Thanks to earbuds you can have calls anywhere while doing anything. The problem: those on the other end of the call hear it all, too, from your roommate’s vacuum cleaner to background conversations at the cafe you’re working from. Now, work by a trio of graduate students at the University of Washington who spent the Read article >

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<Incoming Transmission> Epic Games is bringing a new Fortnite reward to GeForce NOW, available to all members. Drop from the Battle Bus in Fortnite on GeForce NOW between today and Thursday, Aug. 4, to earn “The Dish-stroyer Pickaxe” in game for free. <Transmission continues> Members can earn this item by streaming Fortnite on GeForce NOW Read article >

The post Get Battle Ready With New GeForce NOW Fortnite Reward appeared first on NVIDIA Blog.

The acceleration of digital transformation within data centers and the associated application proliferation is exposing new attack surfaces to potential security threats. These new…

The acceleration of digital transformation within data centers and the associated application proliferation is exposing new attack surfaces to potential security threats. These new attacks typically bypass the well-established perimeter security controls such as traditional and web application firewalls, making detection and remediation of cybersecurity threats more challenging.

Defending against these threats is becoming more challenging due to modern applications not being built entirely within a single data center—whether physical, virtual, or in the cloud. Today’s applications often span multiple servers in public clouds, CDN networks, edge platforms, and as-a-service components for which the location is not even known. 

On top of this, each service or microservice may have multiple instances for scale-out purposes, straining the ability of traditional network security functions to isolate them from the outside world to protect them. 

Finally, the number of data sources and locations is large and growing both because of the distributed nature of modern applications and the effects of scale-out architecture. There is no longer a single gate in the data center, such as an ingress gateway or firewall, that can observe and secure all data traffic. 

The consequence of these changes is the much larger sheer volume of data that must be collected to provide a holistic view of the application and to detect advanced threats. The number of data sources that must be monitored and the diversity in terms of data types is also growing, making effective cybersecurity data collection extremely challenging.

Detection requires a large amount of contextual information that can be correlated in near real time to determine the advanced threat activity in progress.

F5 is researching techniques to augment well-established security measures for web, application, firewall, and fraud mitigation. Detecting such advanced threats, which require contextual analysis of several of these data points through large-scale telemetry and with near real-time analysis, requires machine learning (ML) and AI algorithms.

ML and AI are used to detect anomalous activity in and around applications, as well as cloud environments, to tackle the risks upfront. This is where the NVIDIA BlueField-2 data processing unit (DPU) real-time telemetry and NVIDIA GPU-powered Morpheus cybersecurity framework come into play.

NVIDIA Morpheus provides an open application framework that enables cybersecurity developers to create optimized AI pipelines for filtering, processing, and classifying large volumes of real-time data. Morpheus offers pretrained AI models that provide powerful tools to simplify workflows and help detect and mitigate security threats.

Cybersecurity poses unique requirements for AI/ML processing

From a solution perspective, a robust telemetry collection strategy is a must and the telemetry data must have specific requirements: 

• A secure—encrypted and authenticated—means of transmitting data to a centralized data collector.
• The ability to ingest telemetry with support for all the commonly used data paradigms: 
  • Asynchronously occurring security-relevant events
  • Application logs
  • Statistics and status-related metrics
  • Entity-specific trace records
• A well-defined vocabulary that can map the data collected from diverse data sources into a canonical consumable representation

Finally, all this must be done in a highly scalable way, agnostic to the source location, which may be from a data center, the edge, a CDN, a client device, or even out-of-band metadata, such as threat intelligence feeds.

NVIDIA Morpheus-optimized AI pipelines

With a unique history and expertise in building networking software capable of harnessing the benefits of hardware, F5 is one of the first to join the NVIDIA Morpheus Early Access program.

Morpheus is an open application framework that enables cybersecurity developers to create optimized AI pipelines for filtering, processing, and classifying large volumes of real-time data.

F5 is leveraging Morpheus, which couples BlueField DPUs with NVIDIA certified EGX servers, to provide a powerful solution to detect and eliminate security threats.

Morpheus allows F5 to accelerate access to embedded analytics and provide security across the cloud and emerging edge from their Shape Enterprise Defense application. The joint solution brings a new level of security to data centers and enables dynamic protection, real-time telemetry, and an adaptive defense for detecting and remediating cybersecurity threats.

