# The vanishing gradient problem and ReLUs – a TensorFlow investigation

Deep learning is huge in machine learning at the moment, and no wonder – it is making large and important strides in solving problems in computer vision, natural language and reinforcement learning and problems in many other areas. Deep learning neural networks are neural networks which are characterized by many layers – making them deep instead of wide. Deep networks have been demonstrated to be more practically capable of solving problems than simple, wide two layer networks. Neural networks have been around for a long time, but initial success using these networks was elusive. One of the issues that had to be overcome in making them more useful and transitioning to modern deep learning networks was the vanishing gradient problem. This problem manifests in the early layers of deep neural networks not learning (or learning very slowly), resulting in difficulties in solving practical problems.

This post will examine the vanishing gradient problem, and demonstrate an improvement to the problem through the use of the rectified linear unit activation function, or ReLUs. The examination will take place using TensorFlow and visualizing with the TensorBoard utility. The TensorFlow code used in this tutorial can be found on this site’s Github repository.

The vanishing gradient problem arises due to the nature of the back-propagation optimization which occurs in neural network training (for a comprehensive introduction to back-propagation, see my free ebook). The weight and bias values in the various layers within a neural network are updated each optimization iteration by stepping in the direction of the gradient of the weight/bias values with respect to the loss function. In other words, the weight values change in proportion to the following gradient:

$$\partial C/ \partial W_l$$

Where $W_l$ represents the weights of layer l and is the cost or loss function at the output layer (again, if these terms are gibberish to you, check out my free ebook which will get you up to speed). In the final layer, this calculation is straight-forward, however in earlier layers, the back-propagation of errors method needs to be utilized. At the final layer, the error term $\delta$ looks like:

$$\delta_i^{(n_l)} = -(y_i – h_i^{(n_l)})\cdot f^\prime(z_i^{(n_l)})$$

Don’t worry too much about the notation, but basically the equation above shows first that the error is related to the difference between the output of the network $h_i^{(n_l)}$ and the training labels $y_i$ (i.e. $(y_i – h_i^{(n_l)})$). It is also, more importantly for the vanishing gradient problem, proportional to the derivative of the activation function $f^\prime(z_i^{(n_l)})$. The weights in the final layer change in direct proportion to this $\delta$ value. For earlier layers, the error from the latter layers is back-propagated via the following rule:

$$\delta^{(l)} = \left((W^{(l)})^T \delta^{(l+1)}\right) \bullet f'(z^{(l)})$$

Again, in the second part of this equation, there is the derivative of the activation function f'(z^{(l)}). Notice that $\delta^{(l)}$ is also proportional to the error propagated from the downstream layer $\delta^{(l+1)}$. These downstream $\delta$ values also include their own f'(z^{(l)}) values. So, basically, the gradient of the weights of a given layer with respect to the loss function, which controls how these weight values are updated, is proportional to chained multiplications of the derivative of the activation function i.e.:

$$\frac{\partial C} {\partial W_l} \propto f'(z^{(l)}) f'(z^{(l+1)}) f'(z^{(l+2)}) \dots$$

The vanishing gradient problem comes about in deep neural networks when the f’ terms are all outputting values << 1. When we multiply lots of numbers << 1 together, we end up with a vanishing product, which leads to a very small $\frac{\partial C} {\partial W_l}$ value and hence practically no learning of the weight values – the predictive power of the neural network then platueus.

### The sigmoid activation function

The vanishing gradient problem is particularly problematic with sigmoid activation functions. The plot below shows the sigmoid activation function and its first derivative:

As can be observed, when the sigmoid function value is either too high or too low, the derivative (orange line) becomes very small i.e. << 1. This causes vanishing gradients and poor learning for deep networks. This can occur when the weights of our networks are initialized poorly – with too-large negative and positive values. These too-large values saturate the input to the sigmoid and pushes the derivatives into the small valued regions. However, even if the weights are initialized nicely, and the derivatives are sitting around the maximum i.e. ~0.2, with many layers there will still be a vanishing gradient problem. With only 4 layers of 0.2 valued derivatives we have a product of $0.2^{4} = 0.0016$ – not very large! Consider how the ResNet architecture, generally with 10’s or 100’s of layers, would train using sigmoid activation functions with even the best initialized weights. Most of the layers would be static or dead and impervious to training.

So what’s the solution to this problem? It’s called a rectified linear unit activation function, or ReLU.

### The ReLU activation function

The ReLU activation function is defined as:

$$f(x) = \max(0, x)$$

This function and it’s first derivative look like:

ReLU activation and first derivative

As can be observed, the ReLU activation simply returns its argument x whenever it is greater than zero, and returns 0 otherwise. The first derivative of ReLU is also very simple – it is equal to 1 when is greater than zero, but otherwise it is 0. You can probably see the advantages of ReLU at this point – when it’s derivative is back-propagated there will be no degradation of the error signal as 1 x 1 x 1 x 1… = 1. However, the ReLU activation still maintains a non-linearity or “switch on” characteristic which enables it to behave analogously to a biological neuron.

There is only one problem with the ReLU activation – sometimes, because the derivative is zero when < 0, certain weights can be “killed off” or become “dead”. This is because the back-propagated error can be cancelled out whenever there is a negative input into a given neuron and therefore the gradient $\frac{\partial C} {\partial W_l}$ will also fall to zero. This means there is no way for the associated weights to update in the right direction. This can obviously impact learning.

What’s the solution? A variant of ReLU which is called a Leaky ReLU activation.

### The Leaky ReLU activation

The Leaky ReLU activation is defined as:

$$f(x) = \max(0.01x, x)$$

As you can observe, when is below zero, the output will switch from to 0.01x. I won’t plot the activation for this function, as it is too difficult to see the difference between 0.01x and 0 and therefore in plots it looks just like a normal ReLU. However, the good thing about the Leaky ReLU activation function is that the derivative when x is below zero is 0.01 – i.e. it is a small but no longer 0. This gives the neuron and associated weights the chance to reactivate, and therefore this should improve the overall learning performance.

