The Beauty of Bayesian Optimization, Explained in Simple Terms

Here’s a function: f(x). It’s expensive to calculate, not necessarily an analytic expression, and you don’t know its derivative.

Your task: find the global minima.

This is, for sure, a difficult task, one more difficult than other optimization problems within machine learning. Gradient descent, for one, has access to a function’s derivatives and takes advantage of mathematical shortcuts for faster expression evaluation.

Alternatively, in some optimization scenarios the function is cheap to evaluate. If we can get hundreds of results for variants of an input x in a few seconds, a simple grid search can be employed with good results.

Alternatively, an entire host of non-conventional non-gradient optimization methods can be used, like particle swarming or simulated annealing.

Unfortunately, the current task doesn’t have these luxuries. We are limited in our optimization by several fronts, notably:

• It’s expensive to calculate. Ideally we would be able to query the function enough to essentially replicate it, but our optimization method must work with a limited sampling of inputs.
• The derivative is unknown. There’s a reason why gradient descent and its flavors still remain the most popular methods for deep learning, and sometimes, in other machine learning algorithms. Knowing the derivative gives the optimizer a sense of direction — we don’t have this.
• We need to find the global minima, which is a difficult task even for a sophisticated method like gradient descent. Our model somehow will need a mechanism to avoid getting caught in local minima.

#data-science #statistics #mathematics #artificial-intelligence #machine-learning

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Bayesian Optimization

Optimization is at the core of modern Machine Learning. Why? Linear regression, minimizing the error sum of squares. Logistic Regression, minimizing the negative likelihood. Support Vector Machines, minimizing the negative distance between the two support vectors.

Instead of an explanation heavy with math definitions and equations, let us understand about Optimization in simpler terms. “It’s the last day of the month and you are left with only $30. You are craving to have some delicious food. You wanted to have at least one variety among a pizza ($5 per piece), a burger ($3 per piece) and a coke ($1 per piece). You can’t have more than 4 pizzas and should order at least two burgers”. There you go! A real-life optimization problem: Maximizing or minimizing an objective function subject to some constraints”

Is the above problem a Linear or a Non-Linear Optimization? Linear Optimization involves objective function made up of only linear terms (or first-degree terms) and if it has any second-degree terms or nonlinear equations, then it’s a nonlinear optimization.

We shall only worry about the non-linear optimization paradigm in this article — these problems are much more complex to solve than their linear counterpart…

Let’s play a game! There is a function f(x), which I won’t reveal. This function takes real numbers. The task is this: You have to approximate the minimum (lowest) value of the function in the interval [-150, 350]. I will tell you the value of the function at any value x that you ask. But here’s the catch, you have only limited asks and to be exact, only 5. Well in between -150 and 350 there are infinite real numbers, so choose carefully.

Let’s say your first ask is the nice-and-easy 0.

Now anyone rational would choose one positive and one negative number, so let me do that on your behalf. I will share f(x) at x = 150 and x=-50

That’s an interesting set of points, isn’t it? Are you thinking what I am thinking? Choosing next point around -50 to check if our function might have its minimum value around there? In optimization terms it is called Exploitation. Its like,when you know that the function value at a point is close to our desired value when compared to other values, you will keep on exploiting that region. I do not want to waste my last two chances by exploiting. May be in between 0 and 100, I might have our desired value, the region which we haven’t even explored yet. In optimization terminology this is called as Exploration, you already know what your function outputs for few values, so instead of exploiting around the known values, we will explore new regions to find our desired value. A good optimization algorithm should strike a balance between exploration and exploitation.

#data-science #gaussian-process #optimization #bayesian-optimization #machine-learning #deep learning

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The Beauty of Bayesian Optimization, Explained in Simple Terms

Here’s a function: f(x). It’s expensive to calculate, not necessarily an analytic expression, and you don’t know its derivative.

Your task: find the global minima.

This is, for sure, a difficult task, one more difficult than other optimization problems within machine learning. Gradient descent, for one, has access to a function’s derivatives and takes advantage of mathematical shortcuts for faster expression evaluation.

Alternatively, in some optimization scenarios the function is cheap to evaluate. If we can get hundreds of results for variants of an input x in a few seconds, a simple grid search can be employed with good results.

Alternatively, an entire host of non-conventional non-gradient optimization methods can be used, like particle swarming or simulated annealing.

