Practitioners apply neural networks to increasingly complex problems in natural language processing, such as syntactic parsing and semantic role labeling that have rich output structures. Many such structured-prediction problems require deterministic constraints on the output values; for example, in sequence-to-sequence syntactic parsing, we require that the sequential outputs encode valid trees. While hidden units might capture such properties, the network is not always able to learn such constraints from the training data alone, and practitioners must then resort to post-processing. In this paper, we present an inference method for neural networks that enforces deterministic constraints on outputs without performing rule-based post-processing or expensive discrete search. Instead, in the spirit of gradient-based training, we enforce constraints with gradient-based inference (GBI): for each input at test-time, we nudge continuous model weights until the network's unconstrained inference procedure generates an output that satisfies the constraints. We study the efficacy of GBI on three tasks with hard constraints: semantic role labeling, syntactic parsing, and sequence transduction. In each case, the algorithm not only satisfies constraints but improves accuracy, even when the underlying network is state-of-the-art.
Finally, as previously mentioned, our method highly resembles dual decomposition and more generally Lagrangian relaxation for structured prediction @cite_14 @cite_1 @cite_0 . In such techniques, it is assumed that a computationally efficient inference algorithm can maximize over a superset of the feasible region (this assumption parallels our case because unconstrained inference in the neural network is efficient, but might violate constraints). Then, the method employs gradient descent to concentrate this superset onto the feasible region. However, these techniques are not directly applicable to our non-linear problem with global constraints.
@cite_0 @cite_14 @cite_1
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We present a novel dual decomposition approach to MAP inference with highly connected discrete graphical models. Decompositions into cyclic k-fan structured subproblems are shown to significantly tighten the Lagrangian relaxation relative to the standard local polytope relaxation, while enabling efficient integer programming for solving the subproblems. Additionally, we introduce modified update rules for maximizing the dual function that avoid oscillations and converge faster to an optimum of the relaxed problem, and never get stuck in nonoptimal fixed points. Dual decomposition, and more generally Lagrangian relaxation, is a classical method for combinatorial optimization; it has recently been applied to several inference problems in natural language processing (NLP). This tutorial gives an overview of the technique. We describe example algorithms, describe formal guarantees for the method, and describe practical issues in implementing the algorithms. While our examples are predominantly drawn from the NLP literature, the material should be of general relevance to inference problems in machine learning. A central theme of this tutorial is that Lagrangian relaxation is naturally applied in conjunction with a broad class of combinatorial algorithms, allowing inference in models that go significantly beyond previous work on Lagrangian relaxation for inference in graphical models. One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions This paper discusses a new method to perform propagation over a (two-layer, feed-forward) Neural Network embedded in a Constraint Programming model. The method is meant to be employed in Empirical Model Learning, a technique designed to enable optimal decision making over systems that cannot be modeled via conventional declarative means. The key step in Empirical Model Learning is to embed a Machine Learning model into a combinatorial model. It has been showed that Neural Networks can be embedded in a Constraint Programming model by simply encoding each neuron as a global constraint, which is then propagated individually. Unfortunately, this decomposition approach may lead to weak bounds. To overcome such limitation, we propose a new network-level propagator based on a non-linear Lagrangian relaxation that is solved with a subgradient algorithm. The method proved capable of dramatically reducing the search tree size on a thermal-aware dispatching problem on multicore CPUs. The overhead for optimizing the Lagrangian multipliers is kept within a reasonable level via a few simple techniques. This paper is an extended version of [27], featuring an improved structure, a new filtering technique for the network inputs, a set of overhead reduction techniques, and a thorough experimentation.