If you’ve read about how neural networks are trained, you’ve almost certainly come across the term “gradient descent” before. Gradient descent is the primary method of optimizing a neural network’s performance, reducing the network’s loss/error rate. However, gradient descent can be a little hard to understand for those new to machine learning, and this article will endeavor to give you a decent intuition for how gradient descent operates.
Gradient descent is an optimization algorithm. It’s used to improve the performance of a neural network by making tweaks to the parameters of the network such that the difference between the network’s predictions and the actual/expected values of the network (referred to as the loss) is a small as possible. Gradient descent takes the initial values of the parameters and uses operations based in calculus to adjust their values towards the values that will make the network as accurate as it can be. You don’t need to know a lot of calculus to understand how gradient descent works, but you do need to have an understanding of gradients.
What Are Gradients?
Assume that there is a graph that represents the amount of error a neural network makes. The bottom of the graph represents the points of lowest error while the top of the graph is where the error is the highest. We want to move from the top of the graph down to the bottom. A gradient is just a way of quantifying the relationship between error and the weights of the neural network. The relationship between these two things can be graphed as a slope, with incorrect weights producing more error. The steepness of the slope/gradient represents how fast the model is learning.
A steeper slope means large reductions in error are being made and the model is learning fast, whereas if the slope is zero the model is on a plateau and isn’t learning. We can move down the slope towards less error by calculating a gradient, a direction of movement (change in the parameters of the network) for our model.
Let’s shift the metaphor just slightly and imagine a series of hills and valleys. We want to get to the bottom of the hill and find the part of the valley that represents the lowest loss. When we start at the top of the hill we can take large steps down the hill and be confident that we are heading towards the lowest point in the valley.
However, as we get closer to the lowest point in the valley, our steps will need to become smaller, or else we could overshoot the true lowest point. Similarly, it’s possible that when adjusting the weights of the network, the adjustments can actually take it further away from the point of lowest loss, and therefore the adjustments must get smaller over time. In the context of descending a hill towards a point of lowest loss, the gradient is a vector/instructions detailing the path we should take and how large our steps should be.
Now we know that gradients are instructions that tell us which direction to move in (which coefficients should be updated) and how large the steps we should take are (how much the coefficients should be updated), we can explore how the gradient is calculated.
Calculating Gradients and Gradient Descent Procedure
In order to carry out gradient descent, the gradients must first be calculated. In order to calculate the gradient, we need to know the loss/cost function. We’ll use the cost function to determine the derivative. In calculus, the derivative just refers to the slope of a function at a given point, so we’re basically just calculating the slope of the hill based on the loss function. We determine the loss by running the coefficients through the loss function. If we represent the loss function as “f”, then we can state that the equation for calculating the loss is as follows (we’re just running the coefficients through our chosen cost function):
Loss = f(coefficient)
We then calculate the derivative, or determine the slope. Getting the derivative of the loss will tell us which direction is up or down the slope, by giving us the appropriate sign to adjust our coefficients by. We’ll represent the appropriate direction as “delta”.
delta = derivative_function(loss)
We’ve now determined which direction is downhill towards the point of lowest loss. This means we can update the coefficients in the neural network parameters and hopefully reduce the loss. We’ll update the coefficients based on the previous coefficients minus the appropriate change in value as determined by the direction (delta) and an argument that controls the magnitude of change (the size of our step). The argument that controls the size of the update is called the “learning rate” and we’ll represent it as “alpha”.
coefficient = coefficient – (alpha * delta)
We then just repeat this process until the network has converged around the point of lowest loss, which should be near zero.
It’s very important to choose the right value for the learning rate (alpha). The chosen learning rate must be neither too small or too large. Remember that as we approach the point of lowest loss our steps must become smaller or else we will overshoot the true point of lowest loss and end up on the other side. The point of smallest loss is small and if our rate of change is too large the error can end up increasing again. If the step sizes are too large the network’s performance will continue to bounce around the point of lowest loss, overshooting it on one side and then the other. If this happens the network will never converge on the true optimal weight configuration.
In contrast, if the learning rate is too small the network can potentially take an extraordinarily long time to converge on the optimal weights.
Types Of Gradient Descent
Now that we understand how gradient descent works in general, let’s take a look at some of the different types of gradient descent.
Batch Gradient Descent: This form of gradient descent runs through all the training samples before updating the coefficients. This type of gradient descent is likely to be the most computationally efficient form of gradient descent, as the weights are only updated once the entire batch has been processed, meaning there are fewer updates total. However, if the dataset contains a large number of training examples, then batch gradient descent can make training take a long time.