Unfortunately, the current task doesn’t have these luxuries. We are limited in our optimization by several fronts, notably:

• It’s expensive to calculate. Ideally we would be able to query the function enough to essentially replicate it, but our optimization method must work with a limited sampling of inputs.
• The derivative is unknown. There’s a reason why gradient descent and its flavors still remain the most popular methods for deep learning, and sometimes, in other machine learning algorithms. Knowing the derivative gives the optimizer a sense of direction — we don’t have this.
• We need to find the global minima, which is a difficult task even for a sophisticated method like gradient descent. Our model somehow will need a mechanism to avoid getting caught in local minima.

#data-science #statistics #mathematics #artificial-intelligence #machine-learning

Facebook Uses Bayesian Optimization to Conduct Better Experiments

Hyperparameter optimization is a key aspect of the lifecycle of machine learning applications. While methods such as grid search are incredibly effective for optimizing hyperparameters for specific isolated models, they are very difficult to scale across large permutations of models and experiments. A company like Facebook operates thousands of concurrent machine learning models that need to be constantly tuned. To achieve that, Facebook engineering teams need to regularly conduct A/B tests in order to determine the right hyperparameter configuration. Data in those tests is difficult to collect and they are typically conducted in isolation of each other which end up resulting in very computationally expensive exercises. One of the most innovative approaches in this area came from a team of AI researchers from Facebook who published a paper proposing a method based on Bayesian optimization to adaptively design rounds of A/B tests based on the results of prior tests.

Why Bayesian Optimization?

Bayesian optimization is a powerful method for solving black-box optimization problems that involve expensive function evaluations. Recently, Bayesian optimization has evolved as an important technique for optimizing hyperparameters in machine learning models. Conceptually, Bayesian optimization starts by evaluating a small number of randomly selected function values, and fitting a Gaussian process (GP) regression model to the results. The GP posterior provides an estimate of the function value at each point, as well as the uncertainty in that estimate. The GP works well for Bayesian optimization because it provides excellent uncertainty estimates and is analytically tractable. It provides an estimate of how an online metric varies with the parameters of interest.

Let’s imagine an environment in which we are conducting random and regular experiments on machine learning models. In that scenario, Bayesian optimization can be used to construct a statistical model of the relationship between the parameters and the online outcomes of interest and uses that model to decide which experiments to run. The concept is well illustrated in the following figure in which each data marker corresponds to the outcome of an A/B test of that parameter value. We can use the GP to decide which parameter to test next by balancing exploration (high uncertainty) with exploitation (good model estimate). This is done by computing an acquisition function that estimates the value of running an experiment with any given parameter value.

Source: https://projecteuclid.org/download/pdfview_1/euclid.ba/1533866666

The fundamental goal of Bayesian optimization when applied to hyperparameter optimization is to determine how valuable is an experiment for a specific hyperparameter configuration. Conceptually, Bayesian optimization works very efficiently for isolated models but its value proposition is challenged when used in scenarios running random experiments. The fundamental challenge is related to the noise introduced in the observations.

#bayesian #facebook #machine learning #modeling #optimization

Complete Guide to Adam Optimization

In the 1940s, mathematical programming was synonymous with optimization. An optimization problem included an objective function that is to be maximized or minimized by choosing input values from an allowed set of values [1].

Nowadays, optimization is a very familiar term in AI. Specifically, in Deep Learning problems. And one of the most recommended optimization algorithms for Deep Learning problems is Adam.

Disclaimer: basic understanding of neural network optimization. Such as Gradient Descent and Stochastic Gradient Descent is preferred before reading.

In this post, I will highlight the following points:

1. Definition of Adam Optimization
2. The Road to Adam
3. The Adam Algorithm for Stochastic Optimization
4. Visual Comparison Between Adam and Other Optimizers
5. Implementation
6. Advantages and Disadvantages of Adam
7. Conclusion and Further Reading
8. References

1. Definition of Adam Optimization

The Adam algorithm was first introduced in the paper Adam: A Method for Stochastic Optimization [2] by Diederik P. Kingma and Jimmy Ba. Adam is defined as “a method for efficient stochastic optimization that only requires first-order gradients with little memory requirement” [2]. Okay, let’s breakdown this definition into two parts.

First, stochastic optimization is the process of optimizing an objective function in the presence of randomness. To understand this better let’s think of Stochastic Gradient Descent (SGD). SGD is a great optimizer when we have a lot of data and parameters. Because at each step SGD calculates an estimate of the gradient from a random subset of that data (mini-batch). Unlike Gradient Descent which considers the entire dataset at each step.

#machine-learning #deep-learning #optimization #adam-optimizer #optimization-algorithms