Learn more

For more information about how F5 accelerates cybersecurity protection through real-time, DPU-enhanced telemetry and AI-powered analytics using NVIDIA GPU-powered Morpheus, see the Redefining Cybersecurity at the Distributed Cloud Edge with AI and Real-time Telemetry GTC session.

The new ‘Level Up with NVIDIA’ webinar series offers creators and developers the opportunity to learn more about the NVIDIA RTX platform, interact with NVIDIA experts, and ask…

The new ‘Level Up with NVIDIA’ webinar series offers creators and developers the opportunity to learn more about the NVIDIA RTX platform, interact with NVIDIA experts, and ask questions about game integrations. 

Kicking off in early August, the series features one 60-minute webinar each month, with the first half dedicated to NVIDIA experts discussing the session’s topic and the remaining time dedicated to Q&A. 

We’ll focus on the NVIDIA RTX platform within popular game engines, explore what NVIDIA technologies and SDKs are in Unreal Engine 5 and Unity, and how you can successfully leverage the latest tools in your games.

Join us for the first webinar in the series on August 10 at 10 AM, Pacific time, with NVIDIA experts Richard Cowgill and Zach Lo discussing RTX in Unreal Engine 5.

Learn about NVIDIA technologies integrated into Unreal Engine, get insights into available ray tracing technologies, and see how you can get the most out of NVIDIA technologies across all game engines.

Register now. 

Imagine that you have trained your model with PyTorch, TensorFlow, or the framework of your choice, are satisfied with its accuracy, and are considering deploying it as a service….

Imagine that you have trained your model with PyTorch, TensorFlow, or the framework of your choice, are satisfied with its accuracy, and are considering deploying it as a service. There are two important objectives to consider: maximizing model performance and building the infrastructure needed to deploy it as a service. This post discusses both objectives.

You can squeeze better performance out of a model by accelerating it across three stack levels:

• Hardware acceleration
• Software acceleration
• Algorithmic or network acceleration.

NVIDIA GPUs are the leading choice for hardware acceleration among deep learning practitioners, and their merit is widely discussed in the industry.

The conversation about GPU software acceleration typically revolves around libraries like cuDNN, NCCL, TensorRT, and other CUDA-X libraries.

Algorithmic or network acceleration revolves around the use of techniques like quantization and knowledge distillation that essentially make modifications to the network itself, applications of which are highly dependent on your models.

This need for acceleration is driven primarily by business concerns like reducing costs or improving the end-user experience by reducing latency and tactical considerations like deploying on models on edge devices having fewer compute resources.

Serving deep learning models

After the models are accelerated, the next step is to build a serving service to deploy your model, which comes with its own unique set of challenges. This is a nonexhaustive list:

• Will the service work on different hardware platforms?
• Will it handle other models that I have to deploy simultaneously?
• Will the service be robust?
• How do I reduce latency?
• Models are trained with different frameworks and tech stacks; how do I cater to this?
• How do I scale?

These are all valid questions and addressing each of them presents a challenge.

Solution overview

This post discusses using NVIDIA TensorRT, its framework integrations for PyTorch and TensorFlow, NVIDIA Triton Inference Server, and NVIDIA GPUs to accelerate and deploy your models.

NVIDIA TensorRT

NVIDIA TensorRT is an SDK for high-performance deep learning inference. It includes a deep learning inference optimizer and runtime that delivers low latency and high throughput for deep learning inference applications.

With its framework integrations with PyTorch and TensorFlow, you can speed up inference up to 6x faster with just one line of code.

NVIDIA Triton Inference Server

NVIDIA Triton Inference Server is an open-source inference-serving software that provides a single standardized inference platform. It can support running inference on models from multiple frameworks on any GPU or CPU-based infrastructure in the data center, cloud, embedded devices, or virtualized environments.

For more information, see the following videos:

• Introduction to NVIDIA TensorRT for High Performance Deep Learning Inference
• Getting Started with NVIDIA Torch-TensorRT
• Getting Started with TensorFlow-TensorRT
• Top 5 Reasons Why Triton is Simplifying Inference

Workflow overview

Before we dive into the details, here’s the overall workflow. To follow along, see the following resources:

• TensorRT
• Torch-TensorRT
• Tensorflow-TensorRT

Figure 1 shows the steps that you must go through.

