Deep Lattice Networks (DLN)

Frequent problems that we are confronted with can be traced back to a (multi­di­men­sional) regres­sion or a classi­fi­ca­tion. Starting from a set of charac­te­ristics, regres­sion attempts to repre­sent the depen­dency between the charac­te­ristics and a target variable as a function. One example would be the depen­dence of turnover on expen­diture on research and on employee satis­fac­tion in a company. Classi­fi­ca­tion involves estimating the proba­bi­lity that an object belongs to a prede­fined class. A well-known example is the classi­fi­ca­tion of image content (dog, cat, truck, …).

The comple­xity of regres­sion and classi­fi­ca­tion problems depends on the number of features and the degree of corre­la­tion between the features and the target variable. Another aspect is the inter­de­pen­dence of the charac­te­ristics. Simple problems can be solved using classical static methods, which have the advan­tage that the results can be easily inter­preted. Problems of practical relevance are generally described by several charac­te­ristics that influence each other. A well-known and frequently used method for solving such problems is an artifi­cial neural network, which is very powerful in solving these problems.

Deep lattice networks extend the proper­ties of an artifi­cial neural network to include monoto­ni­city condi­tions between indivi­dual features and the target variable. One example of the neces­sity of this is a price model in which the price of a unit is to be reduced with the order quantity. The project, which was initiated and supported by Google developed process consists essen­ti­ally of $n$-dimen­sional Hyper­cube the edge length $1$where $n$ describes the number of charac­te­ristics. Each dimen­sion of the cube there­fore repres­ents a charac­te­ristic. A (multi­di­men­sional) function is placed in the cube, which describes the relati­onship between the charac­te­ristics and the target variable. For this purpose, function values are calcu­lated for the corner points of the cube, which repre­sent the variable parame­ters of the model, on the basis of training data. The target function is inter­po­lated linearly between the corner points. The edges of the cube can be subdi­vided to increase the detail of the function. A grid(lattice) is placed through the division points, the inter­sec­tion points of which are assigned function values as additional parame­ters of the model.

The follo­wing figure shows a lattice with two features (i.e. a square). For the four corner points $\theta [i]$ values were derived from a set of training data as parame­ters of the model. The function to be repre­sented $f(x)$ is appro­xi­mated by linear inter­po­la­tion between the four corner points. No further support points were used between the corner points to compress the grid and increase the level of detail of the appro­xi­mated function.