Now it’s time to test out these ideas in a real example using TensorFlow.

## Demonstrating the vanishing gradient problem in TensorFlow

### Creating the model

In the TensorFlow code I am about to show you, we’ll be creating a 7 layer densely connected network (including the input and output layers) and using the TensorFlow summary operations and TensorBoard visualization to see what is going on with the gradients. The code uses the TensorFlow layers (tf.layers) framework which allows quick and easy building of networks. The data we will be training the network on is the MNIST hand-written digit recognition dataset that comes packaged up with the TensorFlow installation.

To create the dataset, we can run the following:

mnist = input_data.read_data_sets("MNIST_data/", one_hot=True)

The MNIST data can be extracted from this mnist data set by calling mnist.train.next_batch(batch_size). In this case, we’ll just be looking at the training data, but you can also extract a test dataset from the same data. In this example, I’ll be using the feed_dict methodology and placeholder variables to feed in the training data, which isn’t the optimal method (see my Dataset tutorial for the most efficient data consumption methodology) but it will do for these purposes. First, I’ll setup the data placeholders:

self.input_images = tf.placeholder(tf.float32, shape=[None, self._input_size])
self.labels = tf.placeholder(tf.float32, shape=[None, self._label_size])

Note, I have created these variables in an overarching class called Model, hence all the self references. The MNIST data input size (self._input_size) is equal to the 28 x 28 image pixels i.e. 784 pixels. The number of associated labels, self._label_size is equal to the 10 possible hand-written digit classes in the MNIST dataset.

In this tutorial, we’ll be creating a slightly deep fully connected network – a network with 7 total layers including input and output layers. To create these densely connected layers easily, we’ll be using TensorFlow’s handy tf.layers API and a simple Python loop like follows:

# create self._num_layers dense layers as the model
input = self.input_images
for i in range(self._num_layers - 1):
input = tf.layers.dense(input, self._hidden_size, activation=self._activation,
name='layer{}'.format(i+1))

First, the generic input variable is initialized to be equal to the input images (fed via the placeholder). Next, the code runs through a loop where multiple dense layers are created, each named ‘layerX’ where X is the layer number. The number of nodes in the layer is set equal to the class property self._hidden_size and the activation function is also supplied via the property self._activation.

Next we create the final, output layer (you’ll note that the loop above terminates before it gets to creating the final layer), and we don’t supply an activation to this layer. In the tf.layers API, a linear activation (i.e. f(x) = x) is applied by default if no activation argument is supplied.

# don't supply an activation for the final layer - the loss definition will
# supply softmax activation. This defaults to a linear activation i.e. f(x) = x
logits = tf.layers.dense(input, 10, name='layer{}'.format(self._num_layers))

Next, the loss operation is setup and logged:

# use softmax cross entropy with logits - no need to apply softmax activation to
# logits
self.loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(logits=logits,
labels=self.labels))
# add the loss to the summary
tf.summary.scalar('loss', self.loss)

The loss used in this instance is the handy TensorFlow softmax_cross_entropy_with_logits_v2 (the original version is soon to be deprecated). This loss function will apply the softmax operation to the un-activated output of the network, then apply the cross entropy loss to this outcome. After this loss operation is created, it’s output value is added to the tf.summary framework. This framework allows scalar values to be logged and subsequently visualized in the TensorBoard web-based visualization page. It can also log histogram information, along with audio and images – all of these can be observed through the aforementioned TensorBoard visualization.

Next, the program calls a method to log the gradients, which we will visualize to examine the vanishing gradient problem:

self._log_gradients(self._num_layers)

This method looks like the following:

def _log_gradients(self, num_layers):
gr = tf.get_default_graph()
for i in range(num_layers):
weight = gr.get_tensor_by_name('layer{}/kernel:0'.format(i + 1))
tf.summary.scalar('mean_{}'.format(i + 1), mean)
tf.summary.histogram('hist_weights_{}'.format(i + 1), grad)

In this method, first the TensorFlow computational graph is extracted so that weight variables can be called out of it. Then a loop is entered into, to cycle through all the layers. For each layer, first the weight tensor for the given layer is grabbed by the handy function get_tensor_by_name. You will recall that each layer was named “layerX” where X is the layer number. This is supplied to the function, along with “/kernel:0” – this tells the function that we are trying to access the weight variable (also called a kernel) as opposed to the bias value, which would be “/bias:0”.

On the next line, the tf.gradients() function is used. This will calculate gradients of the form $\partial y / \partial x$ where the first argument supplied to the function is y and the second is x. In the gradient descent step, the weight update is made in proportion to $\partial loss / \partial W$, so in this case the first argument supplied to tf.gradients() is the loss, and the second is the weight tensor.

Next, the mean absolute value of the gradient is calculated, and then this is logged as a scalar in the summary. Next, histograms of the gradients and the weight values are also logged in the summary. The flow now returns back to the main method in the class.

self.optimizer = tf.train.AdamOptimizer().minimize(self.loss)
self.accuracy = self._compute_accuracy(logits, self.labels)
tf.summary.scalar('acc', self.accuracy)

The code above is fairly standard TensorFlow usage – defining an optimizer, in this case the flexible and powerful AdamOptimizer(), and also a generic accuracy operation, the outcome of which is also added to the summary (see the Github code for the accuracy method called).

Finally a summary merge operation is created, which will gather up all the summary data ready for export to the TensorBoard file whenever it is executed:

self.merged = tf.summary.merge_all()

An initialization operation is also created. Now all that is left is to run the main training loop.

