[Training Node] algo support, grad acc, optional grad ckpt (#9015)

* Add factorization utils for lokr

* Add lokr train impl

* Add loha train impl

* Add adapter map for algo selection

* Add optional grad ckpt and algo selection

* Update __init__.py

* correct key name for loha

* Use custom fwd/bwd func and better init for loha

* Support gradient accumulation

* Fix bugs of loha

* use more stable init

* Add OFT training

* linting

comfy/weight_adapter/base.py

@@ -133,3 +133,43 @@ def tucker_weight_from_conv(up, down, mid):
 def tucker_weight(wa, wb, t):
     temp = torch.einsum("i j ..., j r -> i r ...", t, wb)
     return torch.einsum("i j ..., i r -> r j ...", temp, wa)
+
+
+def factorization(dimension: int, factor: int = -1) -> tuple[int, int]:
+    """
+    return a tuple of two value of input dimension decomposed by the number closest to factor
+    second value is higher or equal than first value.
+
+    examples)
+    factor
+        -1               2                4               8               16               ...
+    127 -> 1, 127   127 -> 1, 127    127 -> 1, 127   127 -> 1, 127   127 -> 1, 127
+    128 -> 8, 16    128 -> 2, 64     128 -> 4, 32    128 -> 8, 16    128 -> 8, 16
+    250 -> 10, 25   250 -> 2, 125    250 -> 2, 125   250 -> 5, 50    250 -> 10, 25
+    360 -> 8, 45    360 -> 2, 180    360 -> 4, 90    360 -> 8, 45    360 -> 12, 30
+    512 -> 16, 32   512 -> 2, 256    512 -> 4, 128   512 -> 8, 64    512 -> 16, 32
+    1024 -> 32, 32  1024 -> 2, 512   1024 -> 4, 256  1024 -> 8, 128  1024 -> 16, 64
+    """
+
+    if factor > 0 and (dimension % factor) == 0 and dimension >= factor**2:
+        m = factor
+        n = dimension // factor
+        if m > n:
+            n, m = m, n
+        return m, n
+    if factor < 0:
+        factor = dimension
+    m, n = 1, dimension
+    length = m + n
+    while m < n:
+        new_m = m + 1
+        while dimension % new_m != 0:
+            new_m += 1
+        new_n = dimension // new_m
+        if new_m + new_n > length or new_m > factor:
+            break
+        else:
+            m, n = new_m, new_n
+    if m > n:
+        n, m = m, n
+    return m, n