How to improve self-learning as a PhD student? As a doctoral student, you take some classes, but most of what you do needs material that you must gather from papers, conferences, and other sources. With limited possibilities for communication with authors and such, what's the best way to learn these necessary concepts?

For example, I read a paper (from a big name publication) presenting a novel approach to SVMs with new constraints on the objective function. The description of the optimization was vague, but I needed to use this technique for my own work and my optimization skills are limited at best. I'm sure to a reviewer with the right expertise, the description was fine though.

As a PhD student, one can only take so many courses in so many subjects. Nevertheless, some things (like optimization theory in the above example) are not exactly easily self-taught, and surely not in a few weeks.

So I ask you, what is the best strategy to fill these gaps without devoting entire courses to the subject?

Additionally, how do you do that without just learning things piecemeal? Learning something useful for a hyper-specific case is essentially worthless once circumstances change. How do you balance the need to know both these hyper-specific examples and the more general ideas while doing this gap-filling?
During my Ph.D. studies I had to do exactly that: balance my time between learning specific examples and learning where and how to find these examples without wasting too much time. In my case, I did not have that much guidance from my supervisor (he let me be pretty independent) and on top of that my topic was interdisciplinary. Due to the latter, there were gaps in my learning that I had to fill, and taking 20 ultra-specialized courses was not an option because of the time constraint (I had to finish my PhD within 4 years and also publish some papers in that time). 


From my experience, a top-down approach is the most effective one. It sounds like you started reading a very specialized paper; try reading something more broad about the subject, ideally a tutorial/video/article/book chapter with emphasis on applications (since you mentioned you need to apply a method to your own work) that will give you a good overview in a shorter amount of time. Then, not only you will gain more insight as to which method is the appropriate one to use in your research and can devote more time to it, but you will also have kept in the back of your head the other ones, in case conditions change and you need to switch methods. Online courses (MOOCs) are a good source of information, as you can adjust them to your time schedule and often are not a semester long but rather a few weeks. I am sure you can find something on what you are looking for on www.edx.org and/or www.coursera.com.
Communicate with people around you. This is crucial. If you are not already in an environment where knowledgeable people are available, put in the effort to find one. Look up people and email them; most of the time professors are passionate about helping out students. I was lucky to have an office one door down from a retired professor, who had time to talk and share his expertise and advice - some insights are difficult to find in books. 


Try to get creative with your learning - never is one source enough, try to diversify. It is easy to become overwhelmed by the information, so be organized, take notes, and try to be aware of what you are learning and how it fits in your purpose, with a calm mind. Take breaks when you feel overworked (trust me, this comes from experience).

From the organizational viewpoint, I think sketching out objectives every month or week, and then preparing a summary of what you learned and how it applies to your research (like a progress report) helps a lot in seeing the big picture. You don't want to get carried away too much reading about things you won't use. Of course, that's inevitable, as it is part of research - it is part of navigating in the vast realm of knowledge, sometimes you end up sidestepping.