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Turing Award-winner Leslie Lamport shares the key lessons he has learned about concurrent and distributed computing over decades of writing and reasoning about their algorithms. Algorithms are not programs, and they shouldn't be written in a programming language. Instead, this book explores how to write them and reason about them by using mathematics. It explains the principles underlying abstract programs and understanding those principles helps to avoid concurrency errors. Designing an abstract program before writing any code can lead to better, more reliable programs. The book has very few mathematical prerequisites, with an appendix summarizing the necessary knowledge. Many of the examples are available online, written in the formal language TLA+, and can be checked with the TLA+ tools. This is a fascinating read for any graduate students and researchers in theoretical computer science, concurrency, and distributed systems.
We can now present a simple method for obtaining O(d1.5 n) regret for losses in Fb with the limitation that the analysis only works in the stochastic setting where ft = f for all rounds.
The tool-chest for convex bandits and zeroth-order optimisation has been steadily growing in recent decades. Nevertheless, there are many intriguing open questions, both theoretical and practical. The purpose of this short chapter is to highlight some of the most important (in the author’s view, of course) open problems.
We start with a simple but instructive algorithm for the one-dimensional stochastic setting. The next assumption is considered global throughout the chapter:
Because the optimistic Gaussian surrogate is only well-behaved on a shrinking ellipsoidal focus region, algorithms that use it are most naturally analysed in the stochastic setting, where it is already a challenge to prove that the optimal action remains in the focus region. In the adversarial setting there is limited hope to ensure the optimal action in hindsight remains in the focus region. The plan is to use a mechanism that detects when the minimiser leaves the focus region and restarts the algorithm. This is combined with an argument that the regret is negative whenever a restart occurs. The formal setting studied in this chapter is characterised by the following assumption: