For original PSGD repo, see psgd_torch.
For PyTorch Kron version, see kron_torch.
Implementations of PSGD optimizers in JAX (optax-style). PSGD is a second-order optimizer originally created by Xi-Lin Li that uses either a hessian-based or whitening-based (gg^T) preconditioner and lie groups to improve training convergence, generalization, and efficiency. I highly suggest taking a look at Xi-Lin's PSGD repo's readme linked to above for interesting details on how PSGD works and experiments using PSGD. There are also paper resources listed near the bottom of this readme.
The most versatile and easy-to-use PSGD optimizer is kron, which uses a Kronecker-factored
preconditioner. It has less hyperparameters that need tuning than adam, and can generally act as a
drop-in replacement.
pip install psgd-jaxKron schedules the preconditioner update probability by default to start at 1.0 and anneal to 0.03 at the beginning of training, so training will be slightly slower at the start but will speed up by around 4k steps.
For basic usage, use kron optimizer like any other optax optimizer:
from psgd_jax.kron import kron
optimizer = kron()
opt_state = optimizer.init(params)
updates, opt_state = optimizer.update(grads, opt_state)
params = optax.apply_updates(params, updates)Basic hyperparameters:
TLDR: Learning rate and weight decay act similarly to adam's, start with adam-like settings and go from there. Maybe use slightly lower learning rate (like /2). There is no b2 or epsilon.
These next 3 settings control whether a dimension's preconditioner is diagonal or triangular.
For example, for a layer with shape (256, 128), triagular preconditioners would be shapes (256, 256)
and (128, 128), and diagonal preconditioners would be shapes (256,) and (128,). Depending on how
these settings are chosen, kron can balance between memory/speed and effectiveness. Defaults lead
to most precoditioners being triangular except for 1-dimensional layers and very large dimensions.
max_size_triangular: Any dimension with size above this value will have a diagonal preconditioner.
min_ndim_triangular: Any tensor with less than this number of dims will have all diagonal
preconditioners. Default is 2, so single-dim layers like bias and scale will use diagonal
preconditioners.
memory_save_mode: Can be None, 'one_diag', or 'all_diag'. None is default and lets all
preconditioners be triangular. 'one_diag' sets the largest or last dim per layer as diagonal
using np.argsort(shape)[::-1][0]. 'all_diag' sets all preconditioners to be diagonal.
preconditioner_update_probability: Preconditioner update probability uses a schedule by default
that works well for most cases. It anneals from 1 to 0.03 at the beginning of training, so training
will be slightly slower at the start but will speed up by around 4k steps. PSGD generally benefits
from more preconditioner updates at the start of training, but once the preconditioner is learned
it's okay to do them less often. An easy way to adjust update frequency is to define your own schedule
using the precond_update_prob_schedule function in kron.py (just changing the min_prob value
is easiest) and pass this into kron through the preconditioner_update_probability hyperparameter.
This is the default schedule defined in the precond_update_prob_schedule function at the top of kron.py:
Sharding:
Kron contains einsums, and in general the first axis of the preconditioner matrices is the contracting axis.
If using only FSDP, I usually shard the last axis of each preconditioner matrix and call it good.
However, if using tensor parallelism in addition to FSDP, you may think more carefully about how the preconditioners are sharded in train_state. For example, with grads of shape (256, 128) and kron preconditioners of shapes (256, 256) and (128, 128), if the grads are sharded as (fsdp, tensor), then you may want to shard the (256, 256) preconditioner as (fsdp, tensor) and the (128, 128) preconditioner as (tensor, fsdp) so the grads and its preconditioners have similar contracting axes.
Scanned layers:
If you are scanning layers in your network, you can also have kron scan over these layers while
updating and applying the preconditioner. Simply pass in a pytree through scanned_layers with
the same structure as your params with bool values indicating which layers are scanned. PSGD will
vmap over the first dims of those layers. If you need a more advanced scanning setup, please open
an issue.
For very large models, the preconditioner update may use too much memory all at once when scanning,
in which case you can set lax_map_scanned_layers to True and set lax_map_batch_size to a
reasonable batch size for your setup (lax.map scans over batches of vmap, see JAX docs). If
your net is 32 layers and you're hitting OOM during the optimizer step, you can break the model into
2 or 4 and set lax_map_batch_size to 16 or 8 respectively.
Other forms of PSGD include XMat, LRA, and Affine. PSGD defaults to a gradient whitening type preconditioner (gg^T). In this case, you can use PSGD like any other optax optimizer:
import jax
import jax.numpy as jnp
import optax
from psgd_jax.xmat import xmat # or low_rank_approximation, affine
def loss_fn(params, x):
return jnp.sum((params - x) ** 2)
params = jnp.array([1.0, 2.0, 3.0])
x = jnp.array([0.0, 0.0, 0.0])
# make optimizer and init state
opt = xmat(
learning_rate=1.0,
b1=0.0,
preconditioner_update_probability=1.0, # preconditioner update frequency
)
opt_state = opt.init(params)
def step(params, x, opt_state):
loss_val, grad = jax.value_and_grad(loss_fn)(params, x)
updates, opt_state = opt.update(grad, opt_state)
params = optax.apply_updates(params, updates)
return params, opt_state, loss_val
while True:
params, opt_state, loss_val = step(params, x, opt_state)
print(loss_val)
if loss_val < 1e-4:
print("yay")
break
# Expected output:
# 14.0
# 5.1563816
# 1.7376599
# 0.6118454
# 0.18457186
# 0.056664664
# 0.014270116
# 0.0027846962
# 0.00018843572
# 4.3836744e-06
# yayHowever, PSGD can also be used with a hessian vector product. If values are provided for PSGD's extra
update function arguments Hvp, vector, and update_preconditioner, PSGD automatically
uses hessian-based preconditioning. Hvp is the hessian vector product, vector is the random
vector used to calculate the hessian vector product, and update_preconditioner is a boolean
that tells PSGD whether we're updating the preconditioner this step (passed in real hvp and
vector) or not (passed in dummy hvp and vector).
The hessian_helper function can help with this and generally replace jax.value_and_grad:
import jax
import jax.numpy as jnp
import optax
from psgd_jax.xmat import xmat # or low_rank_approximation, affine
from psgd_jax import hessian_helper
def loss_fn(params, x):
return jnp.sum((params - x) ** 2)
params = jnp.array([1.0, 2.0, 3.0])
x = jnp.array([0.0, 0.0, 0.0])
# make optimizer and init state
# no need to set 'preconditioner_update_probability' here, it's handled by hessian_helper
opt = xmat(
learning_rate=1.0,
b1=0.0,
)
opt_state = opt.init(params)
def step(key, params, x, opt_state):
# replace jax.value_and_grad with the hessian_helper:
key, subkey = jax.random.split(key)
loss_fn_out, grad, hvp, vector, update_precond = hessian_helper(
subkey,
loss_fn,
params,
loss_fn_extra_args=(x,),
has_aux=False,
preconditioner_update_probability=1.0, # update frequency handled in hessian_helper
)
loss_val = loss_fn_out
# Pass hvp, random vector, and whether we're updating the preconditioner
# this step into the update function. PSGD will automatically switch to
# hessian-based preconditioning when these are provided.
updates, opt_state = opt.update(
grad,
opt_state,
Hvp=hvp,
vector=vector,
update_preconditioner=update_precond
)
params = optax.apply_updates(params, updates)
return key, params, opt_state, loss_val
key = jax.random.PRNGKey(0)
while True:
key, params, opt_state, loss_val = step(key, params, x, opt_state)
print(loss_val)
if loss_val < 1e-4:
print("yay")
break
# Expected output:
# 14.0
# 7.460699e-14
# yayIf preconditioner_update_probability is lowered, time is saved by calculating the hessian less
often, but convergence could be slower.
psgd_jax.kron - psgd_jax.xmat - psgd_jax.low_rank_approximation - psgd_jax.affine
There are four variants of PSGD: Kron, which uses Kronecker-factored preconditioners for tensors of any number of dimensions, XMat, which uses an x-shaped global preconditioner, LRA, which uses a low-rank approximation global preconditioner, and Affine, which uses kronecker-factored preconditioners for matrices.
Kron:
Kron uses Kronecker-factored preconditioners for tensors of any number of dimensions. It's very versatile, has less hyperparameters that need tuning than adam, and can generally act as a drop-in replacement for adam.
XMat:
XMat is very simple to use, uses global hessian information for its preconditioner, and has memory use of only n_params * 3 (including momentum which is optional, set b1 to 0 to disable).
LRA:
Low rank approximation uses a low rank hessian for its preconditioner and can give very strong results. It has memory use of n_params * (2 * rank + 1) (n_params * (2 * rank) without momentum).
Affine:
Affine does not use global hessian information, but can be powerful nonetheless and possibly use
less memory than xmat, LRA, or adam. max_size_triangular and max_skew_triangular determine whether
a dimension's preconditioner is triangular or diagonal. Affine and Kron are nearly identical for matrices.
PSGD papers and resources listed from Xi-Lin's repo
- Xi-Lin Li. Preconditioned stochastic gradient descent, arXiv:1512.04202, 2015. (General ideas of PSGD, preconditioner fitting losses and Kronecker product preconditioners.)
- Xi-Lin Li. Preconditioner on matrix Lie group for SGD, arXiv:1809.10232, 2018. (Focus on preconditioners with the affine Lie group.)
- Xi-Lin Li. Black box Lie group preconditioners for SGD, arXiv:2211.04422, 2022. (Mainly about the LRA preconditioner. See these supplementary materials for detailed math derivations.)
- Xi-Lin Li. Stochastic Hessian fittings on Lie groups, arXiv:2402.11858, 2024. (Some theoretical works on the efficiency of PSGD. The Hessian fitting problem is shown to be strongly convex on set
${\rm GL}(n, \mathbb{R})/R_{\rm polar}$ .) - Omead Pooladzandi, Xi-Lin Li. Curvature-informed SGD via general purpose Lie-group preconditioners, arXiv:2402.04553, 2024. (Plenty of benchmark results and analyses for PSGD vs. other optimizers.)
This work is licensed under a Creative Commons Attribution 4.0 International License.
2024 Evan Walters, Omead Pooladzandi, Xi-Lin Li