Recently, a number of learning-based optimization methods that combine data-driven architectures with the classical optimization algorithms have been proposed and explored, showing superior empirical performance in solving various ill-posed inverse problems, but there is still a scarcity of rigorous analysis about the convergence behaviors of learning-based optimization. In particular, most existing analyses are specific to unconstrained problems but cannot apply to the more general cases where some variables of interest are subject to certain constraints. In this paper, we propose Differentiable Linearized ADMM (D-LADMM) for solving the problems with linear constraints. Specifically, D-LADMM is a K-layer LADMM inspired deep neural network, which is obtained by firstly introducing some learnable weights in the classical Linearized ADMM algorithm and then generalizing the proximal operator to some learnable activation function. Notably, we rigorously prove that there exist a set of learnable parameters for D-LADMM to generate globally converged solutions, and we show that those desired parameters can be attained by training D-LADMM in a proper way. To the best of our knowledge, we are the first to provide the convergence analysis for the learning-based optimization method on constrained problems.

This paper proposes a novel landscape smoothing method for the symmetric Traveling Salesman Problem (TSP). We first define the homotopic convex (HC) transformation of a TSP as a convex combination of a well-constructed simple TSP and the original TSP. We observe that controlled by the coefficient of the convex combination, (i) the landscape of the HC transformed TSP is smoothed in terms that its number of local optima is reduced compared to the original TSP; (ii) the fitness distance correlation of the HC transformed TSP is increased. We then propose an iterative algorithmic framework in which the proposed HC transformation is combined with a heuristic TSP solver. It works as an escaping scheme from local optima for improving the global search ability of the combined heuristic. A case study with the 3-Opt local search as the heuristic solver shows that the resultant algorithm significantly outperforms iterated local search and two other smoothing-based TSP heuristic solvers on most of commonly-used test instances.

Events in the world may be caused by other, unobserved events. We consider sequences of events in continuous time. Given a probability model of complete sequences, we propose particle smoothing---a form of sequential importance sampling---to impute the missing events in an incomplete sequence. We develop a trainable family of proposal distributions based on a type of bidirectional continuous-time LSTM: Bidirectionality lets the proposals condition on future observations, not just on the past as in particle filtering. Our method can sample an ensemble of possible complete sequences (particles), from which we form a single consensus prediction that has low Bayes risk under our chosen loss metric. We experiment in multiple synthetic and real domains, using different missingness mechanisms, and modeling the complete sequences in each domain with a neural Hawkes process (Mei & Eisner 2017). On held-out incomplete sequences, our method is effective at inferring the ground-truth unobserved events, with particle smoothing consistently improving upon particle filtering.

In this paper, we propose TauRieL and target Traveling Salesman Problem (TSP) since it has broad applicability in theoretical and applied sciences. TauRieL utilizes an actor-critic inspired architecture that adopts ordinary feedforward nets to obtain a policy update vector $v$. Then, we use $v$ to improve the state transition matrix from which we generate the policy. Also, the state transition matrix allows the solver to initialize from precomputed solutions such as nearest neighbors. In an online learning setting, TauRieL unifies the training and the search where it can generate near-optimal results in seconds. The input to the neural nets in the actor-critic architecture are raw 2-D inputs, and the design idea behind this decision is to keep neural nets relatively smaller than the architectures with wide embeddings with the tradeoff of omitting any distributed representations of the embeddings. Consequently, TauRieL generates TSP solutions two orders of magnitude faster per TSP instance as compared to state-of-the-art offline techniques with a performance impact of 6.1\% in the worst case.

In this paper, we study the sum rate maximization for successive zero-forcing dirty-paper coding (SZFDPC) with per-antenna power constraint (PAPC). Although SZFDPC is a low-complexity alternative to the optimal dirty paper coding (DPC), efficient algorithms to compute its sum rate are still open problems especially under practical PAPC. The existing solution to the considered problem is computationally inefficient due to employing high-complexity interior-point method. In this study, we propose two new low-complexity approaches to this important problem. More specifically, the first algorithm achieves the optimal solution by transforming the original problem in the broadcast channel into an equivalent problem in the multiple access channel, then the resulting problem is solved by alternating optimization together with successive convex approximation. We also derive a suboptimal solution based on machine learning to which simple linear regressions are applicable. The approaches are analyzed and validated extensively to demonstrate their superiors over the existing approach.