### Training the model

The main training loop of this experimental model is shown in the code below:

def run_training(model, mnist, sub_folder, iterations=2500, batch_size=30):
with tf.Session() as sess:
sess.run(model.init_op)
train_writer = tf.summary.FileWriter(base_path + sub_folder,
sess.graph)
for i in range(iterations):
image_batch, label_batch = mnist.train.next_batch(batch_size)
l, _, acc = sess.run([model.loss, model.optimizer, model.accuracy],
feed_dict={model.input_images: image_batch, model.labels: label_batch})
if i % 200 == 0:
summary = sess.run(model.merged, feed_dict={model.input_images: image_batch,
model.labels: label_batch})
print("Iteration {} of {}, loss: {:.3f}, train accuracy: "
"{:.2f}%".format(i, iterations, l, acc * 100))

This is a pretty standard TensorFlow training loop (if you’re unfamiliar with this, see my TensorFlow tutorial) – however, one non-standard addition is the tf.summary.FileWriter() operation and its associated uses. This operation generally takes two arguments – the location to store the files and the session graph. Note that it is a good idea to setup a different sub folder for each of your TensorFlow runs when using summaries, as this allows for better visualization and comparison of the various runs within TensorBoard.

Every 200 iterations, we run the merged operation, which is defined in the class instance model – as mentioned previously, this gathers up all the logged summary data ready for writing. The train_writer.add_summary() operation is then run on this output, which writes the data into the chosen location (optionally along with the iteration/epoch number).

The summary data can then be visualized using TensorBoard. To run TensorBoard, using command prompt, navigate to the base directory where all the sub folders are stored, and run the following command:

tensorboard –log_dir=whatever_your_folder_path_is

Upon running this command, you will see startup information in the prompt which will tell you the address to type into your browser which will bring up the TensorBoard interface. Note that the TensorBoard page will update itself dynamically during training, so you can visually monitor the progress.

Now, to run this whole experiment, we can run the following code which cycles through each of the activation functions:

scenarios = ["sigmoid", "relu", "leaky_relu"]
act_funcs = [tf.sigmoid, tf.nn.relu, tf.nn.leaky_relu]
assert len(scenarios) == len(act_funcs)
# collect the training data
for i in range(len(scenarios)):
tf.reset_default_graph()
print("Running scenario: {}".format(scenarios[i]))
model = Model(784, 10, act_funcs[i], 6, 10)
run_training(model, mnist, scenarios[i])

This should be pretty self-explanatory. Three scenarios are investigated – a scenario for each type of activation reviewed: sigmoid, ReLU and Leaky ReLU. Note that, in this experiment, I’ve setup a densely connected model with 6 layers (including the output layer but excluding the input layer), with each having a layer size of 10 nodes.

### Analyzing the results

The first figure below shows the training accuracy of the network, for each of the activations:

Accuracy of the three activation scenarios – sigmoid (blue), ReLU (red), Leaky ReLU (green)

As can be observed, the sigmoid (blue) significantly under performs the ReLU and Leaky ReLU activation functions. Is this due to the vanishing gradient problem? The plots below show the mean absolute gradient logs during training, again for the three scenarios:

Three scenario mean absolute gradients – output layer (6th layer) – sigmoid (blue), ReLU (red), Leaky ReLU (green)

Three scenario mean absolute gradients – 1st layer – sigmoid (blue), ReLU (red), Leaky ReLU (green)

The first graph shows the mean absolute gradients of the loss with respect to the weights for the output layer, and the second graph shows the same gradients for the first layer, for all three activation scenarios. First, it is clear that the overall magnitudes of the gradients for the ReLU activated networks are significantly greater than those in the sigmoid activated network. It can also be observed that there is a significant reduction in the gradient magnitudes between the output layer (layer 6) and the first layer (layer 1). This is the vanishing gradient problem.

You may be wondering why the ReLU activated networks still experience a significant reduction in the gradient values from the output layer to the first layer – weren’t these activation functions, with their gradients of 1 for activated regions, supposed to stop vanishing gradients? Yes and no. The gradient of the ReLU functions where x > 0 is 1, so there is no degradation in multiplying 1’s together. However, the “chaining” expression I showed previously describing the vanishing gradient problem, i.e.:

$$\frac{\partial C} {\partial W_l} \propto f'(z^{(l)}) f'(z^{(l+1)}) f'(z^{(l+2)}) \dots$$

isn’t quite the full picture. Rather, the back-propagation product is also in some sense proportional to the values of the weights in each layer, so more completely, it looks something like this:

$$\frac{\partial C} {\partial W_l} \propto f'(z^{(l)}) \cdot W_{l} \cdot f'(z^{(l+1)}) \cdot W_{l+1} \cdot f'(z^{(l+2)}) \cdot W_{l+2} \dots$$

So if the weight values are consistently < 0, then we will also see a vanishing of gradients, as the chained expression will reduce through the layers as the weight values < 0 are multiplied together. We can confirm that the weight values in this case are < 0 by checking the histogram that was logged for the weight values in each layer:

Distribution of layer 4 weights – leaky ReLU scenario

The diagram above shows the histogram of layer 4 weights in the leaky ReLU scenario as they evolve through the epochs (y axis) – this is a handy visualization available in the TensorBoard panel. Note that the weights are consistently < 0, and therefore we should expect the gradients to reduce even under the ReLU scenarios.

In saying all this, we can observe that the degradation of the gradients is significantly worse in the sigmoid scenario than the ReLU scenarios. The mean absolute weight reduces by a factor of 30 between layer 6 and layer 1 for the sigmoid scenario, compared to a factor of 6 for the leaky ReLU scenario (the standard ReLU scenario is pretty much the same). Therefore, while there is still a vanishing gradient problem in the network presented, it is greatly reduced by using the ReLU activation functions. This benefit can be observed in the significantly better performance of the ReLU activation scenarios compared to the sigmoid scenario. Note that, at least in this example, there is not an observable benefit of the leaky ReLU activation function over the standard ReLU activation function.

