Lipschitz Bandits: Regret Lower Bounds and Optimal Algorithms

We consider stochastic multi-armed bandit problems where the expected reward is a Lipschitz function of the arm, and where the set of arms is either discrete or continuous. For discrete Lipschitz bandits, we derive asymptotic problem specific lower bounds for the regret satisfied by any algorithm, and propose OSLB and CKL-UCB, two algorithms that efficiently exploit the Lipschitz structure of the problem. In fact, we prove that OSLB is asymptotically optimal, as its asymptotic regret matches the lower bound. The regret analysis of our algorithms relies on a new concentration inequality for weighted sums of KL divergences between the empirical distributions of rewards and their true distributions. For continuous Lipschitz bandits, we propose to first discretize the action space, and then apply OSLB or CKL-UCB, algorithms that provably exploit the structure efficiently. This approach is shown, through numerical experiments, to significantly outperform existing algorithms that directly deal with the continuous set of arms. Finally the results and algorithms are extended to contextual bandits with similarities.Comment: COLT 201

Studies of five dimensional superconformal field theories, localization and geometry

In this thesis, we study the U-plane of rank-one 5d N = 1 superconformal field theories (SCFTs), which is the one-dimensional Coulomb branch (CB) of such theories on $R^4 \times S^1$. This circle compactification gives us 4d N = 2 supersymmetric field theories of ‘Kaluza-Klein (KK) type’, and thus the Coulomb branch physics can be described by its Seiberg-Witten (SW) geometry. The total space of the SW geometry, which consists of the SW curve fibered over the CB, can be modelled as a rational elliptic surface. As such, a classification of all possible Coulomb branch configurations, for the 5d field theories and their 4d descendants, is given by Persson’s classification of rational elliptic surfaces. This formalism trivializes renormalization group flows, while also containing information about the global form of the flavour symmetry group and the one-form symmetry, through the Mordell-Weil group of the SW geometry. Moreover, in a number of important instances, the U-plane is a modular curve, and we use its beautiful modular properties to investigate aspects of the low-energy physics, such as the spectrum of light particles at strong coupling and the associated BPS quivers. A related approach to studying the strong-coupling dynamics of these theories is to consider them on curved backgrounds M5. Our focus is on closed five-manifolds M5 which are principal circle bundles over simply-connected K\"ahler four-manifolds, $M_4$. Starting with the topologically twisted 4d N = 2 KK theory on $M_4$, we propose a new approach to compute the supersymmetric partition function on $M_5$ through the insertion of a fibering operator in the 4d theory, which introduces a non-trivial fibration over $M_4$. We determine the so-called Coulomb branch partition function on any such $M_5$, which is conjectured to be the holomorphic ‘integrand’ of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When $M_4$ is toric, we also reconstruct our CB partition function by appropriately gluing Nekrasov partition functions

Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

We study the Coulomb branches of the rank-one 4d $\mathcal{N} = 2$ quantum field theories, including the KK theories obtained from the circle compactification of the 5d $\mathcal{N}= 1$ $E_n$ Seiberg theories. The focus is set on the relation between their Seiberg-Witten geometries and rational elliptic surfaces, with more attention being given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group ${\rm PSL}(2,\mathbb{Z})$. We derive closed-form expressions for the modular functions for all congruence and some of the non-congruence subgroups associated with these geometries. Moreover, we show how the BPS quivers of these theories can be determined directly from the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This approach can also be generalized to theories whose Coulomb branches contain `undeformable' singularities, leading to known quivers of such theories.Comment: 63 pages plus appendix, many figures; v2: added references, corrected typo

The UU-plane of rank-one 4d N=2\mathcal{N}=2 KK theories

The simplest non-trivial 5d superconformal field theories (SCFT) are the famous rank-one theories with $E_n$ flavour symmetry. We study their $U$-plane, which is the one-dimensional Coulomb branch of the theory on $\mathbb{R}^4 \times S^1$. The total space of the Seiberg-Witten (SW) geometry -- the $E_n$ SW curve fibered over the $U$-plane -- is described as a rational elliptic surface with a singular fiber of type $I_{9-n}$ at infinity. A classification of all possible Coulomb branch configurations, for the $E_n$ theories and their 4d descendants, is given by Persson's classification of rational elliptic surfaces. We show that the global form of the flavour symmetry group is encoded in the Mordell-Weil group of the SW elliptic fibration. We study in detail many special points in parameters space, such as points where the flavour symmetry enhances, and/or where Argyres-Douglas and Minahan-Nemeschansky theories appear. In a number of important instances, including in the massless limit, the $U$-plane is a modular curve, and we use modularity to investigate aspects of the low-energy physics, such as the spectrum of light particles at strong coupling and the associated BPS quivers. We also study the gravitational couplings on the $U$-plane, matching the infrared expectation for the couplings $A(U)$ and $B(U)$ to the UV computation using the Nekrasov partition function.Comment: 137 pages plus appendix, many figures. v2: added references, corrected typos and imprecision

Reading between the rational sections: Global structures of 4d N=2\mathcal{N}=2 KK theories

