The volume samples parameter handles the number of samples used to integrate the in-scattering from direct light. Like the "samples" parameter for surfaces, it is also a squared number.
Controls the quality of the noise in the soft shadows and direct specular highlight. The higher the number of samples, the lower the noise, and the longer it takes to render. The exact number of shadow rays sent to the light is the square of this value multiplied by the AA samples.
Noise from lights can sometimes be difficult to diagnose, particularly if the light source is broad in comparison to the scene and the shadows have an extremely wide penumbra. In these cases, it can sometimes be mistaken for indirect diffuse noise. It highlights the necessity for testing noise raytype. This diagram shows how a light is traced in Arnold.
The example below shows specular highlights from area light sources. There are four spherical mesh light sources of varying size and color temperature. Underneath are four cubes with Standard shaders assigned to them with varying degrees of specular roughness. Note that more noise is apparent with smaller light sources. Increasing the number of light samples resolves the noise.
If the issue is noise in a specular highlight, you will need to confirm that the source is the direct light and not a secondary ray type (such as specular). This is easy to achieve by turning off global illumination; set the Diffuse Depth and Specular Depth to zero (this essentially turns off all global illumination). If the noise is still there, we know it is the specular component of the illumination model. If the issue is shadow noise, then we can simply toggle ignore shadows in the Arnold render settings, and the noise will completely resolve.
The key, again, is to modify the sampling and observe the changes. Increasing the number of light samples should have an immediate and quantitive effect on the smoothness of the specular highlight and the shadow. If there is no change, light samples are not responsible for the noise.
Multiple importance sampling (MIS) is enabled by default in Arnold. The images below show the difference when rendering with and without multiple importance sampling.