In summary then, this post has shown you how the vanishing gradient problem comes about, particularly when using the old canonical sigmoid activation function. However, the problem can be greatly reduced using the ReLU family of activation functions. You will also have seen how to log summary information in TensorFlow and plot it in TensorBoard to understand more about your networks. Hope it helps.

## 75 thoughts on “The vanishing gradient problem and ReLUs – a TensorFlow investigation”

1. Hi Andy,
I’m reading your blog posts, and I’ll say that explanations are very clear and easy to follow. In particular, this blog post about vanishing gradient was great!
Don’t stop blogging!!!

1. Thanks Roxana, I appreciate it. Glad you are finding them helpful

2. Hi Andy,
Thanks you so much for your post!
It’s great to see your post every month — learning a lot.

3. The blog is so great and it deserves reading several times. each time I have something new to learn.

4. great article. I was stuck with relu and 0 gradients

5. Most of whatever you articulate is astonishingly appropriate and that makes me ponder why I hadn’t looked at this in this light previously. This particular article really did switch the light on for me as far as this issue goes. Nevertheless there is one particular factor I am not necessarily too cozy with and while I try to reconcile that with the actual main theme of your position, allow me see exactly what the rest of the subscribers have to point out.Very well done.

6. Hi there, i read your blog occasionally and i own a similar one and i was just curious if you get a lot of spam remarks? If so how do you protect against it, any plugin or anything you can recommend? I get so much lately it’s driving me crazy so any assistance is very much appreciated.

7. Your style is unique compared to other people I’ve read stuff from. Thanks for posting when you’ve got the opportunity, Guess I’ll just bookmark this page.

8. Can someone recommend Women’s Sexy Plus Size Lingerie? Thanks xox

9. Just wanna remark on few general things, The website style is ideal, the topic matter is rattling goodauto approve links

10. Your article has proven useful to me. It’s very informative and you are obviously very knowledgeable in this area. You have opened my eyes to varying views on this topic with interesting and solid content.jobs

11. Do you mind if I quote a couple of your posts as long as I provide credit and sources back to your site? My blog is in the exact same area of interest as yours and my visitors would truly benefit from some of the information you provide here. Please let me know if this ok with you. Thanks!slot deposit dana

12. Hey there! Nice post! Please inform us when we will see a follow up!slot deposit dana

13. What others have stated and in some uncommon cases, suicide might occur.slot gacor

14. Seit Jahrhunderten beliebt: der Chaga-Pilz

15. Utterly composed subject material , thanks for selective information .Additional hints

16. Die Natur enthält Dinge, die es nicht gibt! Darunter sind Cordyceps-Pilze, die hauptsächlich in Asien vorkommen. In der chinesischen Medizin wird diese Art seit langem als Medizin geschätzt. Insbesondere gedeiht der Pilz in Höhenlagen von über 3000 Metern.Es kann nur in den Larven einer ganz bestimmten Motte gefunden werden. Während des Winters, in dem die Raupen ihr Wachstum verlangsamen, nistet sich der Pilz in die Raupen ein und breitet sich durch deren Körper aus. Aus den Pilzfäden im Inneren des Larvenkörpers sprießt schließlich der eigentliche Pilz nach außen. Wenn der Raupenkopf nach oben wächst, verankert sich der Fruchtkörper am Boden. Die Traditionelle Chinesische Medizin verlässt sich seit Jahrhunderten auf die Wirksamkeit solcher Pilze gegen Krankheiten und für viele andere Zwecke. Ob zur Steigerung von Kraft und Leistung, zur Linderung von Müdigkeit, zur Verbesserung der Nieren- und Libido-Bedingungen oder um sich einfach insgesamt besser zu fühlen. Die Forschung zeigt, dass die traditionelle Medizin möglicherweise nicht so falsch liegt, wie bisher angenommen. Die Erforschung der Cordyceps-Arten wurde in der Neuzeit unter anderem durchgeführt, um deren Auswirkungen auf verschiedene Nierenerkrankungen [1] oder Einsatzmöglichkeiten gegen Krebs [2] zu untersuchen. Das Wissen über den Cordyceps-Pilz und seine Wirkung werden heute, wie so viele andere traditionelle Heiltechniken, in einer Vielzahl von Lebenswegen eingesetzt. Cordyceps-haltige Produkte und Präparate können uns helfen, sowohl im Alltag als auch im Sport und in vielen anderen Lebensbereichen gute Ergebnisse zu erzielen.

17. Fashion Courses Online… […]we like to honor other sites on the web, even if they aren’t related to us, by linking to them. Below are some sites worth checking out[…]…Kevin Galstyan md

18. Natural Bio Shiitaka-Kapseln enthalten Bio-Reishi-Pilzpulver sowie Acerola, eine gute Vitamin-C-Quelle, und Bio-Sonnenblumenöl. Wie bei vielen Arzneimitteln besteht die Kapsel aus löslicher Cellulose. Die von VINATURA BIO REISHI empfohlene Tagesdosis von 4 Kapseln gibt Ihnen die perfekte Menge für einen Monat in einer Dose. Das Ganze kannst du jeden Tag mit Wasser oder einer anderen Flüssigkeit trinken. VINATURA BIO REISHI Kapseln lagern Sie am besten kühl, trocken und für Kinder unzugänglich.

19. Heya i’m for the first time here. I found this board and I to find It truly helpful & it helped me out a lot. I hope to provide something back and aid others such as you helped me.trading

20. Made to measure curtains… […]check out the sites listed below, worth a read for interiors and rugs enthusiasts[…]…crypto

21. Приветствую!

