In machine learning models, we often need to manually set various hyperparameters such as the number of trees in random forest and learning rate in neural network. In traditional optimization problems, we can rely on gradient-based approaches to compute optimum. However, hyperparameter tuning is a black box problem and we usually do not have an expression for the objective function and we do not know its gradient. In this post, I will discuss different approaches for hyperparameter tuning and how we can learn to learn. 

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Now what we have discussed unconstrained optimization problems in previous post, it is now time to come to the reality. In the real world, we often have limitations, such as the total budget, motion angles, and some arbitrary desirable range of values. Life would be so easy (and boring) without boundary and conditions. Adding constraints certainly makes optimization problems less easy, but more interesting. 

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Give me a descent direction and a step length to move and I will find the optimum.

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Why do we care about eigenvalues, eigenvectors, and singular values?  Intuitively, what do they tell us about a matrix? When I first studied eigenvalues in college, I regarded it as yet another theoretical math trick that is hardly applicable to my life. Once I passed the final exam, I shelved all my eigen-knowledge to a corner in my memory. Years have passed, and I gradually realize the importance and brilliance of eigenvalues, particularly in the realm of machine learning. In this post, I will discuss how and why we perform eigendecomposition and singular value decomposition in machine learning.

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In machine learning, we are often dealing with high-dimension data. For convenience, we often use matrix to represent data. Numerical optimization in machine learning often involves matrix transformation and computation. To make matrix computation more efficiently, we always factorize a matrix into several special matrices such as triangular matrices and orthogonal matrices. In this post, I will review essential concepts of matrix used in machine learning. 

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During logistic regression, in order to compute the optimal parameters in the model, we have to use an iterative numerical optimization approach (Newton method or Gradient descent method, instead of a simple analytical approach). Numerical optimization is a crucial mathematical concept in machine learning and function fitting, and it is deeply integrated in model training, regularization, support vector machine, neural network, and so on. In the next few posts, I will summarize key concepts and approaches in numerical optimization, and its application in machine learning.

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In the previous post, I have discussed loss function in regression. In this post, I will elaborate how we develop loss function in classification when the output is discrete, rather than continuous. 

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