The formal language for mathematical expressions to which our certification algorithm is applied is specified by the grammar depicted in Figure 1. The language is rich enough to cover all the examples in the main paper and this supplement. In this grammar, number is a placeholder for an arbitrary floating point number, variable is a placeholder for variable names starting with a Latin character and function is a placeholder for the supported elementary differentiable functions like exp,log and sum. Here, is used for transposition and a preceding . Here are some examples from the language (the fist example uses a transposition and the fifth and seventh example use elementwise operations): 2-norm Xw y 2: (X*w-y)'*(X*w-y) logistic log(1+exp(x)): log(1+exp(x)) 1 quadratic x2: x^2 relative entropy xlog(x/y): x*log(x/y), x>0, y>0 logistic regression Our implementation of the Hessian approach works on vectorized and normalized expression DAGs (directed acyclic graphs) for Hessians that contain every subexpression exactly once.

The Hessian of a differentiable convex function is positive semidefinite. Therefore, checking the Hessian of a given function is a natural approach to certify convexity. However, implementing this approach is not straightforward since it requires a representation of the Hessian that allows its analysis. Here, we implement this approach for a class of functions that is rich enough to support classical machine learning. For this class of functions, it was recently shown how to compute computational graphs of their Hessians. We show how to check these graphs for positive semidefiniteness. We compare our implementation of the Hessian approach with the well-established disciplined convex programming (DCP) approach and prove that the Hessian approach is at least as powerful as the DCP approach for differentiable functions. Furthermore, we show for a state-of-the-art implementation of the DCP approach that, for differentiable functions, the Hessian approach is actually more powerful. That is, it can certify the convexity of a larger class of differentiable functions.

Compositional generalization refers to a model's capability to generalize to newly composed input data based on the data components observed during training. It has triggered a series of compositional generalization analysis on different tasks as generalization is an important aspect of language and problem solving skills. However, the similar discussion on math word problems (MWPs) is limited. In this manuscript, we study compositional generalization in MWP solving. Specifically, we first introduce a data splitting method to create compositional splits from existing MWP datasets. Meanwhile, we synthesize data to isolate the effect of compositions. To improve the compositional generalization in MWP solving, we propose an iterative data augmentation method that includes diverse compositional variation into training data and could collaborate with MWP methods. During the evaluation, we examine a set of methods and find all of them encounter severe performance loss on the evaluated datasets. We also find our data augmentation method could significantly improve the compositional generalization of general MWP methods. Code is available at https://github.com/demoleiwang/CGMWP.