Sensitivity to the scaling of variables about synthdid HOT 2 OPEN

Thanks for bringing this to our attention and for the clear example.

To clarify, attr(tau_hat, "weights")$vals is the sequence of values of the loss function we minimize estimate the unit weights omega, the time weights lambda, and the regression coefficients beta of Y on X
that we see as we run the optimizer --- it's not the weights themselves. It's the curve synthdid_rmse_plot(tau_hat) displays to help diagnose problems like this one.

What seems to be happening is that when the covariates are large, the gradient steps we take when estimating beta are so large that they increase our loss function. The scaling issue comes up because the step size used by sc.weight.fw.covariates is from a pre-specified schedule --- it takes steps of the form beta_{t+1} = beta_t + (1/t)*gradient --- instead of something more appropriate to the scale of the covariates and therefore of the gradient. We should be able to fix this by incorporating some curvature information, either by using something like a quasi-newton step instead of a gradient step or choosing the step size via line search. Let's look into doing this.

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Thanks Taro and David. I am experiencing a similar issue. Would you recommend as an interim solution just doing a linear transformation - e.g. dividing covariates by some "large" constant?

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