Correspondence between m and PyTorch call form
| m | call |
|---|---|
| 24 | argsort(rand((batch_count,K))) |
| 32 | argsort(randint(low=-2**31,high=2**31-1,(batch_count,K),dtype=integer32)) |
| 64 | argsort(randint(low=-2**63,high=2**63-1,(batch_count,K),dtype=integer64)) |
Example 1:
Call Validities for log2(K)=15.0000, P_crit=0.55, log2(N)=0.0000
| m | valid | max allowed log2(K) |
|---|---|---|
| 24 | ✓ | 15.3384 |
| 32 | ✓ | 20.6717 |
| 64 | ✓ | 42.0051 |
Given parameters P_crit=0.55 and N=1 a length 32768 permutation is allowed for any bit count random number.
Example 2:
Call Validities for log2(K)=15.0000, P_crit=0.55, log2(N)=16.0000
| m | valid | max allowed log2(K) |
|---|---|---|
| 24 | ✗ | 10.0051 |
| 32 | ✓ | 15.3384 |
| 64 | ✓ | 36.6717 |
Given parameters P_crit=0.55 and N=65536 a length 32768 permutation is not allowed for bit count 24, but is allowed for bit counts 32 and 64 random numbers.
Example 3. Explore the limits of using randint with 64 bits:
Call Validities for log2(K)=25.0000, P_crit=0.55, log2(N)=26.0000*
| m | valid | max allowed log2(K) |
|---|---|---|
| 24 | ✗ | 6.6717 |
| 32 | ✗ | 12.0051 |
| 64 | ✓ | 33.3384 |
Call Validities for log2(K)=25.0000, P_crit=0.50001, log2(N)=26.0000
| m | valid | max allowed log2(K) |
|---|---|---|
| 24 | ✗ | -1.5226 |
| 32 | ✗ | 3.8107 |
| 64 | ✓ | 25.1441 |
Call Validities for log2(K)=25.0000, P_crit=0.500001, log2(N)=26.0000
| m | valid | max allowed log2(K) |
|---|---|---|
| 24 | ✗ | -3.7372 |
| 32 | ✗ | 1.5961 |
| 64 | ✗ | 22.9295 |
It is evident that used 64-bit random numbers will produce unbiased result for all but the most extreme cases.