Exponentiated gradient algorithms for conditional random fields and max-margin Markov Networks

Log-linear and maximum-margin models are two commonly-used methods in supervised machine learning, and are frequently used in structured prediction problems. Efficient learning of parameters in these models is therefore an important problem, and becomes a key factor when learning from very large data sets. This paper describes exponentiated gradient (EG) algorithms for training such models, where EG updates are applied to the convex dual of either the log-linear or max-margin objective function; the dual in both the log-linear and max-margin cases corresponds to minimizing a convex function with simplex constraints. We study both batch and online variants of the algorithm, and provide rates of convergence for both cases. In the max-margin case, O(1/ε) EG updates are required to reach a given accuracy ε in the dual; in contrast, for log-linear models only O(log(1/ε)) updates are required. For both the max-margin and log-linear cases, our bounds suggest that the online EG algorithm requires a factor of n less computation to reach a desired accuracy than the batch EG algorithm, where n is the number of training examples. Our experiments confirm that the online algorithms are much faster than the batch algorithms in practice. We describe how the EG updates factor in a convenient way for structured prediction problems, allowing the algorithms to be efficiently applied to problems such as sequence learning or natural language parsing. We perform extensive evaluation of the algorithms, comparing them to L-BFGS and stochastic gradient descent for log-linear models, and to SVM-Struct for max-margin models. The algorithms are applied to a multi-class problem as well as to a more complex large-scale parsing task. In all these settings, the EG algorithms presented here outperform the other methods.

Mary and Max

Synopsis and critique of Australian film in animation, comedy, and drama genres.

Pest-damage relationships for Helicoverpa armigera (Hübner) (Lepidoptera: Noctuidae) on soybean (Glycine max) and dry bean (Phaseolus vulgaris) during pod-fill.

The response of soybean (Glycine max) and dry bean (Phaseolus vulgaris) to feeding by Helicoverpa armigera during the pod-fill stage was studied in irrigated field cages over three seasons to determine the relationship between larval density and yield loss, and to develop economic injury levels. H. armigera intensity was calculated in Helicoverpa injury equivalent (HIE) units, where 1 HIE was the consumption of one larva from the start of the infestation period to pupation. In the dry bean experiment, yield loss occurred at a rate 6.00 ± 1.29 g/HIE while the rates of loss in the three soybean experiments were 4.39 ± 0.96 g/HIE, 3.70 ± 1.21 g/HIE and 2.12 ± 0.71 g/HIE. These three slopes were not statistically different (P > 0.05) and the pooled estimate of the rate of yield loss was 3.21 ± 0.55 g/HIE. The first soybean experiment also showed a split-line form of damage curve with a rate of yield loss of 26.27 ± 2.92 g/HIE beyond 8.0 HIE and a rapid decline to zero yield. In dry bean, H. armigera feeding reduced total and undamaged pod numbers by 4.10 ± 1.18 pods/HIE and 12.88 ± 1.57 pods/HIE respectively, while undamaged seed numbers were reduced by 35.64 ± 7.25 seeds/HIE. In soybean, total pod numbers were not affected by H. armigera infestation (out to 8.23 HIE in Experiment 1) but seed numbers (in Experiments 1 and 2) and the number of seeds/pod (in all experiments) were adversely affected. Seed size increased with increases in H. armigera density in two of the three soybean experiments, indicating plant compensatory responses to H. armigera feeding. Analysis of canopy pod profiles indicated that loss of pods occurred from the top of the plant downwards, but with an increase in pod numbers close to the ground at higher pest densities as the plant attempted to compensate for damage. Based on these results, the economic injury levels for H. armigera on dry bean and soybean are approximately 0.74 HIE and 2.31 HIE/m2, respectively (0.67 and 2.1 HIE/row-m for 91 cm rows).

Judge Max Frankenbach of Berlin hiking with his wife Grete Frankenbach nee Bernstein; Sassal-Massone Portraits Couples

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Max and Martha Liebermann with unidentified group at a celebration for Max Liebermann at the Akademie der Kunste, Berlin Portraits; Groups

Verso: "Liebermann Feierlichkeiten in der Akademie der Kuenste. In der Mitte: Max Liebermann und Frau. Becker."

Portrait of Max Meyerhof.

Max Meyerhof was a distinguished opthamologist, author, medical historian. He died in Cairo where he helped establish the Egyptian medical service.

Newspaper obituary for Max Molling in the Hannoverscher Anzeiger; Hannover, Germany.

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Newspaper obituary for Max Molling in the Hannoverscher Anzeiger by the personel of M. Molling & Co.; Hannover, Germany.

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Reproduction of newspaper obituary for Max Molling in the Hannoverscher Anzeiger.

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Reproduction of newspaper obituary for Max Molling in the Hannoverscher Anzeiger.

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Newspaper obituary for Max Molling in the Hannoverscher Anzeiger; Hannover, Germany.

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Tombstone of Max Molling; Strangriede Cemetery (R 29.10); Hannover Cemeteries Tombs

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Tombstones of Sophie Molling and Max Molling (center in shadow); Strangriede Cemetery (R 29.09 and R 29.10); Hannover Cemeteries Tombs

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