It has been proved that (1) it guarantees that it is safe for BLasso to start with an initial α which is the largest α that would allow an ε step away from 0 (i.e., larger α's correspond to T(λ)=0); (2) for each value of α, BLasso performs coordinate descent (i.e., reduces Ex-pLoss by updating the weight of a feature) until there is no descent step; and (3) for each step where the value of α decreases, it guarantees that the lasso loss is reduced.
it can be proved that for a finite number of features and θ = 0, the BLasso algorithm shown in Figure 2 converges to the lasso solution when ε 0.