Algebraic expressions are the idea of expressing numbers using letters or alphabets without specifying their actual values. The basics of algebra taught us how to express an unknown value using letters such as x, y, z, etc. These letters are called here as variables. An algebraic expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is a coefficient.
An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.).
3x + 4y – 7, 4x – 10, etc.
These expressions are represented with the help of unknown variables, constants and coefficients. The combination of these three (as terms) is said to be an expression. It is to be noted that, unlike the algebraic equation, an algebraic expression has no sides or equal to sign. Some of its examples include
Algebra is firstly taken from a book, written by Khwarizmi(780-850 CE), which is about calculation and equations. It is a branch of mathematics in which letters are used instead of numbers. Each letter can represent a specific number in one place, and a completely different number in another. Notations and symbols are also used in algebra to show the relationship between numbers. I remember about 17 years ago when I was an ordinary student in applied mathematics(ordinary graduate today!), I was so curious about some research in algebra, done by Maryam Mirzakhani(1977–2017), at Harvard University about analogous counting problem. This science has evolved a lot throughout history and now includes many branches.
Elementary algebra includes basic operations on four main operations. After defining the signs by which fixed numbers and variables are separated, methods are used to solve the equations. A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers.
Abstract algebra or modern algebra is a group in the algebra family that studies advanced algebraic structures such as groups, rings, and fields. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Abstract algebra is so popular and used in many fields of mathematics and engineering sciences. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers.
I believe that the most influencing branch of algebra in other sciences is linear algebra. Let’s suppose that you went out for jogging, that can’t be easy these days with Covid-19 lockdown, and suddenly a beautiful flower catches all your attention. Please don’t be rush and pick it, just take a picture, others can enjoy it, as well. After a while when you look at this picture, you can recognize the flower in the image, because the human brain has evolved over millions of years and able to detect such a thing. We are unaware of the operations that take place in the background of our brains and enable us to recognize the colors in the image, they are trained to do this for us automatically. But, it’s not easy to do such a thing with a machine, that’s why this is one of the most active research areas in machine learning and deep learning. Actually, the fundamental question is: “How does a machine store this image?” You probably know that today’s computers are designed to process only two numbers, 0 and 1. Now, how can an image like this with different features be stored? This is done by storing the pixel intensity in a structure called a “matrix”.
The main topics in linear algebra are vectors and matrices. Vectors are geometric objects that have length and direction. For example, we can mention speed and force, both of which are vector quantities. Each vector is represented by an arrow whose length and direction indicate the size and direction of the vector.
The addition of two or more vectors can be done based on ease of use using parallelogram methods or the method of images in which each vector is decomposed into its components along the coordinate axes. A vector space is a collection of vectors, which may be added together and multiplied by scalars. Scalars generally can be picked up from any field but normally are real numbers.
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Recently, researchers from Google proposed the solution of a very fundamental question in the machine learning community — What is being transferred in Transfer Learning? They explained various tools and analyses to address the fundamental question.
The ability to transfer the domain knowledge of one machine in which it is trained on to another where the data is usually scarce is one of the desired capabilities for machines. Researchers around the globe have been using transfer learning in various deep learning applications, including object detection, image classification, medical imaging tasks, among others.
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Reinforcement learning (RL) is surely a rising field, with the huge influence from the performance of AlphaZero (the best chess engine as of now). RL is a subfield of machine learning that teaches agents to perform in an environment to maximize rewards overtime.
Among RL’s model-free methods is temporal difference (TD) learning, with SARSA and Q-learning (QL) being two of the most used algorithms. I chose to explore SARSA and QL to highlight a subtle difference between on-policy learning and off-learning, which we will discuss later in the post.
This post assumes you have basic knowledge of the agent, environment, action, and rewards within RL’s scope. A brief introduction can be found here.
The outline of this post include:
We will compare these two algorithms via the CartPole game implementation. This post’s code can be found here :QL code ,SARSA code , and the fully functioning code . (the fully-functioning code has both algorithms implemented and trained on cart pole game)
The TD learning will be a bit mathematical, but feel free to skim through and jump directly to QL and SARSA.
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