Find the phase shift of the graph of $y = 2 \sin \left( 2x + \frac{\pi}{3} \right).$
Since the graph of $y = 2 \sin \left( 2x + \frac{\pi}{3} \right)$ is the same as the graph of $y = 2 \sin 2x$ shifted $\frac{\pi}{6}$ units to the left, the phase shift is $\boxed{-\frac{\pi}{6}}.$

[asy]import TrigMacros;

size(400);

real g(real x)
{
	return 2*sin(2*x + pi/3);
}

real f(real x)
{
	return 2*sin(2*x);
}

draw(graph(g,-2*pi,2*pi,n=700,join=operator ..),red);
draw(graph(f,-2*pi,2*pi,n=700,join=operator ..));
trig_axes(-2*pi,2*pi,-3,3,pi/2,1);
layer();
rm_trig_labels(-4,4, 2);
[/asy]