Dual subgradient algorithms for large-scale nonsmooth learning problems

“Classical” First Order (FO) algorithms of convex optimization, such as Mirror Descent algorithm or Nesterov’s optimal algorithm of smooth convex optimization, are well known to have optimal (theoretical) complexity estimates which do not depend on the problem dimension. However, to attain the optimality, the domain of the problem should admit a “good proximal setup”. The latter essentially means that (1) the problem domain should satisfy certain geometric conditions of “favorable geometry”, and (2) the practical use of these methods is conditioned by our ability to compute at a moderate cost proximal transformation at each iteration. More often than not these two conditions are satisfied in optimization problems arising in computational learning, what explains why proximal type FO methods recently became methods of choice when solving various learning problems. Yet, they meet their limits in several important problems such as multi-task learning with large number of tasks, where the problem domain does not exhibit favorable geometry, and learning and matrix completion problems with nuclear norm constraint, when the numerical cost of computing proximal transformation becomes prohibitive in large-scale problems. We propose a novel approach to solving nonsmooth optimization problems arising in learning applications where Fenchel-type representation of the objective function is available. The approach is based on applying FO algorithms to the dual problem and using the accuracy certificates supplied by the method to recover the primal solution. While suboptimal in terms of accuracy guaranties, the proposed approach does not rely upon “good proximal setup” for the primal problem but requires the problem domain to admit a Linear Optimization oracle—the ability to efficiently maximize a linear form on the domain of the primal problem.     

Nonparametric denoising of signals with unknown local structure, I: Oracle inequalities

We consider the problem of pointwise estimation of multi-dimensional signals s, from noisy observations (y_τ) on the regular grid . Our focus is on the adaptive estimation in the case when the signal can be well recovered using a (hypothetical) linear filter, which can depend on the unknown signal itself. The basic setting of the problem we address here can be summarized as follows: suppose that the signal s is “well-filtered”, i.e. there exists an adapted time-invariant linear filter with the coefficients which vanish outside the “cube” {0,…,T}^d which recovers s₀ from observations with small mean-squared error. We suppose that we do not know the filter q^*, although, we do know that such a filter exists. We give partial answers to the following questions:

– is it possible to construct an adaptive estimator of the value s₀, which relies upon observations and recovers s₀ with basically the same estimation error as the unknown filter ?

– how rich is the family of well-filtered (in the above sense) signals?

We show that the answer to the first question is affirmative and provide a numerically efficient construction of a nonlinear adaptive filter. Further, we establish a simple calculus of “well-filtered” signals, and show that their family is quite large: it contains, for instance, sampled smooth signals, sampled modulated smooth signals and sampled harmonic functions.

Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians and by two equal symplectic balls.

     

Robust optimization – methodology and applications

Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO as applied to uncertain linear, conic quadratic and semidefinite programming. For these cases, computationally tractable robust counterparts of uncertain problems are explicitly obtained, or good approximations of these counterparts are proposed, making RO a useful tool for real-world applications. We discuss some of these applications, specifically: antenna design, truss topology design and stability analysis/synthesis in uncertain dynamic systems. We also describe a case study of 90 LPs from the NETLIB collection. The study reveals that the feasibility properties of the usual solutions of real world LPs can be severely affected by small perturbations of the data and that the RO methodology can be successfully used to overcome this phenomenon. Received: May 24, 2000 / Accepted: September 12, 2001?Published online February 14, 2002     

CO2 activation by metal−ligand-cooperation mediated by iridium pincer complexes

Abstract

Herein we report the reversible activation of CO₂ by the dearomatized complex [(^tBuPNP*)Ir(COE)] (1) and by the aromatized complex [(^tBuPNP)Ir(C₆H₅)] (2) via metal-ligand cooperation (MLC) (^tBuPN = 2,6-bis-(di-tert-butylphosphinomethyl)pyridine; ^tBuPNP* = deprotonated PNP; COE = cyclooctadiene). The [1,3]-addition of CO₂ to 1 and 2 is reversible at ambient temperature. While the dearomatized complex 1 reacts readily at ambient temperature with CO₂ in THF or benzene, complex 2 reacts with CO₂ upon heating in benzene at 80 °C or at ambient temperature in THF. The novel aromatized complex [(^tBuPNP)IrCl] (10) does not react with CO₂. Based on the reactivity patterns of 1, 2, and 10 with CO₂, we suggest that CO₂ activation via MLC takes place only via the dearomatized species, and that in the case of 2 THF plays a role as a polar solvent in facilitating formation of the dearomatized hydrido phenyl complex intermediate (complex II).     