Stochastic Gradient Descent: In Stochastic Gradient Descent only a single training example is processed for every iteration of gradient descent and parameter updating. This occurs for every training example. Because only one training example is processed before the parameters are updated, it tends to converge faster than Batch Gradient Descent, as updates are made sooner. However, because the process must be carried out on every item in the training set, it can take quite a long time to complete if the dataset is large, and so use of one of the other gradient descent types if preferred.
Mini-Batch Gradient Descent: Mini-Batch Gradient Descent operates by splitting the entire training dataset up into subsections. It creates smaller mini-batches that are run through the network, and when the mini-batch has been used to calculate the error the coefficients are updated. Mini-batch Gradient Descent strikes a middle ground between Stochastic Gradient Descent and Batch Gradient Descent. The model is updated more frequently than in the case of Batch Gradient Descent, which means a slightly faster and more robust convergence on the model’s optimal parameters. It’s also more computationally efficient than Stochastic Gradient Descent
What is Bayes Theorem?
If you’ve been learning about data science or machine learning, there’s a good chance you’ve heard the term “Bayes Theorem” before, or a “Bayes classifier”. These concepts can be somewhat confusing, especially if you aren’t used to thinking of probability from a traditional, frequentist statistics perspective. This article will attempt to explain the principles behind Bayes Theorem and how it’s used in machine learning.
Defining Bayes Theorem
Bayes Theorem is a method of calculating conditional probability. The traditional method of calculating conditional probability (the probability that one event occurs given the occurrence of a different event) is to use the conditional probability formula, calculating the joint probability of event one and event two occurring at the same time, and then dividing it by the probability of event two occurring. However, conditional probability can also be calculated in a slightly different fashion by using Bayes Theorem.
When calculating conditional probability with Bayes theorem, you use the following steps:
- Determine the probability of condition B being true, assuming that condition A is true.
- Determine the probability of event A being true.
- Multiply the two probabilities together.
- Divide by the probability of event B occurring.
This means that the formula for Bayes Theorem could be expressed like this:
P(A|B) = P(B|A)*P(A) / P(B)
Calculating the conditional probability like this is especially useful when the reverse conditional probability can be easily calculated, or when calculating the joint probability would be too challenging.
A Practical Example
This might be easier to interpret if we spend some time looking at an example of how you would apply Bayesian reasoning and Bayes Theorem. Let’s assume you were playing a simple game where multiple participants tell you a story and you have to determine which one of the participants is lying to you. Let’s fill in the equation for Bayes Theorem with the variables in this hypothetical scenario.
We’re trying to predict whether each individual in the game is lying or telling the truth, so if there are three players apart from you, the categorical variables can be expressed as A1, A2, and A3. The evidence for their lies/truth is their behavior. Like when playing poker, you would look for certain “tells” that a person is lying and use those as bits of information to inform your guess. Or if you were allowed to question them it would be any evidence their story doesn’t add up. We can represent the evidence that a person is lying as B.
To be clear, we’re aiming to predict Probability(A is lying/telling the truth|given the evidence of their behavior). To do this we’d want to figure out the probability of B given A, or the probability that their behavior would occur given the person genuinely lying or telling the truth. You’re trying to determine under which conditions the behavior you are seeing would make the most sense. If there are three behaviors you are witnessing, you would do the calculation for each behavior. For example, P(B1, B2, B3 * A). You would then do this for every occurrence of A/for every person in the game aside from yourself. That’s this part of the equation above:
P(B1, B2, B3,|A) * P|A
Finally, we just divide that by the probability of B.
If we received any evidence about the actual probabilities in this equation, we would recreate our probability model, taking the new evidence into account. This is called updating your priors, as you update your assumptions about the prior probability of the observed events occurring.
Machine Learning Applications
The most common use of Bayes theorem when it comes to machine learning is in the form of the Naive Bayes algorithm.
Naive Bayes is used for the classification of both binary and multi-class datasets, Naive Bayes gets its name because the values assigned to the witnesses evidence/attributes – Bs in P(B1, B2, B3 * A) – are assumed to be independent of one another. It’s assumed that these attributes don’t impact each other in order to simplify the model and make calculations possible, instead of attempting the complex task of calculating the relationships between each of the attributes. Despite this simplified model, Naive Bayes tends to perform quite well as a classification algorithm, even when this assumption probably isn’t true (which is most of the time).
There are also commonly used variants of the Naive Bayes classifier such as Multinomial Naive Bayes, Bernoulli Naive Bayes, and Gaussian Naive Bayes.
Multinomial Naive Bayes algorithms are often used to classify documents, as it is effective at interpreting the frequency of words within a document.
