We show that for constraint satisfaction problems (CSPs), sub-exponential size linear programming relaxations are as powerful as nΩ(1)-rounds of the Sherali-Adams linear programming hierarchy. As a corollary, we obtain sub-exponential size lower bounds for linear programming relaxations that beat random guessing for many CSPs such as MAX-CUT and MAX-3SAT. This is a nearly-exponential improvement over previous results; previously, the best known lower bounds were quasi-polynomial in n (Chan, Lee, Raghavendra, Steurer 2013). Our bounds are obtained by exploiting and extending the recent progress in communication complexity for "lifting" query lower bounds to communication problems. The main ingredient in our results is a new structural result on "high-entropy rectangles" that may of independent interest in communication complexity.