Application of derivatives flashcards

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This document discusses key concepts in derivatives and optimization. It covers local minimums occurring when the second derivative is negative, indicating a concave down graph. Inflection points are identified by a change in concavity. Optimization problems aim to find minimums and maximums.

2 When the second derivative yields a negative number….

In recent years, the generated and collected data is increasing at an almost exponential rate. At the same time, the data’s value has been identified in terms of insights that can be provided. However, retrieving the value requires powerful analysis tools, since valuable insights are buried deep in large amounts of noise. Unfortunately, analytic capacities did not scale well with the growing data. Many existing tools run only on a single computer and are limited in terms of data size by its memory. A very promising solution to deal with large-scale data is scaling systems and exploiting parallelism. In this presentation, we propose Gilbert, a distributed sparse linear algebra system, to decrease the imminent lack of analytic capacities. Gilbert offers a MATLAB-like programming language for linear algebra programs, which are automatically executed in parallel. Transparent parallelization is achieved by compiling the linear algebra operations first into an intermediate representation. This language-independent form enables high-level algebraic optimizations. Different optimization strategies are evaluated and the best one is chosen by a cost-based optimizer. The optimized result is then transformed into a suitable format for parallel execution. Gilbert generates execution plans for Apache Spark and Apache Flink, two massively parallel dataflow systems. Distributed matrices are represented by square blocks to guarantee a well-balanced trade-off between data parallelism and data granularity. An exhaustive evaluation indicates that Gilbert is able to process varying amounts of data exceeding the memory of a single computer on clusters of different sizes. Two well known machine learning (ML) algorithms, namely PageRank and Gaussian non-negative matrix factorization (GNMF), are implemented with Gilbert. The performance of these algorithms is compared to optimized implementations based on Spark and Flink. Even though Gilbert is not as fast as the optimized algorithms, it simplifies the development process significantly due to its high-level programming abstraction.

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1 Introduction Monte Carlo methods (or Monte Carlo experiments) are a broad class of computa- tional algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. Monte Carlo methods vary, but tend to follow a particular pattern • Define a domain of possible inputs • Generate inputs randomly from a probability distribution over the domain • Perform a deterministic computation on the inputs • Aggregate the results

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