Before you start following along, be ready with your trained model.

• Step 1: Optimize the models. You can do this with either TensorRT or its framework integrations. If you choose TensorRT, you can use the trtexec command line interface. For the framework integrations with TensorFlow or PyTorch, you can use the one-line API.
• Step 2: Build a model repository. Spinning up an NVIDIA Triton Inference Server requires a model repository. This repository contains the models to serve, a configuration file that specifies the details, and any required metadata.
• Step 3: Spin up the server.
• Step 4: Finally, we provide simple and robust HTTP and gRPC APIs that you can use to query the server!

Throughout this post, use the Docker containers from NGC. You may need to create an account and get the API key to access these containers. Now, here are the details!

Accelerating models with TensorRT

TensorRT accelerates models through graph optimization and quantization. You can access these benefits in any of the following ways:

• trtexec CLI tool
• TensorRT Python/C++ API
• Torch-TensorRT (integration with PyTorch)
• TensorFlow-TensorRT (integration with TensorFlow)

While TensorRT natively enables greater customization in graph optimizations, the framework integration provides ease of use for developers new to the ecosystem. As choosing the route a user might adopt is subject to the specific needs of their network, we would like to lay out all the options. For more information, see Speeding Up Deep Learning Inference Using NVIDIA TensorRT (Updated).

For TensorRT, there are several ways to build a TensorRT engine. For this post, use the trtexec CLI tool. If you want a script to export a pretrained model to follow along, use the export_resnet_to_onnx.py example. For more information, see the TensorRT documentation.

docker run -it --gpus all -v /path/to/this/folder:/trt_optimize nvcr.io/nvidia/tensorrt:-py3

trtexec --onnx=resnet50.onnx 
        --saveEngine=resnet50.engine 
        --explicitBatch 
        --useCudaGraph

To use FP16, add --fp16 in the command. Before proceeding to the next step, you must know the names of your network’s input and output layers, which is required while defining the config for the NVIDIA Triton model repository. One easy way is to use polygraphy, which comes packaged with the TensorRT container.

polygraphy inspect model resnet50.engine --mode=basic

ForTorch-TensorRT, pull the NVIDIA PyTorch container, which has both TensorRT and Torch TensorRT installed. To follow along, use the sample. For more examples, visit the Torch-TensorRT GitHub repo.

#  is the yy:mm for the publishing tag for NVIDIA's Pytorch 
# container; eg. 21.12

docker run -it --gpus all -v /path/to/this/folder:/resnet50_eg nvcr.io/nvidia/pytorch:-py3

python torch_trt_resnet50.py

To expand on the specifics, you are essentially using Torch-TensorRT to compile your PyTorch model with TensorRT. Behind the scenes, your model gets converted to a TorchScript module, and then TensorRT-supported ops undergo optimizations. For more information, see the Torch-TensorRT documentation.

model = torch.hub.load('pytorch/vision:v0.10.0', 'resnet50', pretrained=True).eval().to("cuda")

# Compile with Torch TensorRT;
trt_model = torch_tensorrt.compile(model, 
    inputs= [torch_tensorrt.Input((1, 3, 224, 224))],
    enabled_precisions= { torch_tensorrt.dtype.float32} # Runs with FP32; can use FP16
)

# Save the model
torch.jit.save(trt_model, "model.pt")

For TensorFlow-TensorRT, the process is pretty much the same. First, pull the NVIDIA TensorFlow container, which comes with TensorRT and TensorFlow-TensorRT. We made a short script tf_trt_resnet50.py as an example. For more examples, see the TensorFlow TensorRT GitHub repo.

#  is the yy:mm for the publishing tag for the NVIDIA Tensorflow
# container; eg. 21.12

docker run -it --gpus all -v /path/to/this/folder:/resnet50_eg nvcr.io/nvidia/tensorflow:-tf2-py3

python tf_trt_resnet50.py

Again, you are essentially using TensorFlow-TensorRT to compile your TensorFlow model with TensorRT. Behind the scenes, your model gets segmented into subgraphs containing operations supported by TensorRT, which then undergo optimizations. For more information, see the TensorFlow-TensorRT documentation.

# Load model
model = ResNet50(weights='imagenet')
model.save('resnet50_saved_model') 

# Optimize with tftrt

converter = trt.TrtGraphConverterV2(input_saved_model_dir='resnet50_saved_model')
converter.convert()

# Save the model
converter.save(output_saved_model_dir='resnet50_saved_model_TFTRT_FP32')

Now that you have optimized your model with TensorRT, you can proceed to the next step, setting up NVIDIA Triton.

Setting up NVIDIA Triton Inference Server

NVIDIA Triton Inference Server is built to simplify the deployment of a model or a collection of models at scale in a production environment. To achieve ease of use and provide flexibility, using NVIDIA Triton revolves around building a model repository that houses the models, configuration files for deploying those models, and other necessary metadata.

Look at the simplest case. Figure 4 has four key points. The config.pbtxt file (a) is the previously mentioned configuration file that contains, well, configuration information for the model.

There are several key points to note in this configuration file:

• Name: This field defines the model’s name and must be unique within the model repository.
• Platform: (c)This field is used to define the type of the model: is it a TensorRT engine, PyTorch model, or something else.
• Input and Output: (d)These fields are required as NVIDIA Triton needs metadata about the model. Essentially, it requires the names of your network’s input and output layers and the shape of said inputs and outputs. In the case of TorchScript, as the name of input and output layers are absent, use input__0. Datatype is set to FP32, and the input format is specified as (Channel, Height, Width) of 3, 224, 224.

There are minor differences between TensorRT, Torch-TensorRT, and TensorFlow-TensorRT workflows in this set, which boils down to specifying the platform and changing the name for the input and output layers. We made sample config files for all three (TensorRT, Torch-TensorRT, or TensorFlow-TensorRT). Lastly, you add the trained model (b).

Now that the model repository has been built, you spin up the server. For this, all you must do is pull the container and specify the location of your model repository. For more Information about scaling this solution with Kubernetes, see Deploying NVIDIA Triton at Scale with MIG and Kubernetes.

docker run --gpus=1 --rm -p 8000:8000 -p 8001:8001 -p 8002:8002 -v /full/path/to/docs/examples/model_repository:/models nvcr.io/nvidia/tritonserver:-py3 tritonserver --model-repository=/models

With your server up and running, you can finally build a client to fulfill inference requests!

Setting up NVIDIA Triton Client

The final step in the pipeline is to query the NVIDIA Triton Inference Server. You can send inference requests to the server through an HTTP or a gRPC request. Before diving into the specifics, install the required dependencies and download a sample image.

pip install torchvision
pip install attrdict
pip install nvidia-pyindex
pip install tritonclient[all]

wget  -O img1.jpg "https://bit.ly/3phN2jy"

In this post, use Torchvision to transform a raw image into a format that would suit the ResNet-50 model. It isn’t necessarily needed for a client. We have a much more comprehensive image client and a plethora of varied clients premade for standard use cases available in the triton-inference-server/client GitHub repo. However, for this explanation, we are going over a much simpler and skinny client to demonstrate the core of the API.

Okay, now you are ready to look at an HTTP client (Figure 5). Download the client script:

• TensorRT
• Torch-TensorRT
• TensorFlow-TensorRT

Building the client has the following steps. First, establish a connection between the NVIDIA Triton Inference Server and the client.

triton_client = httpclient.InferenceServerClient(url="localhost:8000")

Second, pass the image and specify the names of the input and output layers of the model. These names should be consistent with the specifications defined in the config file that you built while making the model repository.

test_input = httpclient.InferInput("input__0", transformed_img.shape, datatype="FP32")
test_input.set_data_from_numpy(transformed_img, binary_data=True)

test_output = httpclient.InferRequestedOutput("output__0", binary_data=True, class_count=1000)

Finally, send an inference request to the NVIDIA Triton Inference Server.

results = triton_client.infer(model_name="resnet50", inputs=[test_input], outputs=[test_output])

These code examples discuss the specifics of the Torch-TensorRT models. The only differences among different models (when building a client) would be the input and output layer names. We have built NVIDIA Triton clients with Python, C++, Go, Java, and JavaScript. For more examples, see the triton-inference-server/client GitHub repo.

Conclusion

This post covered an end-to-end pipeline for inference where you first optimized trained models to maximize inference performance using TensorRT, Torch-TensorRT, and TensorFlow-TensorRT. You then proceeded to model serving by setting up and querying an NVIDIA Triton Inference Server. All the software, including TensorRT, Torch-TensorRT, TensorFlow-TensorRT, and Triton discussed in this tutorial, are available today to download as a Docker container from NGC.

Linear regression is one of the simplest machine learning models out there. It is often the starting point not only for learning about data science but also for building quick and…

Linear regression is one of the simplest machine learning models out there. It is often the starting point not only for learning about data science but also for building quick and simple minimum viable products (MVPs), which then serve as benchmarks for more complex algorithms.

In general, linear regression fits a line (in two dimensions) or a hyperplane (in three and more dimensions) that best describes the linear relationship between the features and the target value. The algorithm also assumes that the probability distributions of the features are well-behaved; for example, they follow the Gaussian distribution.

Outliers are values that are located far outside of the expected distribution. They cause the distributions of the features to be less well-behaved. As a consequence, the model can be skewed towards the outlier values, which, as I’ve already established, are far away from the central mass of observations. Naturally, this leads to the linear regression finding a worse and more biased fit with inferior predictive performance.

It is important to remember that the outliers can be found both in the features and the target variable, and all the scenarios can worsen the performance of the model.

There are many possible approaches to dealing with outliers: removing them from the observations, treating them (capping the extreme observations at a reasonable value, for example), or using algorithms that are well-suited for dealing with such values on their own. This post focuses on these robust methods.

Setup

I use fairly standard libraries: numpy, pandas, scikit-learn. All the models I work with here are imported from the linear_model module of scikit-learn.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

from sklearn import datasets
from sklearn.linear_model import (LinearRegression, HuberRegressor,
                              	RANSACRegressor, TheilSenRegressor)

Data

Given that the goal is to show how different robust algorithms deal with outliers, the first step is to create a tailor-made dataset to show clearly the differences in the behavior. To do so, use the functionalities available in scikit-learn. 

Start with creating a dataset of 500 observations, with one informative feature. With only one feature and the target, plot the data, together with the models’ fits. Also, specify the noise (standard deviation applied to the output) and create a list containing the coefficient of the underlying linear model; that is, what the coefficient would be if the linear regression model was fit to the generated data. In this example, the value of the coefficient is 64.6. Extract those coefficients for all the models and use them to compare how well they fit the data.

Next, replace the first 25 observations (5% of the observations) with outliers, far outside of the mass of generated observations. Bear in mind that the coefficient stored earlier comes from the data without outliers. Including them makes a difference.

N_SAMPLES = 500
N_OUTLIERS = 25

X, y, coef = datasets.make_regression(
	n_samples=N_SAMPLES,
	n_features=1,
	n_informative=1,
	noise=20,
	coef=True,
	random_state=42
)
coef_list = [["original_coef", float(coef)]]

# add outliers          	 
np.random.seed(42)
X[:N_OUTLIERS] = 10 + 0.75 * np.random.normal(size=(N_OUTLIERS, 1))
y[:N_OUTLIERS] = -15 + 20 * np.random.normal(size=N_OUTLIERS)

plt.scatter(X, y);

Linear regression

Start with the good old linear regression model, which is likely highly influenced by the presence of the outliers. Fit the model to the data using the following example:

lr = LinearRegression().fit(X, y)
coef_list.append(["linear_regression", lr.coef_[0]])

Then prepare an object to use for plotting the fits of the models. The plotline_X object is a 2D array containing evenly spaced values within the interval dictated by the generated data set. Use this object for getting the fitted values for the models. It must be a 2D array, given it is the expected input of the models in scikit-learn. Then create a fit_df DataFrame in which to store the fitted values, created by fitting the models to the evenly spaced values.

plotline_X = np.arange(X.min(), X.max()).reshape(-1, 1)

fit_df = pd.DataFrame(
	index = plotline_X.flatten(),
	data={"linear_regression": lr.predict(plotline_X)}
)

Having prepared the DataFrame, plot the fit of the linear regression model to the data with outliers.

fix, ax = plt.subplots()
fit_df.plot(ax=ax)
plt.scatter(X, y, c="k")
plt.title("Linear regression on data with outliers");

Figure 2 shows the significant impact that outliers have on the linear regression model.

The benchmark model has been obtained using linear regression. Now it is time to move toward robust regression algorithms.

Huber regression

Huber regression is an example of a robust regression algorithm that assigns less weight to observations identified as outliers. To do so, it uses the Huber loss in the optimization routine. Here’s a better look at what is actually happening in this model.

Huber regression minimizes the following loss function:

Where denotes the standard deviation, represents the set of features, is the regression’s target variable, is a vector of the estimated coefficients and is the regularization parameter. The formula also indicates that outliers are treated differently from the regular observations according to the Huber loss:

The Huber loss identifies outliers by considering the residuals, denoted by . If the observation is considered to be regular (because the absolute value of the residual is smaller than some threshold , then apply the squared loss function. Otherwise, the observation is considered to be an outlier and you apply the absolute loss. Having said that, Huber loss is basically a combination of the squared and absolute loss functions.

An inquisitive reader might notice that the first equation is similar to Ridge regression, that is, including the L2 regularization. The difference between Huber regression and Ridge regression lies in the treatment of outliers.

You might recognize this approach to loss functions from analyzing the differences between two of the popular regression evaluation metrics: mean squared error (MSE) and mean absolute error (MAE). Similar to what the Huber loss implies, I recommend using MAE when you are dealing with outliers, as it does not penalize those observations as heavily as the squared loss does. 

Connected to the previous point is the fact that optimizing the squared loss results in an unbiased estimator around the mean, while the absolute difference leads to an unbiased estimator around the median. The median is much more robust to outliers than the mean, so expect this to provide a less biased estimate.

Use the default value of 1.35 for , which determines the regression’s sensitivity to outliers. Huber (2004) shows that when the errors follow a normal distribution with = 1 and = 1.35, an efficiency of 95% is achieved relative to the OLS regression.

For your own use cases, I recommend tuning the hyperparameters alpha and epsilon, using a method such as grid search. 

Fit the Huber regression to the data using the following example:

huber = HuberRegressor().fit(X, y)
fit_df["huber_regression"] = huber.predict(plotline_X)
coef_list.append(["huber_regression", huber.coef_[0]])

Figure 3 presents the fitted model’s best fit line.

RANSAC regression

Random sample consensus (RANSAC) regression is a non-deterministic algorithm that tries to separate the training data into inliers (which may be subject to noise) and outliers. Then, it estimates the final model only using the inliers.

RANSAC is an iterative algorithm in which iteration consists of the following steps:

1. Select a random subset from the initial data set.
2. Fit a model to the selected random subset. By default, that model is a linear regression model; however, you can change it to other regression models.
3. Use the estimated model to calculate the residuals for all the data points in the initial data set. All observations with absolute residuals smaller than or equal to the selected threshold are considered inliers and create the so-called consensus set. By default, the threshold is defined as the median absolute deviation (MAD) of the target values.
4. The fitted model is saved as the best one if sufficiently many points have been classified as part of the consensus set. If the current estimated model has the same number of inliers as the current best one, it is only considered to be better if it has a better score.

The steps are performed iteratively either a maximum number of times or until a special stop criterion is met. Those criteria can be set using three dedicated hyperparameters. As I mentioned earlier, the final model is estimated using all inlier samples.

Fit the RANSAC regression model to the data.

ransac = RANSACRegressor(random_state=42).fit(X, y)
fit_df["ransac_regression"] = ransac.predict(plotline_X)
ransac_coef = ransac.estimator_.coef_
coef_list.append(["ransac_regression", ransac.estimator_.coef_[0]])

As you can see, the procedure for recovering the coefficient is a bit more complex, as it’s first necessary to access the final estimator of the model (the one trained using all the identified inliers) using estimator_. As it is a LinearRegression object, proceed to recover the coefficient as you did earlier. Then, plot the fit of the RANSAC regression (Figure 4).

With RANSAC regression, you can also inspect the observations that the model considered to be inliers and outliers. First, check how many outliers the model identified in total and then how many of those that were manually introduced overlap with the models’ decision. The first 25 observations of the training data are all the outliers that have been introduced.

inlier_mask = ransac.inlier_mask_
outlier_mask = ~inlier_mask
print(f"Total outliers: {sum(outlier_mask)}")
print(f"Outliers you added yourself: {sum(outlier_mask[:N_OUTLIERS])} / {N_OUTLIERS}")

Running the example prints the following summary:

Total outliers: 51
Outliers you added yourself: 25 / 25

Roughly 10% of data was identified as outliers and all the observations introduced were correctly classified as outliers. It’s then possible to quickly visualize the inliers compared to outliers to see the remaining 26 observations flagged as outliers.

plt.scatter(X[inlier_mask], y[inlier_mask], color="blue", label="Inliers")
plt.scatter(X[outlier_mask], y[outlier_mask], color="red", label="Outliers")
plt.title("RANSAC - outliers vs inliers");

Figure 5 shows that the observations located farthest from the hypothetical best-fit line of the original data are considered outliers.

Theil-Sen regression

The last of the robust regression algorithms available in scikit-learn is the Theil-Sen regression. It is a non-parametric regression method, which means that it makes no assumption about the underlying data distribution. In short, it involves fitting multiple regression models on subsets of the training data and then aggregating the coefficients at the last step.

Here’s how the algorithm works. First, it calculates the least square solutions (slopes and intercepts) on subsets of size p (hyperparameter n_subsamples) created from all the observations in the training set X. If you calculate the intercept (it is optional), then the following condition must be satisfied p >= n_features + 1. The final slope of the line (and possibly the intercept) is defined as the (spatial) median of all the least square solutions.

A possible downside of the algorithm is its computational complexity, as it can consider a total number of least square solutions equal to n_samples choose n_subsamples, where n_samples is the number of observations in X. Given that this number can quickly explode in size, there are a few things that can be done:

• Use the algorithm only for small problems in terms of the number of samples and features. However, for obvious reasons, this might not always be feasible.
• Tune the n_subsamples hyperparameter. A lower value leads to higher robustness to outliers at the cost of lower efficiency, while a higher value leads to lower robustness and higher efficiency.
• Use the max_subpopulation hyperparameter. If the total value of n_samples choose n_subsamples is larger than max_subpopulation, the algorithm only considers a stochastic subpopulation of a given maximal size. Naturally, using only a random subset of all the possible combinations leads to the algorithm losing some of its mathematical properties.

Also, be aware that the estimator’s robustness decreases quickly with the dimensionality of the problem. To see how that works out in practice, estimate the Theil-Sen regression using the following example:

theilsen = TheilSenRegressor(random_state=42).fit(X, y)
fit_df["theilsen_regression"] = theilsen.predict(plotline_X)
coef_list.append(["theilsen_regression", theilsen.coef_[0]])

Comparison of the models

So far, three robust regression algorithms have been fitted to the data containing outliers and the individual best fit lines have been identified. Now it is time for a comparison.

Start with the visual inspection of Figure 7. To show too many lines, the fit line of the original data is not printed. However, it is quite easy to imagine what it looks like, given the direction of the majority of the data points. Clearly, the RANSAC and Theil-Sen regressions have resulted in the most accurate best fit lines.

To be more precise, look at the estimated coefficients. Table 1 shows that the RANSAC regression results in the fit closest to the one of the original data. It is also interesting to see how big of an impact the 5% of outliers had on the regular linear regression’s fit.

You might ask which robust regression algorithm is the best? As is often the case, the answer is, “It depends.” Here are some guidelines that might help you find the right model for your specific problem:

• In general, robust fitting in a high-dimensional setting is difficult. 
• In contrast to Theil-Sen and RANSAC, Huber regression is not trying to completely filter out the outliers. Instead, it lessens their effect on the fit.
• Huber regression should be faster than RANSAC and Theil-Sen, as the latter ones fit on smaller subsets of the data. 
• Theil-Sen and RANSAC are unlikely to be as robust as the Huber regression using the default hyperparameters.
• RANSAC is faster than Theil-Sen and it scales better with the number of samples.
• RANSAC should deal better with large outliers in the y-direction, which is the most common scenario.

Taking all the preceding information into consideration, you might also empirically experiment with all three robust regression algorithms and see which one fits your data best. 

You can find the code used in this post in my /erykml GitHub repo. I look forward to hearing from you in the comments.

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