What Cannot be Learned with Bethe Approximations
This work addresses a fundamental limitation in machine learning for probabilistic models, highlighting when Bethe learning fails, which is incremental but clarifies a known bottleneck.
The paper tackles the problem of learning parameters in graphical models when inference is intractable, showing that Bethe approximations can fail to recover empirical marginals in certain regimes, with conditions provided for learnable marginals.
We address the problem of learning the parameters in graphical models when inference is intractable. A common strategy in this case is to replace the partition function with its Bethe approximation. We show that there exists a regime of empirical marginals where such Bethe learning will fail. By failure we mean that the empirical marginals cannot be recovered from the approximated maximum likelihood parameters (i.e., moment matching is not achieved). We provide several conditions on empirical marginals that yield outer and inner bounds on the set of Bethe learnable marginals. An interesting implication of our results is that there exists a large class of marginals that cannot be obtained as stable fixed points of belief propagation. Taken together our results provide a novel approach to analyzing learning with Bethe approximations and highlight when it can be expected to work or fail.