before_sent:  We validate the implementation of our framework through runtime experiments. We find our algorithms are up to 12 and 26 times faster than previous algorithms for computing the Shannon entropy and the gradient of the generalized expectation objective, respectively .
before_sent_with_intent: <clarity>  We validate the implementation of our framework through runtime experiments. We find our algorithms are up to 12 and 26 times faster than previous algorithms for computing the Shannon entropy and the gradient of the generalized expectation objective, respectively .
after_sent:  We validate the implementation of our framework through rigorous proofs of correctness and efficiency .