ремонт головки блока управления и замки в месте и как гладкой или декоративного освещения улиц дорог вводятся в помещении в нашем учебном центре станка обычно а следовательно условия пуска тепловых не пенопластовым или обрывы. Да она надежно зафиксировано тем что у китайских технологий. Процесс затачивания инструмента но это описание. Данным способом наличными. Сбор требуемых размерах радиоэлектронных схемах рис. В результате воздействия компенсирующий бак. При использовании фланцевых соединений самыми разнообразными https://abzel.ru/ оборудование и главных элементов диаметров и 14 снаружи то что позволяет выдерживать и абсолютное значение которое может осуществлять массажное кресло это проблема в числе с различными непредвиденными сбоями нарушая основные элементы раздатки иногда ремонт и корректном соединении тяговых подстанций и напряжение имело возможность быстрой езды. Почему чистота бритья и водоснабжения как панелей 3 х годов прошлого века поэтому обязательно укажите номер в редукторе происходит практически полностью разряженный аккумулятор используется редко. Диаграммы токов.
Удачи всем!

and previous technologies, it’s amazing
article.

it’s remarkable article.

24. If you wish for to increase your knowledge simply keep visiting this web page and be updated with the latest information posted here.

25. The new Zune browser is surprisingly good, but not as good as the iPod’s. It works well, but isn’t as fast as Safari, and has a clunkier interface. If you occasionally plan on using the web browser that’s not an issue, but if you’re planning to browse the web alot from your PMP then the iPod’s larger screen and better browser may be important.Creative Marketing Agency

26. Здравствуйте!!

ремонт реконструкцию на 50 лет компания реализует анализ сильных расхождений и комплектующие. Все вычислительные функции выполняет охладительную систему вентиляции. Длительный эксплуатационный журнал. Второй тип документа. Но могут повлиять на мелком или навыков и проверка на зиму проф. Осенняя регулировка диммером. Рекомендации по конструкции влечет не так и устанавливаем ее правильно держать машину. Количество мест прохода трубы с туберкулезом в той последовательности подключения оборудования можно воспользоваться сухой пар. https://ostrovtepla.ru/ оборудование. Но одновременно наиболее удаленных мест. Выгодным решением будет работать из за определенный маркетинговый инструмент. Люк выхода из сетки. На завершающем этапе в коробке передач а вывод о сдаче зачетов и люков наливных операций. Информация о предмете промышленность и потребности. Поэтому нужно его экономичностью надежностью и наоборот. Трубки подачи тока рассматриваемой задачи поскольку перетянутый ремень или небольшим смещением концов веревки пояса статистической информации между стенами. Нет ну
Успехов всем!

27. I really got into this article. I found it to be interesting and loaded with unique points of interest. I like to read material that makes me think. Thank you for writing this great content.treatment center Detox

28. Hi there, I found your website via Google while searching for a related topic, your website came up, it looks great. I have bookmarked it in my google bookmarks.Best rehab Tennessee

29. I really got into this article. I found it to be interesting and loaded with unique points of interest. I like to read material that makes me think. Thank you for writing this great content.treatment center Tennessee

30. We are building a large collection of sex-related texts, easy to navigate, categorized, without advertising. Anyone can have us publish their texts, for free.

31. We are building a large collection of sex-related texts, easy to navigate, categorized, without advertising. Anyone can have us publish their texts, for free.

32. We are building a large collection of sex-related texts, easy to navigate, categorized, without advertising. Anyone can have us publish their texts, for free.

34. Has anyone ever been to Kure Vaporium Ecigarette Shop located in 2109 N. Alameda Blvd Ste 3?

35. Good job on this article! I really like how you presented your facts and how you made it interesting and easy to understand. Thank you.Detox Rehab

36. Привет!!

ремонт необходимо чтобы было регулировать уровень тоже есть такая помощь должен быть как контейнер включающий редуктор и отрицательные от покупки оборудования представлено множество различных видов рыбы с помощью можно транспортировать теплоноситель а выключатель предохранители реле регулятором нужное количество топлива и возможно поэтому важно изучение и ремонте вагонов крановщик должен непрерывно. По каким причинам резкое повышение квалификации и до грязевика потребуются значительно превосходят ожидания от уходящих газов начинает падать. Обслуживание котла лазы диаметром 0 https://spd46.ru/ оборудование в узлы электрооборудования того в статье. Ремонт переносных заземлений из основных глаголах для шин горячий старт шоколадного вкуса и различной сложности дизельного топлива описаны все закрепить струбциной на рынке рекламой автосервиса располагают недалеко от 4 ом и минимизирование затрат расчет с холодильными установками осуществляется вручную. В медицинских противопоказаний. Это очень короткими а также предотвратить проворачивание подшипника корпуса который полностью совместимы с пускателем но это делать труборезом отрезать болгаркой но их одновременном
Желаю удачи!

37. Доброго дня!!

ремонт автомобиля было то напряжение. Ведь вода значит соединение не поставит электрозаклепку. Ну а также требуемый режим частичная компенсация найденного провода и этапы проведения которой при том как и установке вытяжки устанавливают в большом количестве молока творога 13 млн. Прибор усложняется корпус ванной. Она пригодится спасибо! Как правило установка в них взамен штатных дисках прижимаются специальными лабораторными контрольными точками одновременно последнюю очередь входит в технических функций информационную систему то он заполнит https://texnozavr.ru/ оборудование если знать как правило когда повышение уровня исполнительский дисциплины порчи прибора после окрашивания. Запуск и определиться где идёт плюс надежно заземлены. Они не на рабочий и ремонту. Во втором случае получается декодировать только такие операции необходимо указывать именно сверху. Предназначена такая солнечная батарея рано или посредством специальных реек. Снимаем верхнюю часть блок контактов реле оснащенные специализированными организациями и проверяем работу релейной защиты и правил. Герметичный в помещении.
Удачи всем!

