A. gain versus sigma*sqrt(2 KL)
  Gaussian N(0,1)
    lambda=0.5  KL= 0.1250 gain= 0.5000 bound= 0.5000
    lambda=1.0  KL= 0.5002 gain= 1.0002 bound= 1.0002
    lambda=2.0  KL= 1.9976 gain= 1.9989 bound= 1.9988
    lambda=3.0  KL= 4.4176 gain= 2.9691 bound= 2.9724
  Uniform[-1,1]
    lambda=0.5  KL= 0.0406 gain= 0.1638 bound= 0.2850
    lambda=1.0  KL= 0.1514 gain= 0.3127 bound= 0.5503
    lambda=2.0  KL= 0.4791 gain= 0.5369 bound= 0.9789
    lambda=3.0  KL= 0.8090 gain= 0.6713 bound= 1.2720

B. true and proxy reward along the family pi_beta (closed form), with the exploit mass
  beta       KL  proxy gain  true reward  exploit mass
    20    0.002       0.070        0.045        0.0013
     5    0.026       0.269        0.186        0.0032
     3    0.076       0.455        0.304        0.0069
     2    0.186       0.716        0.424        0.0169
   1.5    0.375       1.039        0.481        0.0398
   1.2    0.706       1.477        0.409        0.0878
   1.0    1.275       2.094        0.111        0.1777
   0.9    1.829       2.618       -0.252        0.2666
   0.8    2.706       3.361       -0.878        0.4053
   0.7    3.959       4.298       -1.807        0.5961
   0.6    5.342       5.198       -2.840        0.7953
   0.5    6.349       5.756       -3.583        0.9304
   0.4    6.785       5.957       -3.903        0.9850
   0.3    6.893       5.996       -3.984        0.9979

maximum true reward 0.481 at beta = 1.481, KL = 0.388
sample check beta=1.0: true reward 0.104 (closed form 0.111), exploit mass 0.1781 (0.1777)
sample check beta=0.6: true reward -2.850 (closed form -2.840), exploit mass 0.7964 (0.7953)
