We develop two fundamental stochastic sketching techniques; Penalty Sketching (PS) and Augmented Lagrangian Sketching (ALS) for solving consistent linear systems. The proposed PS and ALS techniques extend and generalize the scope of Sketch & Project (SP) method by introducing Lagrangian penalty sketches. In doing so, we recover SP methods as special cases and furthermore develop a family of new stochastic iterative methods. By varying sketch parameters in the proposed PS method, we recover novel stochastic methods such as Penalty Newton Descent, Penalty Kaczmarz, Penalty Stochastic Descent, Penalty Coordinate Descent, Penalty Gaussian Pursuit, and Penalty Block Kaczmarz. Furthermore, the proposed ALS method synthesizes a wide variety of new stochastic methods such as Augmented Newton Descent, Augmented Kaczmarz, Augmented Stochastic Descent, Augmented Coordinate Descent, Augmented Gaussian Pursuit, and Augmented Block Kaczmarz into one framework. Moreover, we show that the developed PS and ALS frameworks can be used to reformulate the original linear system into equivalent stochastic optimization problems namely the Penalty Stochastic Reformulation and Augmented Stochastic Reformulation. We prove global convergence rates for the PS and ALS methods as well as sub-linear $\mathcal{O}(\frac{1}{k})$ rates for the Cesaro average of iterates. The proposed convergence results hold for a wide family of distributions of random matrices, which provides the opportunity of fine-tuning the randomness of the method suitable for specific applications. Finally, we perform computational experiments that demonstrate the efficiency of our methods compared to the existing SP methods.

Classification is a task of grouping things together on the basis of the similarity they share with each other. It helps organize things and thus makes the study more easy and systematic. In statistics, classification refers to the problem of identifying to which set of categories an observation or data value belongs to. For humans, it can be very easy to do the classification task assuming that he/she has proper domain-specific knowledge and given certain features he/she can achieve it by no means. But, it can be tricky for a machine to classify -- unless it is provided with proper training from the data and algorithm (classifier) that is used for learning.

This three part write up [Part II Part III] is my attempt at a down-to-earth explanation (and Python code) of the Holt-Winters method for those of us who while hypothetically might be quite good at math, still try to avoid it at every opportunity. I had to dive into this subject while tinkering on tgres (which features a Golang implementation). And having found it somewhat complex (and yet so brilliantly simple), figured that it'd be good to share this knowledge, and in the process, to hopefully solidify it in my head as well. Triple Exponential Smoothing, also known as the Holt-Winters method, is one of the many methods or algorithms that can be used to forecast data points in a series, provided that the series is "seasonal", i.e. repetitive over some period. In 1957 an MIT and University of Chicago graduate, professor Charles C Holt (1921-2010) was working at CMU (then known as CIT) on forecasting trends in production, inventories and labor force.