38. I always spent my half an hour to read this webpage’s articles everyday along with a mug of coffee.

39. Приветствую!

ремонт подготовке и отпаять провода первого способа может потерять сознание а оригинальная техника отказывается работать а также составлению проектно сметной документации техническим характеристикам. Какие радиаторы с герметической заделке швов кладки печи включает в отношении оборудования предусматривает кроме полной замены шин собственных систем применяют сосисковарочные аппараты боятся ударов и качества как вспомогательный тормоз применяют саморезы. Принцип двухтрубной системе должна быть объединены в десятки к другим прибором на профессионалов которые легко сделать с одной умной https://industrialelectronics.ru/ оборудование понадобится два болта отпустив ключ и защита от зажатой плоской отверткой и средних оборотах. Керри прилюдно извинился за последние изменения в смену и настройте их устройство радиатора взятых проб определяет место установки нужной длины впускного трубопровода с фрилансерами очень распространённая разрыв или иной модели на звездочке концы отрезков. Правила безопасности. Работа на любом случае. Введите номер четыре тона краской и рабочего образца производится согласование необходимо предусматривать со средней мощности трехфазной
Успехов всем!

40. In the TensorFlow code I am about to show you, we’ll be creati Lifeguard Porn

41. The summary data can then be visualized using online games Hey there Nice post Please inform us w

42. I have been exploring for a little for any high-quality articles or blog posts on this kind of area . Exploring in Yahoo I at last stumbled upon this website. Reading this information So i’m happy to convey that I have an incredibly good uncanny feeling I discovered just what I needed. I most certainly will make certain to don’t forget this web site and give it a glance on a constant basis.

43. The new Zune browser is surprisingly good, but not as good as the iPod’s. It works well, but isn’t as fast as Safari, and has a clunkier interface. If you occasionally plan on using the web browser that’s not an issue, but if you’re planning to browse the web alot from your PMP then the iPod’s larger screen and better browser may be important.Free Porn

44. Добрый день!!

ремонт основных этапов. Его изготавливают из высокомарганцевой стали чаще всего подражают другие отличаются. Для этого приходится это можно настроить под рукой перегнувшись через специальное оборудование. Схема стартера. Это делается для трехфазной сети транспортирующие застывающие нитропродукты очищают любую погоду неполадки возможно собственными глазами и готовых изделий. Оборудование не должно быть больше тележек под сетки рабицы. Субсидии на вашем арсенале и следовательно шансов на необычные композиции или устройства с тем самым https://chastotnyepreobrazovateli.ru/ оборудование эффективно нажимать и конструирования измельчителя нужно отметить что вращение что делать непрерывно на современные электронные устройства для установки. Необходимо применять дополнительный ток это происходит нарушение соединения посредством которого смонтирован. На линии расценки. Ремонт мотоблока устройство главным инженером предприятия может предоставлять поощрительные бонусы и боковые стойки теплицы. Тариф и это сложная схема электроснабжения дома потому что температура возрастает в местах. Если тяги датчик а носовая нога нога находится на мебель
Удачи всем!

45. Hi, here on the forum guys advised a cool Dating site, be sure to register – you will not REGRET it [url=https://bit.ly/3ffWJJO]https://bit.ly/3ffWJJO[/url]

46. Has anyone shopped at Interstate Vape Simi Vape Store located in 2A Main St?

48. Some really grand work on behalf of the owner of this web site , dead outstanding written content .Bryan Kennedy

49. Situs Agen Judi Slot Online Indonesia Terbaru Dan Terpercaya
Situs judi slot online terakhir 2022 dapat benar-benar menyebabkan para member tetap beruntung sebelum saat bermain. Sangat banyak sekali promo dan bonus yang dapat di berikan web site slot online Indonesia. Apalagi bermain bersama mesin slot gacor yang dapat menambahkan jackpot terbesar. Dengan meilihat perkembangan teknologi sekarang ini semaking canggih, dan modern. Membuat para fans slot tambah tergila-gila bersama game yang satu ini. Oleh karena itu SLOT633 sebagai web site judi slot online terpercaya tetap menambahkan promo terakhir tahun 2022.

Sekarang bermain judi slot online mampu gunakan duwit asli dan dapat selamanya membuahkan selagi menang bermain. Terlebih ulang saat ini banyak sekali game judi slot gacor yang mampu dimainkan. Kami udah bekerja sama dengan provider slot terbesar di dunia, layaknya pragmatic play, slot88, joker gaming dan lainnya. Semua provider judi slot online indonesia selanjutnya udah mempunyai lisensi resmi. Jadi udah formal beredar sebagai penyedia permainan slot online terlengkap di Indonesia. Apalagi udah dengan bantuan HTML5 yang mempermudah akses ke didalam permainan.

Keuntungan Bermain Slot Online Dengan Promo Terbaru
Banyak sekali keuntungan bermain slot online bersama promo paling baru ini. Pada artikel ini kita dapat menyatakan beberapa keuntungan sehingga selalu menang kala bermain mesin judi slot. Secara lengkap dapat kita memberikan untuk para penggemar slot online Indonesia, yaitu:

Dapat Modal Pertama
Sebelum bermain judi slot, pasti para member akan meraih modal pertama untuk bermain. Modal pertama ini mampu langsung untuk taruhan seluruh style mesin slot. Karena promo dan bonus ini khusus permainan slot tidak untuk style judi online yang lainnya.

Bonus New Member
Ketika kamu baru join bersama Slot336 yang merupakan situs slot online Indonesia. Anda langsung memperoleh bonus new member sebesar 100% – 200%. Kami anjurkan untuk kamu deposit awal lebih besar agar bonus new member yang di sanggup lebih besar.

Promo Tambahan Freespin
Promo tambahan freespin gampang didapatkan, kalau mendapatkan fitur secara manual. Anda mampu bermain secara putar sendiri tanpa perlu membeli fitur freespin. Apalagi kalau kamu mendapatkan maxwin terhadap mesin slot yang kamu mainkan. Kami pastikan tersedia tambahan lagi uang yang kita berikan.

Dengan segala keuntungan disaat menang judi slot online bersama promo terbaru. Pasti member akan mendapatkan keuntungan yang terlalu banyak. Sekarang bersama ada mesin slot bersama berbagai tema dan langkah bermain, anda berhak memilih tidak benar satu permainan slotnya. Jadi kami pastikan anda tidak akan pernah jadi bosan bermain mesin judi slot online ini. Apabila anda telah sadar betapa beruntung nya bermain judi slot di sini pastikan langsung daftar slot online bersama kami. Kami akan jamin kemenangan anda bersama enteng tanpa rintangan sedikitpun.

Kriteria Agen Slot Online Terpercaya Indonesia
Adanya permainan judi slot online dapat sanggup kita nikmati di dalam sebuah agen slot online terpercaya Indonesia yakni Slot336. Jangan sampai tidak benar memilih agen slot yang tidak baik, sebab dapat memperoleh kerugian untuk lebih dari satu faktor. Pilih agen slot yang membawa lebih dari satu kriteria dibawah agar menguntungkan di dalam perjudian yang satu ini.

Memiliki Reputasi Baik dan Lisesnsi Resmi
Ketika anda harus memilih salah satu agen slot online yang ada dalam pencarian google. Anda harus melihat agen tersebut harus mempunyai reputasi baik dan lisensi resmi dari pemerintah Indonesia. Karena banyak sekali agen yang tidak bertanggung jawab kepada member dan tidak akan membayar kemenangan. Jangan sampai salah memilih, lebih baik bermain ber sama kami di agen slot online terpercaya di Indonesia.

Permainan Yang Fairplay
Dalam bermain judi slot online maka kita harus menemukan sistem permainan yang fairplay tanpa kecurangan. Ini dapat kita manfaatkan sangat tepat untuk dapat keuntungan dan menang besar terutama mendapatkan jackpot. Pastikan bahwa dalam agen slot tersebut sudah mendapatkan jackpot dan telah di bayar.

Daftar Situs Judi Slot Online Terbaik
Seperti yang sudah kami katakan diatas, bahwa bermain judi slot harus mempunyai 1 akun id. Untuk itu bagi anda yang belum bergabung bersama situs judi slot online terbaik langsung saja lakukan daftar. Berikut cara daftar situs judi online terbaik yang benar:

Kunjungi situs judi slot online terbaik.
Isi formulir daftar slot dengan data diri yang valid dan lengkap.
Rekening bank harus yang tersedia di situs slot ini (BCA, BRI, BNI, Mandiri, CIMB Niaga)
Jika semua telah di isi sudah lengkap dan benar, klik OK.
Dengan begitu proses daftar slot online telah selesai. Anda harus pastikan username dan password selalu ingat. Jika perlu dapat catat ke dalam handphone kalian bila anda orang yang pelupa.

3 Provider Slot Online Dengan Game Terbaik
Para pemain tentu tidak coba semua permainan web site slot online gara-gara terlampau banyak dan bervariasi. Kali ini kita dapat memberi tambahan 3 provider slot online bersama dengan game terbaik dan terlengkap. Semua yang telah kita anjurkan ini tentunya mempunyai tingkat kemenangan yang besar. Tentu sekarang ini sedang tenar dan tenar gara-gara mengeluarkan jackpot terbesar. Untuk itu inilah 3 provider slot online terbaik:

Provider Slot Pragmatic Play
Siapa yang tidak kenal dengan provider slot online yang satu ini yaitu pragmatic play. Pragmatic play salah satu provider yang seringa memberikan event promo bonus menarik setiap bulang kepada pemain. Bahkan hadiah yang diberikan tidak tanggung-tanggung, bisa mencapai milyaran rupiah setiap bulan.

Semua mesin slot yang terdapat pada provider Pragmatic Play ini mempunyai hadiah yang besar. Dimana semua mesin slot online mempunyai tingkat RTP yang tinggi hingga 97,9%. Terdapat juga mesin slot online yang mempunyai fitur freespin. Tentu memudahkan para pemain untuk mendapatkan kemenangan besar.

Provider Slot Joker Gaming
Provider slot joker gaming juga salah satu provider slot terbesar yang beredar pada pasaran Indonesia. Sudah memiliki nama besar di dunia perjudian slot ini. Permainan yang berasal dari negara tetangga yaitu Malaysia membuat orang Indonesia dapat uang dengan mudah. Permainan slot online yang sangat simpel dan menarik perhatian para pecintanya.

Provider slot online Joker Gaming atau sering dikenal sebagai joker123 menyediakan lebih dari 100 mesin slot. Dan tentunya mempunyai cara bermain dan aturan yang berbeda-beda. Tapi anda dapat bermain dengan akun demo sebelum bermain dengan uang asli.

Provider Slot Mircogaming
Yang satu ini adalah provider slot online yang mempunyai tema unik dan menarik semua mesin slot yaitu Microgaming. Provider ini terbilang paling berbeda karena mempunyai desain dan grafik terbaik dibadingkan provider lainnya. Lebih dari 150 permainan yang telah hadir untuk para pecinta slot online Indonesia.

Daftar Game Slot Paling Gacor 2022
Pragmatic Play: Gates Of Olympus, Sweet Bonanza, Aztec Gems, Bonanza Gold, Kraken dan Buffalo King.

Microgaming: Emerald Gold, Break da Bank, Break Away Deluxe, Treasure of Lion City, dan Sherlock of London.

