We propose in this paper a general framework for deriving loss functions for structured prediction. In our framework, the user chooses a convex set including the output space and provides an oracle for projecting onto that set. Given that oracle, our framework automatically generates a corresponding convex and smooth loss function. As we show, adding a projection as output layer provably makes the loss smaller. We identify the marginal polytope, the output space's convex hull, as the best convex set on which to project. However, because the projection onto the marginal polytope can sometimes be expensive to compute, we allow to use any convex superset instead, with potentially cheaper-to-compute projection. Since efficient projection algorithms are available for numerous convex sets, this allows us to construct loss functions for a variety of tasks. On the theoretical side, when combined with calibrated decoding, we prove that our loss functions can be used as a consistent surrogate for a (potentially non-convex) target loss function of interest. We demonstrate our losses on label ranking, ordinal regression and multilabel classification, confirming the improved accuracy enabled by projections.

We identify the marginal polytope, the output space's convex hull,

We consider the problem of learning the canonical parameters specifying an undirected graphical model (Markov random field) from the mean parameters. For graphical models representing a minimal exponential family, the canonical parameters are uniquely determined by the mean parameters, so the problem is feasible in principle. The goal of this paper is to investigate the computational feasibility of this statistical task. Our main result shows that parameter estimation is in general intractable: no algorithm can learn the canonical parameters of a generic pair-wise binary graphical model from the mean parameters in time bounded by a polynomial in the number of variables (unless RP = NP). Indeed, such a result has been believed to be true (see [1]) but no proof was known. Our proof gives a polynomial time reduction from approximating the partition function of the hard-core model, known to be hard, to learning approximate parameters. Our reduction entails showing that the marginal polytope boundary has an inherent repulsive property, which validates an optimization procedure over the polytope that does not use any knowledge of its structure (as required by the ellipsoid method and others).

Based on structured SVM, the authors combine the structured prediction and the learning using hinge loss, the results is a novel model, Graph Structured Prediction Energy Networks. Overall the model is novel and the theory is mostly solid. However, I have some concerns about the inference part. 1. Marginal Polytope. The relaxation of the marginal polytope is always tricky for structured prediction. A loose relaxation might result in an efficient algorithm, but the bad quality of the solution.

We propose in this paper a general framework for deriving loss functions for structured prediction. In our framework, the user chooses a convex set including the output space and provides an oracle for projecting onto that set. Given that oracle, our framework automatically generates a corresponding convex and smooth loss function. As we show, adding a projection as output layer provably makes the loss smaller. We identify the marginal polytope, the output space's convex hull, as the best convex set on which to project.

We consider the problem of learning the canonical parameters specifying an undirected graphical model (Markov random field) from the mean parameters. For graphical models representing a minimal exponential family, the canonical parameters are uniquely determined by the mean parameters, so the problem is feasible in principle. The goal of this paper is to investigate the computational feasibility of this statistical task. Our main result shows that parameter estimation is in general intractable: no algorithm can learn the canonical parameters of a generic pair-wise binary graphical model from the mean parameters in time bounded by a polynomial in the number of variables (unless RP = NP). Indeed, such a result has been believed to be true (see [1]) but no proof was known. Our proof gives a polynomial time reduction from approximating the partition function of the hard-core model, known to be hard, to learning approximate parameters. Our reduction entails showing that the marginal polytope boundary has an inherent repulsive property, which validates an optimization procedure over the polytope that does not use any knowledge of its structure (as required by the ellipsoid method and others).

We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the marginal polytope through repeated maximum a posteriori (MAP) calls. This modular structure enables us to leverage black-box MAP solvers (both exact and approximate) for variational inference, and obtains more accurate results than tree-reweighted algorithms that optimize over the local consistency relaxation. Theoretically, we bound the sub-optimality for the proposed algorithm despite the TRW objective having unbounded gradients at the boundary of the marginal polytope. Empirically, we demonstrate the increased quality of results found by tightening the relaxation over the marginal polytope as well as the spanning tree polytope on synthetic and real-world instances.

A number of exact and approximate methods are available for inference calculations in graphical models. Many recent approximate methods for graphs with cycles are based on tractable algorithms for tree structured graphs. Here we base the approximation on a different tractable model, planar graphs with binary variables and pure interaction potentials (no external field). The partition function for such models can be calculated exactly using an algorithm introduced by Fisher and Kasteleyn in the 1960s. We show how such tractable planar models can be used in a decomposition to derive upper bounds on the partition function of non-planar models.

We give a new class of outer bounds on the marginal polytope, and propose a cutting-plane algorithm for efficiently optimizing over these constraints. When combined with a concave upper bound on the entropy, this gives a new variational inference algorithm for probabilistic inference in discrete Markov Random Fields (MRFs). Valid constraints on the marginal polytope are derived through a series of projections onto the cut polytope. As a result, we obtain tighter upper bounds on the log-partition function. We also show empirically that the approximations of the marginals are significantly more accurate when using the tighter outer bounds.