This article presents a new search algorithm for the NP-hard problem of optimizing functions of binary variables that decompose according to a graphical model. It can be applied to models of any order and structure. The main novelty is a technique to constrain the search space based on the topology of the model. When pursued to the full search depth, the algorithm is guaranteed to converge to a global optimum, passing through a series of monotonously improving local optima that are guaranteed to be optimal within a given and increasing Hamming distance. For a search depth of 1, it specializes to Iterated Conditional Modes. Between these extremes, a useful tradeoff between approximation quality and runtime is established. Experiments on models derived from both illustrative and real problems show that approximations found with limited search depth match or improve those obtained by state-of-the-art methods based on message passing and linear programming.
Third, lazy flipping with a limited search depth as a means of approximate optimization competes with message passing algorithms @cite_29 @cite_3 @cite_18 @cite_28 and with algorithms based on convex programming relaxations of the optimization problem @cite_18 @cite_7 @cite_0 @cite_8 , in particular with Tree-reweighted Belief Propagation (TRBP) @cite_13 @cite_28 @cite_21 and sub-gradient descent @cite_35 @cite_1 .
@cite_35 @cite_18 @cite_7 @cite_8 @cite_28 @cite_29 @cite_21 @cite_1 @cite_3 @cite_0 @cite_13
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We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. We consider the problem of finding an approximate second-order stationary point of a constrained non-convex optimization problem. We first show that, unlike the unconstrained scenario, the vanilla projected gradient descent algorithm may converge to a strict saddle point even when there is only a single linear constraint. We then provide a hardness result by showing that checking ( , )-second order stationarity is NP-hard even in the presence of linear constraints. Despite our hardness result, we identify instances of the problem for which checking second order stationarity can be done efficiently. For such instances, we propose a dynamic second order Frank--Wolfe algorithm which converges to ( , )-second order stationary points in O ( ^ -2 , ^ -3 ) iterations. The proposed algorithm can be used in general constrained non-convex optimization as long as the constrained quadratic sub-problem can be solved efficiently. A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance. 1. Automatic code generation for real-time convex optimization J. Mattingley and S. Boyd 2. Gradient-based algorithms with applications to signal recovery problems A. Beck and M. Teboulle 3. Graphical models of autoregressive processes J. Songsiri, J. Dahl and L. Vandenberghe 4. SDP relaxation of homogeneous quadratic optimization Z. Q. Luo and T. H. Chang 5. Probabilistic analysis of SDR detectors for MIMO systems A. Man-Cho So and Y. Ye 6. Semidefinite programming, matrix decomposition, and radar code design Y. Huang, A. De Maio and S. Zhang 7. Convex analysis for non-negative blind source separation with application in imaging W. K. Ma, T. H. Chan, C. Y. Chi and Y. Wang 8. Optimization techniques in modern sampling theory T. Michaeli and Y. C. Eldar 9. Robust broadband adaptive beamforming using convex optimization M. Rubsamen, A. El-Keyi, A. B. Gershman and T. Kirubarajan 10. Cooperative distributed multi-agent optimization A. Nenadic and A. Ozdaglar 11. Competitive optimization of cognitive radio MIMO systems via game theory G. Scutari, D. P. Palomar and S. Barbarossa 12. Nash equilibria: the variational approach F. Facchinei and J. S. Pang. A large number of compiler optimizations are nowadays available to users. These optimizations interact with each other and with the input code in several and complex ways. The sequence of application of optimization passes can have a significant impact on the performance achieved. The effect of the optimizations is both platform and application dependent. The exhaustive exploration of all viable sequences of compiler optimizations for a given code fragment is not feasible. As this exploration is a complex and time-consuming task, several researchers have focused on Design Space Exploration (DSE) strategies both to select optimization sequences to improve the performance of each function of the application and to reduce the exploration time. In this article, we present a DSE scheme based on a clustering approach for grouping functions with similarities and exploration of a reduced search space resulting from the combination of optimizations previously suggested for the functions in each group. The identification of similarities between functions uses a data mining method that is applied to a symbolic code representation. The data mining process combines three algorithms to generate clusters: the Normalized Compression Distance, the Neighbor Joining, and a new ambiguity-based clustering algorithm. Our experiments for evaluating the effectiveness of the proposed approach address the exploration of optimization sequences in the context of the ReflectC compiler, considering 49 compilation passes while targeting a Xilinx MicroBlaze processor, and aiming at performance improvements for 51 functions and four applications. Experimental results reveal that the use of our clustering-based DSE approach achieves a significant reduction in the total exploration time of the search space (20× over a Genetic Algorithm approach) at the same time that considerable performance speedups (41p over the baseline) were obtained using the optimized codes. Additional experiments were performed considering the LLVM compiler, considering 124 compilation passes, and targeting a LEON3 processor. The results show that our approach achieved geometric mean speedups of 1.49 × , 1.32 × , and 1.24 × for the best 10, 20, and 30 functions, respectively, and a global improvement of 7p over the performance obtained when compiling with -O2. This paper provides a systematic study of several proposed measures for online algorithms in the context of a specific problem, namely, the two server problem on three colinear points. Even though the problem is simple, it encapsulates a core challenge in online algorithms which is to balance greediness and adaptability. We examine Competitive Analysis, the Max Max Ratio, the Random Order Ratio, Bijective Analysis and Relative Worst Order Analysis, and determine how these measures compare the Greedy Algorithm, Double Coverage, and Lazy Double Coverage, commonly studied algorithms in the context of server problems. We find that by the Max Max Ratio and Bijective Analysis, Greedy is the best of the three algorithms. Under the other measures, Double Coverage and Lazy Double Coverage are better, though Relative Worst Order Analysis indicates that Greedy is sometimes better. Only Bijective Analysis and Relative Worst Order Analysis indicate that Lazy Double Coverage is better than Double Coverage. Our results also provide the first proof of optimality of an algorithm under Relative Worst Order Analysis. Minimization with orthogonality constraints (e.g., (X^ X = I )) and or spherical constraints (e.g., ( x _2 = 1 )) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop. Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. We present a large-margin formulation and algorithm for structured output prediction that allows the use of latent variables. Our proposal covers a large range of application problems, with an optimization problem that can be solved efficiently using Concave-Convex Programming. The generality and performance of the approach is demonstrated through three applications including motiffinding, noun-phrase coreference resolution, and optimizing precision at k in information retrieval. We introduce a generic scheme for accelerating first-order optimization methods in the sense of Nesterov, which builds upon a new analysis of the accelerated proximal point algorithm. Our approach consists of minimizing a convex objective by approximately solving a sequence of well-chosen auxiliary problems, leading to faster convergence. This strategy applies to a large class of algorithms, including gradient descent, block coordinate descent, SAG, SAGA, SDCA, SVRG, Finito MISO, and their proximal variants. For all of these methods, we provide acceleration and explicit support for non-strongly convex objectives. In addition to theoretical speed-up, we also show that acceleration is useful in practice, especially for ill-conditioned problems where we measure significant improvements. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math .