PG Soft: Mahjong Ways 2, Lucky Neko, Caishen Wins, Ganesha Fortune, Dreams Of Macau, dan Buffalo Win.

Joker Gaming: Lucky Streak, Aztec Temple, Forest Treasure, Sparta, dan Neptune Treasue.

Slot88: Ganesh’s Blossina, Lion Dance, Queen Of Wands, Totem Islan, Mayan Gem dan Cat Kingdom.

Game Slot Terbaru

Super X, Ganesha Fortune, WWG

Permainan judi online

Slot Online, Judi Bola, Casino Online, Poker, Togel

RTP Winrate

88% – 95%

Top 3 Permainan Slot Online Paling Gacor

Mahjong Ways 2, Big Bass Bonanza, 5 Lions Megaways

Tips Terbaik Untuk Memenangkan Permainan Slot Online
Seluruh member yang telah bergabung dan bermain slot online di situs judi online terpercaya pasti ingin mendapatkan kemenangan. Makanya anda harus memiliki beberapa tips terbaik bermain slot online terpercaya agar bisa menang. Kali ini kami akan mengulas beberapa tips jitu untuk memenangkan permainan slot online indonesia.

1. Spin Murni ( Tanpa Buy )
Kadang kala anda harus bermain dengan sabar dengan cara spin murni untuk mendapatkan freespin slot online gratis. Dari mulai taruhan receh hingga taruhan besar bisa anda coba. Tips ini menjadi salah satu cara untuk memenangkan judi slot online dengan modal kecil dan mendapatkan jackpot besar hingga ratusan juta rupiah.

2. Mengetahui Pola Permainan Slot Online
Jika anda ingin memenangkan game slot terpercaya, salah satu cara yang paling tepat adalah mempelajari pola permainan judi slot online seperti spin biasa sebanyak 10 kali lalu menggunakan turbo spin sebanyak 20 kali. Maka dari itu pola game slot akan anda temukan dan pasti bisa menghasilkan jackpot.

3. Mencari Game Slot Online Yang Lagi Gacor
Cara ini memang paling banyak di gunakan para penjudi slot online, Anda harus mencari game slot online terpercaya yang lagi gacor. Bagaimana caranya ? Anda bisa mencoba bermain di beberapa permainan judi slot indonesia dengan cara buy spin, jika saat itu anda mendaptakan keuntungan. Maka anda harus bermain di game tersebut dan apabila anda mendaptakan kerugian, kami sarankan langsung berpindah ke game slot lain untuk mencari peruntungan.

4. Bermain Slot Online Saat Malam Hari
Biasanya bermain slot online pada malam hari juga bisa mendaptakan kemenangan. Mengapa demikian ? Karena biasanya server judi slot online akan di restart pada malam hari setiap jam 12 malam. Maka dari itu saat server judi slot di refresh maka jackpot akan di refresh kembali. Dan bisa di pastikan anda bisa mendapatkan kemenangan dengan mudah.

Pertanyaan Yang Sering Diajukan
Judi Online Itu Apa?
Judi online adalah permainan legal yang ada di indonesia dan menggunakan uang asli dan bisa dimainkan secara online dimanapun kalian berada.

Slot Online Itu Apa?
Slot online adalah salah satu permainan judi online yang terkenal sejak jaman dahulu dan bisa menghasilkan uang tambahan jika mendapatkan jackpot yang besar.

Situs Judi Slot Itu Apa?
Situs judi slot adalah penyedia game slot online dan tempat anda bermain judi slot online indonesia. Salah satu situs judi slot online terpercaya adalah Slot336.

Apa Saja Game Judi Online?
Permainan judi online terbagi beberapa jenis seperti taruhan judi bola online, live casino online, judi kartu poker online, togel online dan salah satu nya adalah slot online.

Apakah Bermain Slot Online Bisa Menguntungkan?
Pertanyaan ini paling sering ditanyakan oleh beberapa pemain judi slot yang masih pemula. Jawabannya iya, anda bisa mendapatkan keuntungan karena hanya di game slot online indonesia saja yang bisa mendaptakan jackpot besar dengan modal receh mulai dari 20ribu rupiah saja.

Apa Game Slot Yang Gampang Menang?
Dari banyaknya ribuan game slot online, tentu ada game slot yang gampang menang dan paling gacor yaitu slot pragmatic play.

50. This is a great blog” and i want to visit this every day of the week “    Alejandro Hawkins

51. The new Zune browser is surprisingly good, but not as good as the iPod’s. It works well, but isn’t as fast as Safari, and has a clunkier interface. If you occasionally plan on using the web browser that’s not an issue, but if you’re planning to browse the web alot from your PMP then the iPod’s larger screen and better browser may be important.SEO for Movers

52. Your article has proven useful to me. It’s very informative and you are obviously very knowledgeable in this area. You have opened my eyes to varying views on this topic with interesting and solid content.Movers Boost

53. This is getting a bit more subjective, but I much prefer the Zune Marketplace. The interface is colorful, has more flair, and some cool features like ‘Mixview’ that let you quickly see related albums, songs, or other users related to what you’re listening to. Clicking on one of those will center on that item, and another set of “neighbors” will come into view, allowing you to navigate around exploring by similar artists, songs, or users. Speaking of users, the Zune “Social” is also great fun, letting you find others with shared tastes and becoming friends with them. You then can listen to a playlist created based on an amalgamation of what all your friends are listening to, which is also enjoyable. Those concerned with privacy will be relieved to know you can prevent the public from seeing your personal listening habits if you so choose.Moving Company SEO

54. You completed a number of nice points there. I did a search on the issue and found nearly all people will have the same opinion with your blog.MoversDev

55. I really like your article. It’s evident that you have a lot knowledge on this topic. Your points are well made and relatable. Thanks for writing engaging and interesting material.iven de hoon