Computing Nash equilibrium (NE) of multi-player games has witnessed renewed interest due to recent advances in generative adversarial networks. However, computing equilibrium efficiently is challenging. To this end, we introduce the Gradient-based Nikaido-Isoda (GNI) function which serves: (i) as a merit function, vanishing only at the first-order stationary points of each player's optimization problem, and (ii) provides error bounds to a stationary Nash point. Gradient descent is shown to converge sublinearly to a first-order stationary point of the GNI function. For the particular case of bilinear min-max games and multi-player quadratic games, the GNI function is convex. Hence, the application of gradient descent in this case yields linear convergence to an NE (when one exists). In our numerical experiments, we observe that the GNI formulation always converges to the first-order stationary point of each player's optimization problem.

This paper deals with estimating model parameters in graphical models. We reformulate it as an information geometric optimization problem and introduce a natural gradient descent strategy that incorporates additional meta parameters. We show that our approach is a strong alternative to the celebrated EM approach for learning in graphical models. Actually, our natural gradient based strategy leads to learning optimal parameters for the final objective function without artificially trying to fit a distribution that may not correspond to the real one. We support our theoretical findings with the question of trend detection in financial markets and show that the learned model performs better than traditional practitioner methods and is less prone to overfitting.

Many optimization problems can be cast into the maximum satisfiability (MAX-SAT) form, and many solvers have been developed for tackling such problems. To evaluate the performance of a MAX-SAT solver, it is convenient to generate difficult MAX-SAT instances with solutions known in advance. Here, we propose a method of generating weighted MAX-2-SAT instances inspired by the frustrated-loop algorithm used by the quantum annealing community to generate Ising spin-glass instances with nearest-neighbor coupling. Our algorithm is extended to instances whose underlying coupling graph is general, though we focus here on the case of bipartite coupling, with the associated energy being the restricted Boltzmann machine (RBM) energy. It is shown that any MAX-2-SAT problem can be reduced to the problem of minimizing an RBM energy over the nodal values. The algorithm is designed such that the difficulty of the generated instances can be tuned through a central parameter known as the frustration index. Two versions of the algorithm are presented: the random- and structured-loop algorithms. For the random-loop algorithm, we provide a thorough theoretical and empirical analysis on its mathematical properties from the perspective of frustration, and observe empirically, using simulated annealing, a double phase transition behavior in the difficulty scaling behavior driven by the frustration index. For the structured-loop algorithm, we show that it offers an improvement in difficulty of the generated instances over the random-loop algorithm, with the improvement factor scaling super-exponentially with respect to the frustration index for instances at high loop density. At the end of the paper, we provide a brief discussion of the relevance of this work to the pre-training of RBMs.

A number of optimization approaches have been proposed for optimizing nonconvex objectives (e.g. deep learning models), such as batch gradient descent, stochastic gradient descent and stochastic variance reduced gradient descent. Theory shows these optimization methods can converge by using an unbiased gradient estimator. However, in practice biased gradient estimation can allow more efficient convergence to the vicinity since an unbiased approach is computationally more expensive. To produce fast convergence there are two trade-offs of these optimization strategies which are between stochastic/batch, and between biased/unbiased. This paper proposes an integrated approach which can control the nature of the stochastic element in the optimizer and can balance the trade-off of estimator between the biased and unbiased by using a hyper-parameter. It is shown theoretically and experimentally that this hyper-parameter can be configured to provide an effective balance to improve the convergence rate.

Differential privacy is concerned about the prediction quality while measuring the privacy impact on individuals whose information is contained in the data. We consider differentially private risk minimization problems with regularizers that induce structured sparsity. These regularizers are known to be convex but they are often non-differentiable. We analyze the standard differentially private algorithms, such as output perturbation, Frank-Wolfe and objective perturbation. Output perturbation is a differentially private algorithm that is known to perform well for minimizing risks that are strongly convex. Previous works have derived excess risk bounds that are independent of the dimensionality. In this paper, we assume a particular class of convex but non-smooth regularizers that induce structured sparsity and loss functions for generalized linear models. We also consider differentially private Frank-Wolfe algorithms to optimize the dual of the risk minimization problem. We derive excess risk bounds for both these algorithms. Both the bounds depend on the Gaussian width of the unit ball of the dual norm. We also show that objective perturbation of the risk minimization problems is equivalent to the output perturbation of a dual optimization problem. This is the first work that analyzes the dual optimization problems of risk minimization problems in the context of differential privacy.