Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates.Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.Comment: Neural Information Processing Systems (2011

Minimizing Finite Sums with the Stochastic Average Gradient

We propose the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG) methods, the SAG method's iteration cost is independent of the number of terms in the sum. However, by incorporating a memory of previous gradient values the SAG method achieves a faster convergence rate than black-box SG methods. The convergence rate is improved from O(1/k^{1/2}) to O(1/k) in general, and when the sum is strongly-convex the convergence rate is improved from the sub-linear O(1/k) to a linear convergence rate of the form O(p^k) for p \textless{} 1. Further, in many cases the convergence rate of the new method is also faster than black-box deterministic gradient methods, in terms of the number of gradient evaluations. Numerical experiments indicate that the new algorithm often dramatically outperforms existing SG and deterministic gradient methods, and that the performance may be further improved through the use of non-uniform sampling strategies.Comment: Revision from January 2015 submission. Major changes: updated literature follow and discussion of subsequent work, additional Lemma showing the validity of one of the formulas, somewhat simplified presentation of Lyapunov bound, included code needed for checking proofs rather than the polynomials generated by the code, added error regions to the numerical experiment

Interactions of Deep-Sea Vent Invertebrates with Their Environment: The Case of Rimicaris exoculata

The vent shrimp Rimicaris exoculata thrives around many hydrothermal vent sites along the Mid-Atlantic Ridge (MAR), where it aggregates into dense swarms. In contrast to hydrothermal vent fields at the East Pacific Rise (EPR), where the biomass is dominated by tubeworms, clams, and mussels, this shrimp is one of the major animal species at MAR vents. These animals are found in the dynamic mixing interface between cold oxygenated seawater and hot, reduced hydrothermal vent fluid. The adaptation of this shrimp to the hostile deep-sea hydrothermal environment and its survival strategy has been investigated since their discovery at the TAG site in the late 1980s. Rimicaris exoculata is now known to colonize black smoker complexes along the MAR in the depth-range of 2,300-3,900 in (Rainbow, Broken Spur, TAG, Snake Pit, Logatchev, 5 degrees S (Rimicaris of exoculata). Although the presence of the Rimicaris genus was first believed to be restricted to the MAR, a related species, Rimicaris kairei, was found recently at the Central Indian Ridge (CIA) (Edmonds and Kairei vent field). This review summarizes the current knowledge of Rimicaris shrimp, focusing on their spatial and temporal distribution, chemical and thermal environment, as well as on possible nutrition strategies and behavioral aspects. Recent studies suggested that iron oxide encrusted bacteria hosted in the branchial chamber of R. exoculata from the Rainbow vent field (MAR) might rely on iron oxidation. Striking results on the occurrence and morphology of iron precipitates, as well as on bacterial-mineral interaction in the gill chamber, have lead to the hypothesis of an iron-based symbiosis between bacteria and the shrimp. Special attention is called to these issues

A Stochastic Gradient Method with an Exponential Convergence Rate for Finite Training Sets

We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard algorithms, both in terms of optimizing the training error and reducing the test error quickly.Comment: The notable changes over the current version: - worked example of convergence rates showing SAG can be faster than first-order methods - pointing out that the storage cost is O(n) for linear models - the more-stable line-search - comparison to additional optimal SG methods - comparison to rates of coordinate descent methods in quadratic cas

Identification of the temperature dependent relation between thermo-optical properties and morphology of semi-crystalline thermoplastics for thermoforming process

International audienceHeating stage of the thermoforming of thermoplastics are critical as it has great effect on their formability under forming and therefore product quality. As radiation heat transfer is widely used for the heating of thermoplastic preforms, physical background of the radiation heating of bulk thermoplastic polymers has to be understood well for an accurate prediction on their temperature profile. In the past, many numerical approaches were developed based on thermo-optical characteristics of thermoplastics whereas little attention was given to the relation between their microstructure and thermo-optical parameters. Considering semi-crystalline thermoplastics the effect of microcrystalline structure is key to identify the thermo-optical properties and develop an accurate numerical radiative heat transfer model for optimization of thermoforming process. Previous studies in literature showed that there is a strong coupling between microstructure of semi-crystalline thermoplastics and their thermo-optical properties in the near-infrared spectral region. In the present work, the relation between thermo-optical characteristics and microstructure of polyolefin-based (PO) polymer was studied considering the change in its morphology at various temperatures. The optical characteristics of the PO were experimentally analyzed under heating conditions using an in-house developed device that is built using a Fourier Transform Infrared spectroscopy, integrating sphere and heating plate. Thanks to the analyses, the changes in the thermo-optical properties of the PO were correlated to its varying morphology under increasing temperature. As semi-crystalline thermoplastics are heated up to melting temperature to soften enough for successful forming process, their microcrystalline structure may show variation above glass transition temperature and this temperature-dependent relation cannot be neglected for building an accurate numerical model for infrared heating assisted thermoforming

A 2-dimensional Geometry for Biological Time

This paper proposes an abstract mathematical frame for describing some features of biological time. The key point is that usual physical (linear) representation of time is insufficient, in our view, for the understanding key phenomena of life, such as rhythms, both physical (circadian, seasonal ...) and properly biological (heart beating, respiration, metabolic ...). In particular, the role of biological rhythms do not seem to have any counterpart in mathematical formalization of physical clocks, which are based on frequencies along the usual (possibly thermodynamical, thus oriented) time. We then suggest a functional representation of biological time by a 2-dimensional manifold as a mathematical frame for accommodating autonomous biological rhythms. The "visual" representation of rhythms so obtained, in particular heart beatings, will provide, by a few examples, hints towards possible applications of our approach to the understanding of interspecific differences or intraspecific pathologies. The 3-dimensional embedding space, needed for purely mathematical reasons, allows to introduce a suitable extra-dimension for "representation time", with a cognitive significance.Comment: Presented in an invited Lecture, conference "Biologie e selezioni naturali", Florence, December 4-8, 200

Detection

This review on second- and third-generation multidetectors devoted to heavy-ion collisions aims to cover the last twenty years. The presented list of devices is not exhaustive but regroups most of the techniques used during this period for nuclear reactions at intermediate energy (≈ 10A MeV to 1A GeV), both for charged-particle and neutron detection. The main part will be devoted to 4π multidetectors, projectile decay fragmentation, high-resolution magnetic spectrometers, auxiliary detectors and neutron detection. The last part will present the progress in electronics and detection in view of the construction of future-generation detectors