MAML Tutorial with JAX¶
Eric Jang
Blog post: https://blog.evjang.com/2019/02/maml-jax.html
21 Feb 2019
Pedagogical tutorial for implementing Model-Agnostic Meta-Learning with JAX's awesome grad and vmap and jit operators.
Overview¶
In this notebook we'll go through:
- how to take gradients, gradients of gradients.
- how to fit a sinusoid function with a neural network (and do auto-batching with vmap)
- how to implement MAML and check its numerics
- how to implement MAML for sinusoid task (single-task objective, batching task instances).
- extending MAML to handle batching at the task-level
In [1]:
# import jax.numpy (almost-drop-in for numpy) and gradient operators.
import jax.numpy as np
from jax import grad
Gradients of Gradients¶
JAX makes it easy to compute gradients of python functions. Here, we thrice-differentiate $e^x$ and $x^2$
In [2]:
f = lambda x : np.exp(x)
g = lambda x : np.square(x)
print(grad(f)(1.)) # = e^{1}
print(grad(grad(f))(1.))
print(grad(grad(grad(f)))(1.))
print(grad(g)(2.)) # 2x = 4
print(grad(grad(g))(2.)) # x = 2
print(grad(grad(grad(g)))(2.)) # x = 0
Sinusoid Regression and vmap¶
To get you familiar with JAX syntax first, we'll optimize neural network params with fixed inputs on a mean-squared error loss to $f_\theta(x) = sin(x)$.
In [3]:
from jax import vmap # for auto-vectorizing functions
from functools import partial # for use with vmap
from jax import jit # for compiling functions for speedup
from jax.experimental import stax # neural network library
from jax.experimental.stax import Conv, Dense, MaxPool, Relu, Flatten, LogSoftmax # neural network layers
import matplotlib.pyplot as plt # visualization
In [4]:
# Use stax to set up network initialization and evaluation functions
net_init, net_apply = stax.serial(
Dense(40), Relu,
Dense(40), Relu,
Dense(1)
)
in_shape = (-1, 1,)
out_shape, net_params = net_init(in_shape)
In [5]:
def loss(params, inputs, targets):
# Computes average loss for the batch
predictions = net_apply(params, inputs)
return np.mean((targets - predictions)**2)
In [6]:
# batch the inference across K=100
xrange_inputs = np.linspace(-5,5,100).reshape((100, 1)) # (k, 1)
targets = np.sin(xrange_inputs)
predictions = vmap(partial(net_apply, net_params))(xrange_inputs)
losses = vmap(partial(loss, net_params))(xrange_inputs, targets) # per-input loss
plt.plot(xrange_inputs, predictions, label='prediction')
plt.plot(xrange_inputs, losses, label='loss')
plt.plot(xrange_inputs, targets, label='target')
plt.legend()
Out[6]:
In [7]:
import numpy as onp
from jax.experimental import optimizers
from jax.tree_util import tree_multimap # Element-wise manipulation of collections of numpy arrays
In [8]:
opt_init, opt_update = optimizers.adam(step_size=1e-2)
opt_state = opt_init(net_params)
# Define a compiled update step
@jit
def step(i, opt_state, x1, y1):
p = optimizers.get_params(opt_state)
g = grad(loss)(p, x1, y1)
return opt_update(i, g, opt_state)
for i in range(100):
opt_state = step(i, opt_state, xrange_inputs, targets)
net_params = optimizers.get_params(opt_state)
In [9]:
# batch the inference across K=100
targets = np.sin(xrange_inputs)
predictions = vmap(partial(net_apply, net_params))(xrange_inputs)
losses = vmap(partial(loss, net_params))(xrange_inputs, targets) # per-input loss
plt.plot(xrange_inputs, predictions, label='prediction')
plt.plot(xrange_inputs, losses, label='loss')
plt.plot(xrange_inputs, targets, label='target')
plt.legend()
Out[9]:
MAML: Optimizing for Generalization¶
Suppose task loss function $\mathcal{L}$ is defined with respect to model parameters $\theta$, input features $X$, input labels $Y$. MAML optimizes the following:
$\mathcal{L}(\theta - \nabla \mathcal{L}(\theta, x_1, y_1), x_2, y_2)$
$x_1, y_2$ and $x_2, y_2$ are identically distributed from $X, Y$. Therefore, MAML objective can be thought of as a differentiable cross-validation error (w.r.t. $x_2, y_2$) for a model that learns (via a single gradient descent step) from $x_1, y_1$. Minimizing cross-validation error provides an inductive bias on generalization.
The following toy example checks MAML numerics via parameter $x$ and input $y$.
In [10]:
# gradients of gradients test for MAML
# check numerics
g = lambda x, y : np.square(x) + y
x0 = 2.
y0 = 1.
print('grad(g)(x0) = {}'.format(grad(g)(x0, y0))) # 2x = 4
print('x0 - grad(g)(x0) = {}'.format(x0 - grad(g)(x0, y0))) # x - 2x = -2
def maml_objective(x, y):
return g(x - grad(g)(x, y), y)
print('maml_objective(x,y)={}'.format(maml_objective(x0, y0))) # x**2 + 1 = 5
print('x0 - maml_objective(x,y) = {}'.format(x0 - grad(maml_objective)(x0, y0))) # x - (2x)
Sinusoid Task + MAML¶
Now let's re-implement the Sinusoidal regression task from Chelsea Finn's MAML paper.
In [11]:
alpha = .1
def inner_update(p, x1, y1):
grads = grad(loss)(p, x1, y1)
inner_sgd_fn = lambda g, state: (state - alpha*g)
return tree_multimap(inner_sgd_fn, grads, p)
def maml_loss(p, x1, y1, x2, y2):
p2 = inner_update(p, x1, y1)
return loss(p2, x2, y2)
In [12]:
x1 = xrange_inputs
y1 = targets
x2 = np.array([0.])
y2 = np.array([0.])
maml_loss(net_params, x1, y1, x2, y2)
Out[12]:
Let's try minimizing the MAML loss (without batching across multiple tasks, which we will do in the next section)
In [34]:
opt_init, opt_update = optimizers.adam(step_size=1e-3) # this LR seems to be better than 1e-2 and 1e-4
out_shape, net_params = net_init(in_shape)
opt_state = opt_init(net_params)
@jit
def step(i, opt_state, x1, y1, x2, y2):
p = optimizers.get_params(opt_state)
g = grad(maml_loss)(p, x1, y1, x2, y2)
l = maml_loss(p, x1, y1, x2, y2)
return opt_update(i, g, opt_state), l
K=20
np_maml_loss = []
# Adam optimization
for i in range(20000):
# define the task
A = onp.random.uniform(low=0.1, high=.5)
phase = onp.random.uniform(low=0., high=np.pi)
# meta-training inner split (K examples)
x1 = onp.random.uniform(low=-5., high=5., size=(K,1))
y1 = A * onp.sin(x1 + phase)
# meta-training outer split (1 example). Like cross-validating with respect to one example.
x2 = onp.random.uniform(low=-5., high=5.)
y2 = A * onp.sin(x2 + phase)
opt_state, l = step(i, opt_state, x1, y1, x2, y2)
np_maml_loss.append(l)
if i % 1000 == 0:
print(i)
net_params = optimizers.get_params(opt_state)
In [14]:
# batch the inference across K=100
targets = np.sin(xrange_inputs)
predictions = vmap(partial(net_apply, net_params))(xrange_inputs)
plt.plot(xrange_inputs, predictions, label='pre-update predictions')
plt.plot(xrange_inputs, targets, label='target')
x1 = onp.random.uniform(low=-5., high=5., size=(K,1))
y1 = 1. * onp.sin(x1 + 0.)
for i in range(1,5):
net_params = inner_update(net_params, x1, y1)
predictions = vmap(partial(net_apply, net_params))(xrange_inputs)
plt.plot(xrange_inputs, predictions, label='{}-shot predictions'.format(i))
plt.legend()
Out[14]:
Batching Meta-Gradient Across Tasks¶
Kind of does the job but not that great. Let's reduce the variance of gradients in outer loop by averaging across a batch of tasks (not just one task at a time).
vmap is awesome it enables nice handling of batching at two levels: inner-level "intra-task" batching, and outer level batching across tasks.
