On the Variance, Admissibility, and Stability of Empirical Risk Minimization

It is well known that Empirical Risk Minimization (ERM) with squared loss may attain minimax suboptimal error rates (Birg\'e and Massart, 1993). The key message of this paper is that, under mild assumptions, the suboptimality of ERM must be due to large bias rather than variance. More precisely, in the bias-variance decomposition of the squared error of the ERM, the variance term necessarily enjoys the minimax rate. In the case of fixed design, we provide an elementary proof of this fact using the probabilistic method. Then, we prove this result for various models in the random design setting. In addition, we provide a simple proof of Chatterjee's admissibility theorem (Chatterjee, 2014, Theorem 1.4), which states that ERM cannot be ruled out as an optimal method, in the fixed design setting, and extend this result to the random design setting. We also show that our estimates imply stability of ERM, complementing the main result of Caponnetto and Rakhlin (2006) for non-Donsker classes. Finally, we show that for non-Donsker classes, there are functions close to the ERM, yet far from being almost-minimizers of the empirical loss, highlighting the somewhat irregular nature of the loss landscape

Superconductor-to-Metal Transitions in Dissipative Chains of Mesoscopic Grains and Nanowires

The interplay of quantum fluctuations and dissipation in chains of mesoscopic superconducting grains is analyzed, and the results are also applied to nanowires. It is shown that in 1-d arrays of resistively shunted Josephson junctions, the superconducting-normal charge relaxation within the grains plays an important role. At zero temperature, two superconducting phases can exist, depending primarily on the strength of the dissipation. In the fully superconducting phase (FSC), each grain acts superconducting, and the coupling to the dissipative conduction is important. In the SC* phase, the dissipation is irrelevant at long wavelengths. The phase transitions between these two superconducting phases and the normal metallic phase may be either local or global, and possess rich and complex critical properties. These are inferred from both weak and strong coupling renormalization group analyses. At intermediate temperatures, near either superconductor-to-normal phase transition, there are regimes of super-metallic behavior, in which the resistivity first decreases gradually with decreasing temperature before eventually increasing as temperature is lowered further. The results on chains of Josephson junctions are extended to continuous superconducting nanowires and the subtle issue of whether these can exhibit an FSC phase is considered. Potential relevance to superconductor-metal transitions in other systems is also discussed.Comment: 42 pages, 14 figure

Building a fault-tolerant quantum computer using concatenated cat codes

We present a comprehensive architectural analysis for a fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout. Using estimated near-term physical parameters for electro-acoustic systems, we perform a detailed error analysis of measurements and gates, including CNOT and Toffoli gates. Having built a realistic noise model, we numerically simulate quantum error correction when the outer code is either a repetition code or a thin rectangular surface code. Our next step toward universal fault-tolerant quantum computation is a protocol for fault-tolerant Toffoli magic state preparation that significantly improves upon the fidelity of physical Toffoli gates at very low qubit cost. To achieve even lower overheads, we devise a new magic-state distillation protocol for Toffoli states. Combining these results together, we obtain realistic full-resource estimates of the physical error rates and overheads needed to run useful fault-tolerant quantum algorithms. We find that with around 1,000 superconducting circuit components, one could construct a fault-tolerant quantum computer that can run circuits which are intractable for classical supercomputers. Hardware with 32,000 superconducting circuit components, in turn, could simulate the Hubbard model in a regime beyond the reach of classical computing

Logical omniscience at the laboratory

This paper investigates the ability of individuals to make complex chains of reasoning, similar to those underlying the logic of iterated deletion of dominated strategies. Controlling for other-regarding preferences and beliefs about the rationality of others, we show, in the laboratory, that the ability of individuals to perform complex chains of iterative reasoning is better than previously thought. We conclude this from comparing our results with those from studies that use the same game without controlling for confounding factors. Subjects were able to perform about two to three iterations of reasoning on average as measured by our version of the Red-Hat Puzzle

Étude d’une orthèse d’avancée mandibulaire dans le traitement du syndrome d’apnées obstructives du sommeil

Introduction Le but de cette étude était d’évaluer l’efficacité et la tolérance d’une orthèse mandibulaire en traction de type optimisation de la retenue mandibulaire (orthèse type ORM) dans le traitement des syndromes d’apnées obstructives du sommeil (SAOS) modérés à sévères avec une prise en charge ambulatoire simple n’imposant pas de protocole de titration de l’avancée. Matériel et méthodes 40 patients, 10 présentant un SAOS sévère en échec ou refus de PPC avec un indice d’apnée-hypopnée (IAH) ≥ g0, et 30 un SAOS modéré (15 ≤ gAH < 30) ont été inclus dans 4 centres. Une polygraphie respiratoire nocturne, des questionnaires de qualité de vie et de qualité du sommeil ont été utilisés pour évaluer l’effet du traitement après 45 jours. Résultats 35 patients ont terminé l’étude. La fréquence des événements respiratoires, la somnolence diurne, la qualité du sommeil évaluée par le patient, le questionnaire SF-36 de qualité de vie, et l’index de qualité du sommeil du questionnaire de Pittsburgh (PSQI) ont été significativement améliorés avec l’orthèse. 60 % des patients étaient répondeurs au traitement avec une diminution de l’IAH d’au moins 50 %. L’observance du traitement était élevée (80 % des patients ont porté leur orthèse toutes les nuits). Les effets indésirables ont été mineurs et transitoires. Discussion L’efficacité sur les paramètres respiratoires et la somnolence de cette orthèse mandibulaire en retenue a été validée avec un taux de réponse similaire à celui publié dans la littérature. Cette étude montre une amélioration constante des paramètres de qualité de vie et de qualité du sommeil sous orthèse. Le traitement par l’orthèse type ORM dans le cadre d’une prise en charge ambulatoire simple sans titration systématique avec contrôle rapide par polygraphie ventilatoire est approprié pour la prise en charge des patients porteurs d’un SAOS, particulièrement si ce SAOS est modéré