0.2735 -0.3481, 0.5490, -0.3484, 0.7797, 0.5034, 0.4379 -0.5785, -0.9170, -0.3563, -0.9258, 0.3877, 0.4121 0.3407, -0.1391, 0.5356, 0.0720, -0.9203, 0.3458 -0.3287, -0.8954, 0.2102, 0.0241, 0.2349, 0.3247 -0.1353, 0.6954, -0.0919, -0.9692, 0.7461, 0.3338 0.9036, -0.8982, -0.5299, -0.8733, -0.1567, 0.1187 0.7277, -0.8368, -0.0538, -0.7489, 0.5458, 0.0830 0.9049, 0.8878, 0.2279, 0.9470, -0.3103, 0.2194 0.7957, -0.1308, -0.5284, 0.8817, 0.3684, 0.2172 0.4647, -0.4931, 0.2010, 0.6292, -0.8918, 0.3371 -0.7390, 0.6849, 0.2367, 0.0626, -0.5034, 0.7039 -0.1567, -0.8711, 0.7940, -0.5932, 0.6525, 0.1710 0.7635, -0.0265, 0.1969, 0.0545, 0.2496, 0.1445 0.7675, 0.1354, -0.7698, -0.5460, 0.1920, 0.1728 -0.5211, -0.7372, -0.6763, 0.6897, 0.2044, 0.5217 0.1913, 0.1980, 0.2314, -0.8816, 0.5006, 0.1998 0.8964, 0.0694, -0.6149, 0.5059, -0.9854, 0.1825 0.1767, 0.7104, 0.2093, 0.6452, 0.7590, 0.2832 -0.3580, -0.7541, 0.4426, -0.1193, -0.7465, 0.5657 -0.5996, 0.5766, -0.9758, -0.3933, -0.9572, 0.6800 0.9950, 0.1641, -0.4132, 0.8579, 0.0142, 0.2003 -0.4717, -0.3894, -0.2567, -0.5111, 0.1691, 0.4266 0.3917, -0.8561, 0.9422, 0.5061, 0.6123, 0.1212 -0.0366, -0.1087, 0.3449, -0.1025, 0.4086, 0.2475 0.3633, 0.3943, 0.2372, -0.6980, 0.5216, 0.1925 -0.5325, -0.6466, -0.2178, -0.3589, 0.6310, 0.3568 0.2271, 0.5200, -0.1447, -0.8011, -0.7699, 0.3128 0.6415, 0.1993, 0.3777, -0.0178, -0.8237, 0.2181 -0.5298, -0.0768, -0.6028, -0.9490, 0.4588, 0.4356 0.6870, -0.1431, 0.7294, 0.3141, 0.1621, 0.1632 -0.5985, 0.0591, 0.7889, -0.3900, 0.7419, 0.2945 0.3661, 0.7984, -0.8486, 0.7572, -0.6183, 0.3449 0.6995, 0.3342, -0.3113, -0.6972, 0.2707, 0.1712 0.2565, 0.9126, 0.1798, -0.6043, -0.1413, 0.2893 -0.3265, 0.9839, -0.2395, 0.9854, 0.0376, 0.4770 0.2690, -0.1722, 0.9818, 0.8599, -0.7015, 0.3954 -0.2102, -0.0768, 0.1219, 0.5607, -0.0256, 0.3949 0.8216, -0.9555, 0.6422, -0.6231, 0.3715, 0.0801 -0.2896, 0.9484, -0.7545, -0.6249, 0.7789, 0.4370 -0.9985, -0.5448, -0.7092, -0.5931, 0.7926, 0.5402
Test data:
# synthetic_test_40.txt # 0.7462, 0.4006, -0.0590, 0.6543, -0.0083, 0.1935 0.8495, -0.2260, -0.0142, -0.4911, 0.7699, 0.1078 -0.2335, -0.4049, 0.4352, -0.6183, -0.7636, 0.5088 0.1810, -0.5142, 0.2465, 0.2767, -0.3449, 0.3136 -0.8650, 0.7611, -0.0801, 0.5277, -0.4922, 0.7140 -0.2358, -0.7466, -0.5115, -0.8413, -0.3943, 0.4533 0.4834, 0.2300, 0.3448, -0.9832, 0.3568, 0.1360 -0.6502, -0.6300, 0.6885, 0.9652, 0.8275, 0.3046 -0.3053, 0.5604, 0.0929, 0.6329, -0.0325, 0.4756 -0.7995, 0.0740, -0.2680, 0.2086, 0.9176, 0.4565 -0.2144, -0.2141, 0.5813, 0.2902, -0.2122, 0.4119 -0.7278, -0.0987, -0.3312, -0.5641, 0.8515, 0.4438 0.3793, 0.1976, 0.4933, 0.0839, 0.4011, 0.1905 -0.8568, 0.9573, -0.5272, 0.3212, -0.8207, 0.7415 -0.5785, 0.0056, -0.7901, -0.2223, 0.0760, 0.5551 0.0735, -0.2188, 0.3925, 0.3570, 0.3746, 0.2191 0.1230, -0.2838, 0.2262, 0.8715, 0.1938, 0.2878 0.4792, -0.9248, 0.5295, 0.0366, -0.9894, 0.3149 -0.4456, 0.0697, 0.5359, -0.8938, 0.0981, 0.3879 0.8629, -0.8505, -0.4464, 0.8385, 0.5300, 0.1769 0.1995, 0.6659, 0.7921, 0.9454, 0.9970, 0.2330 -0.0249, -0.3066, -0.2927, -0.4923, 0.8220, 0.2437 0.4513, -0.9481, -0.0770, -0.4374, -0.9421, 0.2879 -0.3405, 0.5931, -0.3507, -0.3842, 0.8562, 0.3987 0.9538, 0.0471, 0.9039, 0.7760, 0.0361, 0.1706 -0.0887, 0.2104, 0.9808, 0.5478, -0.3314, 0.4128 -0.8220, -0.6302, 0.0537, -0.1658, 0.6013, 0.4306 -0.4123, -0.2880, 0.9074, -0.0461, -0.4435, 0.5144 0.0060, 0.2867, -0.7775, 0.5161, 0.7039, 0.3599 -0.7968, -0.5484, 0.9426, -0.4308, 0.8148, 0.2979 0.7811, 0.8450, -0.6877, 0.7594, 0.2640, 0.2362 -0.6802, -0.1113, -0.8325, -0.6694, -0.6056, 0.6544 0.3821, 0.1476, 0.7466, -0.5107, 0.2592, 0.1648 0.7265, 0.9683, -0.9803, -0.4943, -0.5523, 0.2454 -0.9049, -0.9797, -0.0196, -0.9090, -0.4433, 0.6447 -0.4607, 0.1811, -0.2389, 0.4050, -0.0078, 0.5229 0.2664, -0.2932, -0.4259, -0.7336, 0.8742, 0.1834 -0.4507, 0.1029, -0.6294, -0.1158, -0.6294, 0.6081 0.8948, -0.0124, 0.9278, 0.2899, -0.0314, 0.1534 -0.1323, -0.8813, -0.0146, -0.0697, 0.6135, 0.2386
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