We study how the global structure of rank-one 4d $\mathcal{N}=2$ supersymmetric field theories is encoded into global aspects of the Seiberg-Witten elliptic fibration. Starting with the prototypical example of the $\mathfrak{su}(2)$ gauge theory, we distinguish between relative and absolute Seiberg-Witten curves. For instance, we discuss in detail the three distinct absolute curves for the $SU(2)$ and $SO(3)_\pm$ 4d $\mathcal{N}=2$ gauge theories. We propose that the $1$-form symmetry of an absolute theory is isomorphic to a torsion subgroup of the Mordell-Weil group of sections of the absolute curve, while the full defect group of the theory is encoded in the torsion sections of a so-called relative curve. We explicitly show that the relative and absolute curves are related by isogenies (that is, homomorphisms of elliptic curves) generated by torsion sections -- hence, gauging a one-form symmetry corresponds to composing isogenies between Seiberg-Witten curves. We apply this approach to Kaluza-Klein (KK) 4d $\mathcal{N}=2$ theories that arise from toroidal compactifications of 5d and 6d SCFTs to four dimensions, uncovering an intricate pattern of 4d global structures obtained by gauging discrete $0$-form and/or $1$-form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on $\mathbb{R}^4\times T^2$.Comment: 60 pages plus appendi

Few Aspects Regarding the Building of the Rule of Law

Delegating the sovereignty from the citizens to the Government is necessary, to a certain degree, in order to ensure the efficiency of the constitution and the separation of powers, as well as the process of lawmaking. However the complex structure that we now call a “state” and that already seems to have reached its limitations is often perceived as a cold, impersonal structure, in which citizens are becoming only an insignificant part, an interchangeable wheel in a grand design. Thus the decision tends to become more distant from the citizens, than it was in the Greek democracy for example. The new means of communication, especially the Internet, give the citizen the possibility to be closer to the decisions taken by the legislator. The building of a “virtual agora” in which citizens are asked to participate directly to their own governing seems an achievable dream. Main difficulties towards such an ambitious goal are no longer of a technological nature, but of transforming the traditional law making process, the somehow rigid essential characteristics that the rule of law must have, with the direct exercise of sovereignty, by the citizens, if not in all matters, at least at a local, administrative level

A review of combined advanced oxidation technologies for the removal of organic pollutants from water

Water pollution through natural and anthropogenic activities has become a global problem causing short-and long-term impact on human and ecosystems. Substantial quantity of individual or mixtures of organic pollutants enter the surface water via point and nonpoint sources and thus affect the quality of freshwater. These pollutants are known to be toxic and difficult to remove by mere biological treatment. To date, most researches on the removal of organic pollutants from wastewater were based on the exploitation of individual treatment process. This single-treatment technology has inherent challenges and shortcomings with respect to efficiency and economics. Thus, application of two advanced treatment technologies characterized with high efficiency with respect to removal of primary and disinfection by-products in wastewater is desirable. This review article focuses on the application of integrated technologies such as electrohydraulic discharge with heterogeneous photocatalysts or sonophotocatalysis to remove target pollutants. The information gathered from more than 100 published articles, mostly laboratories studies, shows that process integration effectively remove and degrade recalcitrant toxic contaminants in wastewater better than single-technology processing. This review recommends an improvement on this technology (integrated electrohydraulic discharge with heterogeneous photocatalysts) viz-a-vis cost reduction in order to make it accessible and available in the rural and semi-urban settlement. Further recommendation includes development of an economic model to establish the cost implications of the combined technology. Proper monitoring, enforcement of the existing environmental regulations, and upgrading of current wastewater treatment plants with additional treatment steps such as photocatalysis and ozonation will greatly assist in the removal of environmental toxicants

A new device used in the restoration of kinematics after total facet arthroplasty

Facet degeneration can lead to spinal stenosis and instability, and often requires stabilization. Interbody fusion is commonly performed, but it can lead to adjacent-segment disease. Dynamic posterior stabilization was performed using a total facet arthroplasty system. The total facet arthroplasty system was originally intended to restore the natural motion of the posterior stabilizers, but follow-up studies are lacking due to limited clinical use. We studied the first 14 cases (long-term follow-up) treated with this new device in our clinic. All patients were diagnosed with lumbar stenosis due to hypertrophy of the articular facets on one to three levels (maximum). Disk space was of normal height. The design of this implant allows its use only at levels L3-L4 and L4-L5. We implanted nine patients at the L4-L5 level and four patients at level L3-L4. Postoperative follow-up of the patients was obtained for an average of 3.7 years. All patients reported persistent improvement of symptoms, visual analog scale score, and Oswestry Disability Index score. Functional scores and dynamic radiographic imaging demonstrated the functional efficacy of this new implant, which represents an alternative technique and a new approach to dynamic stabilization of the vertebral column after interventions for spine decompression. The total facet arthroplasty system represents a viable option for dynamic posterior stabilization after spinal decompression. For the observed follow-up, it preserved motion without significant complications or apparent intradisk or adjacent-disk degeneration. © 2014 Vermesan et al