Bernoulli Naive Bayes operates similarly to Multinomial Naive Bayes, but the predictions rendered by the algorithm are booleans. This means that when predicting a class the values will be binary, no or yes. In the domain of text classification, a Bernoulli Naive Bayes algorithm would assign the parameters a yes or no based on whether or not a word is found within the text document.
If the value of the predictors/features aren’t discrete but are instead continuous, Gaussian Naive Bayes can be used. It’s assumed that the values the continuous features have been sampled from a gaussian distribution.
What are RNNs and LSTMs in Deep Learning?
Many of the most impressive advances in natural language processing and AI chatbots are driven by Recurrent Neural Networks (RNNs) and Long Short-Term Memory (LSTM) networks. RNNs and LSTMs are special neural network architectures that are able to process sequential data, data where chronological ordering matters. LSTMs are essentially improved versions of RNNs, capable of interpreting longer sequences of data. Let’s take a look at how RNNs and LSTMS are structured and how they enable the creation of sophisticated natural language processing systems.
Feed-Forward Neural Networks
So before we talk about how Long Short-Term Memory (LSTM) and Convolutional Neural Networks (CNN) work, we should discuss the format of a neural network in general.
A neural network is intended to examine data and learn relevant patterns, so that these patterns can be applied to other data and new data can be classified. Neural networks are divided into three sections: an input layer, a hidden layer (or multiple hidden layers), and an output layer.
The input layer is what takes in the data into the neural network, while the hidden layers are what learn the patterns in the data. The hidden layers in the dataset are connected to the input and output layers by “weights” and “biases” which are just assumptions of how the data points are related to each other. These weights are adjusted during training. As the network trains, the model’s guesses about the training data (the output values) are compared against the actual training labels. During the course of training, the network should (hopefully) get more accurate at predicting relationships between data points, so it can accurately classify new data points. Deep neural networks are networks that have more layers in the middle/more hidden layers. The more hidden layers and more neurons/nodes the model has, the better the model can recognize patterns in the data.
Regular, feed-forward neural networks, like the ones I’ve described above are often called “dense neural networks”. These dense neural networks are combined with different network architectures that specialize in interpreting different kinds of data.
Recurrent Neural Networks
Recurrent Neural Networks take the general principle of feed-forward neural networks and enable them to handle sequential data by giving the model an internal memory. The “Recurrent” portion of the RNN name comes from the fact that the input and outputs loop. Once the output of the network is produced, the output is copied and returned to the network as input. When making a decision, not only the current input and output are analyzed, but the previous input is also considered. To put that another way, if the initial input for the network is X and the output is H, both H and X1 (the next input in the data sequence) are fed into the network for the next round of learning. In this way, the context of the data (the previous inputs) is preserved as the network trains.
The result of this architecture is that RNNs are capable fo handling sequential data. However, RNNs suffer from a couple of issues. RNNs suffer from the vanishing gradient and exploding gradient problems.
The length of sequences that an RNN can interpret are rather limited, especially in comparison to LSTMs.
Long Short-Term Memory Networks
Long Short-Term Memory networks can be considered extensions of RNNs, once more applying the concept of preserving the context of inputs. However, LSTMs have been modified in several important ways that allow them to interpret past data with superior methods. The alterations made to LSTMs deal with the vanishing gradient problem and enable LSTMs to consider much longer input sequences.
LSTM models are made up of three different components, or gates. There’s an input gate, an output gate, and a forget gate. Much like RNNs, LSTMs take inputs from the previous timestep into account when modifying the model’s memory and input weights. The input gate makes decisions about which values are important and should be let through the model. A sigmoid function is used in the input gate, which makes determinations about which values to pass on through the recurrent network. Zero drops the value, while 1 preserves it. A TanH function is used here as well, which decides how important to the model the input values are, ranging from -1 to 1.
After the current inputs and memory state are accounted for, the output gate decides which values to push to the next time step. In the output gate, the values are analyzed and assigned an importance ranging from -1 to 1. This regulates the data before it is carried on to the next time-step calculation. Finally, the job of the forget gate is to drop information that the model deems unnecessary to make a decision about the nature of the input values. The forget gate uses a sigmoid function on the values, outputting numbers between 0 (forget this) and 1 (keep this).
An LSTM neural network is made out of both special LSTM layers that can interpret sequential word data and the densely connected like those described above. Once the data moves through the LSTM layers, it proceeds into the densely connected layers.
What is K-Nearest Neighbors?
