In March of 2016, the computer program AlphaGo defeated Lee Sedol, one of the top 10 Go players in the world, in a five game match.  Never before had a Go computer program beaten a professional Go player on the full size board.  In January of 2017, AlphaGo won 60 consecutive online Go games against many of the best Go players in the world using the online pseudonym Master.  During these games, AlphaGo (Master) played many non-traditional moves—moves that most professional Go players would have considered bad before AlphaGo appeared. These moves are changing the Go community as professional Go players adopt them into their play.

Michael Redmond, one of the highest ranked Go players in the world outside of Asia, reviews most of these games on You Tube.  I have played Go maybe 10 times in my life, but for some reason, I enjoy watching these videos and seeing how AlphGo is changing the way Go is played. Here are some links to the videos by Redmond.

Two Randomly Selected Games from the series of 60 AlphaGo games played in January 2017


Match 1 – Google DeepMind Challenge Match: Lee Sedol vs AlphaGo


The algorithms used by AlphaGo (Deep Learning, Monte Carlo Tree Search, and convolutional neural nets) are similar to the algorithms that I used at Penn State for autonomous vehicle path planning in a dynamic environment.  These algorithms are not specific to Go.  Deep Learning and Monte Carlo Tree Search can be used in any game.  Google Deep Mind has had a lot of success applying these algorithms to Atari video games where the computer learns strategy through self play.  Very similar algorithms created AlphaGo from self play and analysis of professional and amateur Go games.

I often wonder what we can learn about other board games from computers.  We will learn more about Go from AlphaGo in two weeks.  From May 23rd to 27th, AlphaGo will play against several top Go professionals at the “Future of Go Summit” conference.


Gettysburg University has a nice webpage on Counterfactual Regret. Counterfactual Regret was used by the University of Alberta to solve the heads up limit Holdem poker game. (See e.g. the  article).

Dr Xu at Penn State introduced me to Richardson Extrapolation way back in the early 90’s.  We used it to increase the speed of convergence of finite element analysis algorithms for partial differential equations, but it can be used for many numerical methods.

“Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems” by Sébastie Bubeck and Nicolò Cesa-Bianchi is available in pdf format at

The book “Deep Learning” by Ian Goodfellow, Yoshua Bengio, and Aaron Courville (associated with the Google Deep Mind Team) is available in HTML format.

(See also:  Half-precision floating-point format )

Wired has a nice article about the two most brilliant moves in the historic match between AlphaGo and Lee Sedol.

In case you had not gotten the news yet, the Go playing program AlphaGo (developed by the Deep Mind division of Google) has beaten Lee Se-dol who is among the top two or three Go players in the world.  Follow the link below for an informative informal video describing AlphaGo and the victory.

Science magazine has a nice pregame report.

The Data Processing Inequality is a nice, intuitive inequality about Mutual Information.  Suppose X,Y, Z are random variables and Z is independent of X given Y, then 

MI(X,Z) <= MI(X,Y).

See which has an easy one line proof.

We can apply this inequality to a stacked restricted Boltzmann machine (a type of deep neural net).

Let X be a random binary vector consisting of the states of neurons in the first layer.

Let Y be a random binary vector consisting of the states of neurons in the second layer.

And let Z be a random binary vector consisting of the states of neurons in the third layer.


MI(X,Z) <= min( MI(X,Y), MI(Y,Z) ).

Informally, that inequality says that amount of information that can flow from the first layer to the third layer of a stacked RBM deep neural net is less than or equal to the maximum flow rate between the first and second layer.  Also, the amount of information that can flow from the first layer to the third layer is less than or equal to the maximum flow rate between the second and third layer.  This inequality will seem obvious to those who know information theory, but I still think it’s cute.

The above inequality is also sharp in the sense that there are simple examples where the right hand side equals the left hand side.  Consider a Markov Random Field consisting of just three random binary variables X, Y and Z.  Suppose further,  that P(X)=0.5, P(X=Y)=1,  and P(Y=Z)=1.  Then MI(X,Y)=1 bit, MI(Y,Z) =1 bit, and MI(X,Z) =1 bit so both sides of the inequality are 1.

Information theory can also be used to construct a lower bound on the information transfer between the first and third layer.

MI(X,Z) >= MI(Y,X)+MI(Y,Z) – H(Y)

where H(Y) is the entropy of Y (i.e. the information content of the random variable Y).


Intuitively, if the sum of the information from X to Y and from Z to Y  exceeds the information capacity of Y, then there must be some information transfer between X and Z.

Zhoa, Li, Geng, and Ma recently wrote a poorly written but interesting paper “Artificial Neural Networks Based on Fractal Growth”.  The paper describes a neural net architecture that grows in a fractal pattern (Similar to evolutionary artificial neural nets, see e.g. “A review of evolutionary artificial neural networks“ Yao 1993).  The input region assigned to each label by the neural net grows in a fractal like pattern to adapt to new data.  The growth of the nodes suggest that the fractal neural network classifications are similar to k-Nearest Neighbor with k=1 or an SVM with radial basis functions.  They report on application of their method to SEMG (Surface electromyogram signal) classification.

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