Synthesis of oxalamides by acceptorless dehydrogenative coupling of ethylene glycol and amines and the reverse hydrogenation catalyzed by ruthenium

A sustainable, new synthesis of oxalamides, by acceptorless dehydrogenative coupling of ethylene glycol with amines, generating H₂, homogeneously catalyzed by a ruthenium pincer complex, is presented. The reverse hydrogenation reaction is also accomplished using the same catalyst. A plausible reaction mechanism is proposed based on stoichiometric reactions, NMR studies, X-ray crystallography as well as observation of plausible intermediates.

Ruthenium catalyzed acceptorless dehydrogenative coupling of ethylene glycol and amines to oxalamides is reported. The reverse hydrogenation reaction is also accomplished.     

Photolysis of thin films of cypermethrin using in situ FTIR monitoring: Products, rates and quantum yields

Cypermethrin is a synthetic pyrethroid insecticide commonly used both indoors and outdoors. Following its application, cypermethrin residues are exposed to atmospheric oxidants, such as ozone and OH radicals, as well as to solar radiation. The current work focuses on the photochemical degradation of cypermethrin under 254 nm, 310 nm, and solar radiation; including kinetic analysis, quantum yields, and identification of volatile and non-volatile products. The investigation was done using a combined FTIR set-up for parallel analysis of condensed and gas phases. Complementary GC–MS analysis was conducted on the pesticide’s residues. Quantum yields of 0.41 and 0.25 were obtained under 254 and 310 nm, respectively. These values are in agreement with outdoor studies and are higher than the known yields for aqueous solution. The values obtained under all tested conditions suggest that photodegradation of adsorbed cypermethrin is a major outdoor degradation pathway. Oxygen presence was found to increase degradation rate and quantum yield.The identified photoproducts include 3-phenoxybenzaldehyde, 3-phenoxybenzoic acid, acetonitrile (m-phenoxyphenyl), and cypermethrin isomers on the surface and formic acid, CO₂ and CO in the gas phase. The condensed phase products are more polar than the parent molecule, suggesting that they may be susceptible to a greater leaching rate in soils.     

Robust solutions of Linear Programming problems contaminated with uncertain data

Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed. We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization methodology (Ben-Tal and Nemirovski [1–3]; El Ghaoui et al. [5, 6]) to produce “robust” solutions of the above LPs which are in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in optimality. Received: July 1999 / Accepted: May 2000?Published online July 20, 2000     

On maximization of quadratic form over intersection of ellipsoids with common center

We demonstrate that if A ₁,...,A _m are symmetric positive semidefinite n×n matrices with positive definite sum and A is an arbitrary symmetric n×n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxation of the optimization program is not worse than . It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a~feasible solution x to (P) with can be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to (*) is established for the case when all A_i are of rank 1. Received August 13, 1998 / Revised version received May 25, 1999? Published online September 15, 1999     

Dragon Fruits as a Reservoir of Natural Polyphenolics with Chemopreventive Properties

Dragon fruits are a valued source of bioactive compounds with high potential to become a functional food. The aim of the study was to evaluate and compare the chemopreventive potential and chemical composition of fruits harvested in Thailand and Israel. The amount of different compounds in water and methanol extracts and antioxidant activity was investigated. Moreover, cytotoxic activity against cancer and normal cells of skin, prostate, and gastrointestinal origin was performed, accompanied by anti-inflammatory assay based on NO production in RAW 264.7 macrophage model. Additionally, the quenching properties of polyphenols from fruits were determined by the interaction of the main drug carrier in blood human serum (HSA). The chemometric analysis was used to reveal the relationships between the determined parameters. Dragon fruits harvested in Israel revealed higher antioxidant properties and total content of polyphenols and betacyanins when compared to those from Thailand. The examined fruits of both origins showed significant cytotoxic activity toward colon and prostate cancer cells, with no toxic effect on normal cells, but also no anti-inflammatory effect. Moreover, a high binding ability to HSA was observed for water extracts of dragon fruits. All these predestine dragon fruits are the candidates for the attractive and chemopreventive elements of daily diet.