K-Nearest Neighbors is a machine learning technique and algorithm that can be used for both regression and classification tasks. K-Nearest Neighbors examines the labels of a chosen number of data points surrounding a target data point, in order to make a prediction about the class that the data point falls into. K-Nearest Neighbors (KNN) is a conceptually simple yet very powerful algorithm, and for those reasons, it’s one of the most popular machine learning algorithms. Let’s take a deep dive into the KNN algorithm and see exactly how it works. Having a good understanding of how KNN operates will let you appreciated the best and worst use cases for KNN.
An Overview Of KNN
Let’s visualize a dataset on a 2D plane. Picture a bunch of data points on a graph, spread out along the graph in small clusters. KNN examines the distribution of the data points and, depending on the arguments given to the model, it separates the data points into groups. These groups are then assigned a label. The primary assumption that a KNN model makes is that data points/instances which exist in close proximity to each other are highly similar, while if a data point is far away from another group it’s dissimilar to those data points.
A KNN model calculates similarity using the distance between two points on a graph. The greater the distance between the points, the less similar they are. There are multiple ways of calculating the distance between points, but the most common distance metric is just Euclidean distance (the distance between two points in a straight line).
KNN is a supervised learning algorithm, meaning that the examples in the dataset must have labels assigned to them/their classes must be known. There are two other important things to know about KNN. First, KNN is a non-parametric algorithm. This means that no assumptions about the dataset are made when the model is used. Rather, the model is constructed entirely from the provided data. Second, there is no splitting of the dataset into training and test sets when using KNN. KNN makes no generalizations between a training and testing set, so all the training data is also used when the model is asked to make predictions.
How The KNN Algorithm Operates
A KNN algorithm goes through three main phases as it is carried out:
- Setting K to the chosen number of neighbors.
- Calculating the distance between a provided/test example and the dataset examples.
- Sorting the calculated distances.
- Getting the labels of the top K entries.
- Returning a prediction about the test example.
In the first step, K is chosen by the user and it tells the algorithm how many neighbors (how many surrounding data points) should be considered when rendering a judgment about the group the target example belongs to. In the second step, note that the model checks the distance between the target example and every example in the dataset. The distances are then added into a list and sorted. Afterward, the sorted list is checked and the labels for the top K elements are returned. In other words, if K is set to 5, the model checks the labels of the top 5 closest data points to the target data point. When rendering a prediction about the target data point, it matters if the task is a regression or classification task. For a regression task, the mean of the top K labels is used, while the mode of the top K labels is used in the case of classification.
The exact mathematical operations used to carry out KNN differ depending on the chosen distance metric. If you would like to learn more about how the metrics are calculated, you can read about some of the most common distance metrics, such as Euclidean, Manhattan, and Minkowski.
Why The Value Of K Matters
The main limitation when using KNN is that in an improper value of K (the wrong number of neighbors to be considered) might be chosen. If this happen, the predictions that are returned can be off substantially. It’s very important that, when using a KNN algorithm, the proper value for K is chosen. You want to choose a value for K that maximizes the model’s ability to make predictions on unseen data while reducing the number of errors it makes.
Lower values of K mean that the predictions rendered by the KNN are less stable and reliable. To get an intuition of why this is so, consider a case where we have 7 neighbors around a target data point. Let’s assume that the KNN model is working with a K value of 2 (we’re asking it to look at the two closest neighbors to make a prediction). If the vast majority of the neighbors (five out of seven) belong to the Blue class, but the two closest neighbors just happen to be Red, the model will predict that the query example is Red. Despite the model’s guess, in such a scenario Blue would be a better guess.
If this is the case, why not just choose the highest K value we can? This is because telling the model to consider too many neighbors will also reduce accuracy. As the radius that the KNN model considers increases, it will eventually start considering data points that are closer to other groups than they are the target data point and misclassification will start occurring. For example, even if the point that was initially chosen was in one of the red regions above, if K was set too high, the model would reach into the other regions to consider points. When using a KNN model, different values of K are tried to see which value gives the model the best performance.
KNN Pros And Cons
Let’s examine some of the pros and cons of the KNN model.
KNN can be used for both regression and classification tasks, unlike some other supervised learning algorithms.
KNN is highly accurate and simple to use. It’s easy to interpret, understand, and implement.
KNN doesn’t make any assumptions about the data, meaning it can be used for a wide variety of problems.
KNN stores most or all of the data, which means that the model requires a lot of memory and its computationally expensive. Large datasets can also cause predictions to be take a long time.
KNN proves to be very sensitive to the scale of the dataset and it can be thrown off by irrelevant features fairly easily in comparison to other models.
K-Nearest Neighbors is one of the simplest machine learning algorithms. Despite how simple KNN is, in concept, it’s also a powerful algorithm that gives fairly high accuracy on most problems. When you use KNN, be sure to experiment with various values of K in order to find the number that provides the highest accuracy.