Seems like a lifetime ago, but I suddenly wanted to pull up some reference books I had used when preparing for INMO-2000 (class 10), INMO-2002 (class 11) and INMO-2003 (class 12)
Resources:
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math
Video Lectures on Discrete Mathematics and Algorithms
Stony Brook [my Alma Mater 🙂 ]
- http://www3.cs.stonybrook.edu/~algorith/math-video/
- http://www3.cs.stonybrook.edu/~algorith/video-lectures/
IMSc (Indian Institute of Mathematical Sciences):
- https://www.youtube.com/watch?v=Lv3o4OOHG_w
- https://www.youtube.com/watch?v=fH0V-sQrBqE
- Check out the channel! Some very cool videos
MIT
- https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/index.htm
- https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/index.htm
- https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-006-introduction-to-algorithms-fall-2011/index.htm
- https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-design-and-analysis-of-algorithms-spring-2015/index.htm
Alias method
“You are given an n-sided die where side i has probability pi of being rolled. What is the most efficient data structure for simulating rolls of the die?”
A very similar question was posted to me recently :
- http://stackoverflow.com/questions/4511331/randomly-selecting-an-element-from-a-weighted-list
- http://stackoverflow.com/questions/3120035/indexing-count-of-buckets/3120179#3120179
The approaches above are very cool, and illustrate the use of augmented search trees.
However, it seems there is a better method for this problem – and it has been out there for a while now. This was a fascinating read :
Additional pointers for the alias method:
Bandits for Online Recommendations
I came across this interesting set of blog posts by Sergei Feldman on the use of bandit approaches in online recommendation.
In particular, the one I really enjoyed reading was the comparison of the approaches needed to solve the multi armed bandit problem. Need to play around with his code someday
References:
Balanced Ternary
I was solving this math problem which had to do with representing every Natural number as a summation/subtraction of distinct power of 3
Interestingly this led me to this branch of mathematics called ‘Balanced Ternary’. Check it out!
Exploration of this problem gave me interesting insights about base representation of a number, something that I have been keeping in the backburner for a long while now. Finally got a chance to follow up on this.
References:
- https://en.wikipedia.org/wiki/Balanced_ternary
- http://math.stackexchange.com/questions/119606/represent-every-natural-number-as-a-summation-subtraction-of-distinct-power-of
- http://userpages.wittenberg.edu/bshelburne/BalancedTernaryTalkSu09.pdf
Problem:
Code:
Bandit Problems: an ‘Experiment Strategy’ or a ‘ML Algorithm’ ?
Do a simple search on Google – ‘how do bandit algorithms work’ ?
Do the results look confusing ? Some links (here1, here2) say they are better than A/B. Then there are other links which say otherwise (here3, here4).
In fact, when one hears about Bandit Problems, there are couple of questions to think about:
Questions:
1.Is it an ‘Experiment Strategy’ ?
- MAB gets compared with A/B tests. So is it an ‘experiment strategy’ like A/B testing ?
2. Is it an ‘ML Algorithm’ ?
- Bandit algorithms select the most optimal ‘action’. So is it fundamentally an ML Algorithm ?
- If yes, whats the relation between these ‘bandit problems’ v/s supervised ML algos like Logistic Regression and Decision Trees.
3. Where do the algorithms like epsilon-greedy, UCB etc fit into ?
Thoughts:
- The correct way of looking at bandit problems is to think of it as an optimization problem for online interactive systems.
- The goal of bandit algorithms is to select the best policy that will maximize rewards. The space of policies is extremely large (or infinite)
- In literature, people have treated bandit problems in different settings:
- Multi Armed Bandit setting
- Contextual Bandit
- Multi Armed Bandit setting.
- In the MAB setting, there are a few known approaches for selecting the best policy.
- Naive
- Epsilon-Greedy
- Upper Confidence Bounds.
- In the MAB setting, there are a few known approaches for selecting the best policy.
- Contextual Bandit.
- In one of my previous posts I noted the ML reduction stack in VW for the contextual bandits problem. In a separate post, I have also noted some thoughts on the use of the IPS score for conterfactual evaluation.
- In the Full Information Setting, the task of selecting the best policy is mapped to a cost-sensitive classification problem where:
- context <-> example
- action <-> label/class
- policy <-> classifier
- reward <-> gain / (negative) cost
- Thereby, we can use known supervised techniques like Decision Trees, Logistic Regression etc. to solve the cost-sensitive classification problem.
- This was an interesting insight for me, and helped me answer the question #2 above
- In the Partial Information aka. Bandit setting, there would be two more issues we would like to handle
- Filling in missing data.
- Overcoming Bias.
- The Partial Information aka. Bandit setting can further be looked into in 2 different ways:
- Online.
- In the online setting the problem has been solved in different ways
- Epsilon-Greedy / Epoch Greedy [Langford & Zhang].
- “Monster” Algorithm [Dudik, Hsu, Kale, Langford]
- They mostly vary in how they optimize regret. And/Or computational efficiency.
- Offline.
- This is where Counterfactual evaluation and Learning comes in..
- Online.
- Bandit algorithms are not just an alternate ‘experiment strategy’ that is ‘better’ or ‘worse’ than A/B tests.
- The objectives behind doing an A/B test are different from the objectives of using a bandit system (which is to do continuous optimization).
- Typic issues to consider for bandit problems:
- Explore-Exploit
- exploit what has been learned
- explore to learn which behaviour might give best results.
- Context
- In the contextual setting (‘contextual bandit’) there are many more choice available. unlikely to see the same context twice.
- Selection bias
- the exploit introduces bias that must be accounted for
- Efficiency.
- Explore-Exploit
References:
- https://support.google.com/analytics/answer/2844870?hl=en&ref_topic=1745207
- This post gives a nice overview of how Google’s Analytics Content Experiment platform uses the multi-armed bandit approach for managing online experiments
- https://www.youtube.com/watch?v=gzxRDw3lXv8
- Rob Schapire explains the fundamentals of the bandits problem. Very Cool!
- http://www.cs.cornell.edu/~adith/CfactSIGIR2016/
- Tutorial by Thorsten Joachims on the bandits problem
- http://engineering.richrelevance.com/bandits-recommendation-systems/
- Nice read!
- http://stevehanov.ca/blog/index.php?id=132
- https://www.chrisstucchio.com/blog/2012/bandit_algorithms_vs_ab.html
- https://vwo.com/blog/multi-armed-bandit-algorithm/https://www.chrisstucchio.com/blog/2015/dont_use_bandits.html
Euclid’s algorithm and GCD
Euclid’s algorithm for computing the GCD has a lot of interesting extensions for exploration.
References:
- http://www.cut-the-knot.org/blue/Euclid.shtml#
- https://en.wikipedia.org/wiki/Euclidean_algorithm
- Factor tree
- Bézout’s identity
- Extended Euclidean Algorithm
Code:
