How to set common axes labels for subplots

plt.setp() will do the job:

# plot something
fig, axs = plt.subplots(3,3, figsize=(15, 8), sharex=True, sharey=True)
for i, ax in enumerate(axs.flat):
    ax.scatter(*np.random.normal(size=(2,200)))
    ax.set_title(f'Title {i}')

# set labels
plt.setp(axs[-1, :], xlabel='x axis label')
plt.setp(axs[:, 0], ylabel='y axis label')

New in matplotlib 3.4.0

There are now built-in methods to set common axis labels:

• supxlabel

  fig.supxlabel('common x label')
  

• supylabel

  fig.supylabel('common y label')
  

To reproduce OP's loglog plots (common labels but separate titles):

x = np.arange(0.01, 10.01, 0.01)
y = 2 ** x

fig, (ax1, ax2) = plt.subplots(2, 1, constrained_layout=True)
ax1.loglog(y, x)
ax2.loglog(x, y)

# separate subplot titles
ax1.set_title('ax1.title')
ax2.set_title('ax2.title')

# common axis labels
fig.supxlabel('fig.supxlabel')
fig.supylabel('fig.supylabel')

# list loss and acc are your data
fig = plt.figure()
ax1 = fig.add_subplot(121)
ax2 = fig.add_subplot(122)

ax1.plot(iteration1, loss)
ax2.plot(iteration2, acc)

ax1.set_title('Training Loss')
ax2.set_title('Training Accuracy')

ax1.set_xlabel('Iteration')
ax1.set_ylabel('Loss')

ax2.set_xlabel('Iteration')
ax2.set_ylabel('Accuracy')

The methods in the other answers will not work properly when the yticks are large. The ylabel will either overlap with ticks, be clipped on the left or completely invisible/outside of the figure.

I've modified Hagne's answer so it works with more than 1 column of subplots, for both xlabel and ylabel, and it shifts the plot to keep the ylabel visible in the figure.

def set_shared_ylabel(a, xlabel, ylabel, labelpad = 0.01, figleftpad=0.05):
    """Set a y label shared by multiple axes
    Parameters
    ----------
    a: list of axes
    ylabel: string
    labelpad: float
        Sets the padding between ticklabels and axis label"""

    f = a[0,0].get_figure()
    f.canvas.draw() #sets f.canvas.renderer needed below

    # get the center position for all plots
    top = a[0,0].get_position().y1
    bottom = a[-1,-1].get_position().y0

    # get the coordinates of the left side of the tick labels
    x0 = 1
    x1 = 1
    for at_row in a:
        at = at_row[0]
        at.set_ylabel('') # just to make sure we don't and up with multiple labels
        bboxes, _ = at.yaxis.get_ticklabel_extents(f.canvas.renderer)
        bboxes = bboxes.inverse_transformed(f.transFigure)
        xt = bboxes.x0
        if xt < x0:
            x0 = xt
            x1 = bboxes.x1
    tick_label_left = x0

    # shrink plot on left to prevent ylabel clipping
    # (x1 - tick_label_left) is the x coordinate of right end of tick label,
    # basically how much padding is needed to fit tick labels in the figure
    # figleftpad is additional padding to fit the ylabel
    plt.subplots_adjust(left=(x1 - tick_label_left) + figleftpad)

    # set position of label, 
    # note that (figleftpad-labelpad) refers to the middle of the ylabel
    a[-1,-1].set_ylabel(ylabel)
    a[-1,-1].yaxis.set_label_coords(figleftpad-labelpad,(bottom + top)/2, transform=f.transFigure)

    # set xlabel
    y0 = 1
    for at in axes[-1]:
        at.set_xlabel('')  # just to make sure we don't and up with multiple labels
        bboxes, _ = at.xaxis.get_ticklabel_extents(fig.canvas.renderer)
        bboxes = bboxes.inverse_transformed(fig.transFigure)
        yt = bboxes.y0
        if yt < y0:
            y0 = yt
    tick_label_bottom = y0

    axes[-1, -1].set_xlabel(xlabel)
    axes[-1, -1].xaxis.set_label_coords((left + right) / 2, tick_label_bottom - labelpad, transform=fig.transFigure)

It works for the following example, while Hagne's answer won't draw ylabel (since it's outside of the canvas) and KYC's ylabel overlaps with the tick labels:

import matplotlib.pyplot as plt
import itertools

fig, axes = plt.subplots(3, 4, sharey='row', sharex=True, squeeze=False)
fig.subplots_adjust(hspace=.5)
for i, a in enumerate(itertools.chain(*axes)):
    a.plot([0,4**i], [0,4**i])
    a.set_title(i)
set_shared_ylabel(axes, 'common X', 'common Y')
plt.show()

Alternatively, if you are fine with colorless axis, I've modified Julian Chen's solution so ylabel won't overlap with tick labels.

Basically, we just have to set ylims of the colorless so it matches the largest ylims of the subplots so the colorless tick labels sets the correct location for the ylabel.

Again, we have to shrink the plot to prevent clipping. Here I've hard coded the amount to shrink, but you can play around to find a number that works for you or calculate it like in the method above.

import matplotlib.pyplot as plt
import itertools

fig, axes = plt.subplots(3, 4, sharey='row', sharex=True, squeeze=False)
fig.subplots_adjust(hspace=.5)
miny = maxy = 0
for i, a in enumerate(itertools.chain(*axes)):
    a.plot([0,4**i], [0,4**i])
    a.set_title(i)
    miny = min(miny, a.get_ylim()[0])
    maxy = max(maxy, a.get_ylim()[1])

# add a big axes, hide frame
# set ylim to match the largest range of any subplot
ax_invis = fig.add_subplot(111, frameon=False)
ax_invis.set_ylim([miny, maxy])

# hide tick and tick label of the big axis
plt.tick_params(labelcolor='none', top=False, bottom=False, left=False, right=False)
plt.xlabel("common X")
plt.ylabel("common Y")

# shrink plot to prevent clipping
plt.subplots_adjust(left=0.15)
plt.show()