From a software engineering perspective, it is nice because the "task-batched" MAML implementation simply re-uses code from the non-task batched MAML algorithm, without losing any vectorization benefits.
In [15]:
def sample_tasks(outer_batch_size, inner_batch_size):
# Select amplitude and phase for the task
As = []
phases = []
for _ in range(outer_batch_size):
As.append(onp.random.uniform(low=0.1, high=.5))
phases.append(onp.random.uniform(low=0., high=np.pi))
def get_batch():
xs, ys = [], []
for A, phase in zip(As, phases):
x = onp.random.uniform(low=-5., high=5., size=(inner_batch_size, 1))
y = A * onp.sin(x + phase)
xs.append(x)
ys.append(y)
return np.stack(xs), np.stack(ys)
x1, y1 = get_batch()
x2, y2 = get_batch()
return x1, y1, x2, y2
In [16]:
outer_batch_size = 2
x1, y1, x2, y2 = sample_tasks(outer_batch_size, 50)
for i in range(outer_batch_size):
plt.scatter(x1[i], y1[i], label='task{}-train'.format(i))
for i in range(outer_batch_size):
plt.scatter(x2[i], y2[i], label='task{}-val'.format(i))
plt.legend()
Out[16]:
In [17]:
x2.shape
Out[17]:
In [35]:
opt_init, opt_update = optimizers.adam(step_size=1e-3)
out_shape, net_params = net_init(in_shape)
opt_state = opt_init(net_params)
# vmapped version of maml loss.
# returns scalar for all tasks.
def batch_maml_loss(p, x1_b, y1_b, x2_b, y2_b):
task_losses = vmap(partial(maml_loss, p))(x1_b, y1_b, x2_b, y2_b)
return np.mean(task_losses)
@jit
def step(i, opt_state, x1, y1, x2, y2):
p = optimizers.get_params(opt_state)
g = grad(batch_maml_loss)(p, x1, y1, x2, y2)
l = batch_maml_loss(p, x1, y1, x2, y2)
return opt_update(i, g, opt_state), l
np_batched_maml_loss = []
K=20
for i in range(20000):
x1_b, y1_b, x2_b, y2_b = sample_tasks(4, K)
opt_state, l = step(i, opt_state, x1_b, y1_b, x2_b, y2_b)
np_batched_maml_loss.append(l)
if i % 1000 == 0:
print(i)
net_params = optimizers.get_params(opt_state)
In [36]:
# batch the inference across K=100
targets = np.sin(xrange_inputs)
predictions = vmap(partial(net_apply, net_params))(xrange_inputs)
plt.plot(xrange_inputs, predictions, label='pre-update predictions')
plt.plot(xrange_inputs, targets, label='target')
x1 = onp.random.uniform(low=-5., high=5., size=(10,1))
y1 = 1. * onp.sin(x1 + 0.)
for i in range(1,3):
net_params = inner_update(net_params, x1, y1)
predictions = vmap(partial(net_apply, net_params))(xrange_inputs)
plt.plot(xrange_inputs, predictions, label='{}-shot predictions'.format(i))
plt.legend()
Out[36]:
In [48]:
# Comparison of maml_loss for task batch size = 1 vs. task batch size = 8
plt.plot(onp.convolve(np_maml_loss, [.05]*20), label='task_batch=1')
plt.plot(onp.convolve(np_batched_maml_loss, [.05]*20), label='task_batch=4')
plt.ylim(0., 1e-1)
plt.legend()
Out[48]:
