While online algorithms for SVMs do exist, it has become important to specify if you want kernel or linear SVMs, as many efficient algorithms have been developed for the special case of linear SVMs.
For the linear case, if you use the SGD classifier in scikit-learn with the hinge loss and L2 regularization you will get an SVM that can be updated online/incrementall. You can combine this with feature transforms that approximate a kernel to get similar to an online kernel SVM.
One of my specifications is that it should continuously update to changing trends.
This is referred to as concept drift, and will not be handled well by a simple online SVM. Using the PassiveAggresive classifier will likely give you better results, as it's learning rate does not decrease over time.
Assuming you get feedback while training / running, you can attempt to detect decreases in accuracy over time and begin training a new model when the accuracy starts to decrease (and switch to the new one when you believe that it has become more accurate). JSAT has 2 drift detection methods (see jsat.driftdetectors) that can be used to track accuracy and alert you when it has changed.
It also has more online linear and kernel methods.
(bias note: I'm the author of JSAT).
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