Linear-in-Δ\Delta Lower Bounds in the LOCAL Model

By prior work, there is a distributed algorithm that finds a maximal fractional matching (maximal edge packing) in $O(\Delta)$ rounds, where $\Delta$ is the maximum degree of the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in $o(\Delta)$ rounds. Our work gives the first linear-in-$\Delta$ lower bound for a natural graph problem in the standard model of distributed computing---prior lower bounds for a wide range of graph problems have been at best logarithmic in $\Delta$.Comment: 1 + 21 pages, 10 figure

Federated Linear Contextual Bandits with User-level Differential Privacy

This paper studies federated linear contextual bandits under the notion of user-level differential privacy (DP). We first introduce a unified federated bandits framework that can accommodate various definitions of DP in the sequential decision-making setting. We then formally introduce user-level central DP (CDP) and local DP (LDP) in the federated bandits framework, and investigate the fundamental trade-offs between the learning regrets and the corresponding DP guarantees in a federated linear contextual bandits model. For CDP, we propose a federated algorithm termed as \robin and show that it is near-optimal in terms of the number of clients $M$ and the privacy budget $\varepsilon$ by deriving nearly-matching upper and lower regret bounds when user-level DP is satisfied. For LDP, we obtain several lower bounds, indicating that learning under user-level $(\varepsilon,\delta)$-LDP must suffer a regret blow-up factor at least {$\min\{1/\varepsilon,M\}$ or $\min\{1/\sqrt{\varepsilon},\sqrt{M}\}$} under different conditions.Comment: Accepted by ICML 202

Constraining primordial non-Gaussianity with cosmological weak lensing: shear and flexion

We examine the cosmological constraining power of future large-scale weak lensing surveys on the model of \emph{Euclid}, with particular reference to primordial non-Gaussianity. Our analysis considers several different estimators of the projected matter power spectrum, based on both shear and flexion, for which we review the covariances and Fisher matrices. The bounds provided by cosmic shear alone for the local bispectrum shape, marginalized over $\sigma_8$, are at the level of $\Delta f_\mathrm{NL} \sim 100$. We consider three additional bispectrum shapes, for which the cosmic shear constraints range from $\Delta f_\mathrm{NL}\sim 340$ (equilateral shape) up to $\Delta f_\mathrm{NL}\sim 500$ (orthogonal shape). The competitiveness of cosmic flexion constraints against cosmic shear ones depends on the galaxy intrinsic flexion noise, that is still virtually unconstrained. Adopting the very high value that has been occasionally used in the literature results in the flexion contribution being basically negligible with respect to the shear one, and for realistic configurations the former does not improve significantly the constraining power of the latter. Since the flexion noise decreases with decreasing scale, by extending the analysis up to $\ell_\mathrm{max} = 20,000$ cosmic flexion, while being still subdominant, improves the shear constraints by $\sim 10$ when added. However on such small scales the highly non-linear clustering of matter and the impact of baryonic physics make any error estimation uncertain. By considering lower, and possibly more realistic, values of the flexion intrinsic shape noise results in flexion constraining power being a factor of $\sim 2$ better than that of shear, and the bounds on $\sigma_8$ and $f_\mathrm{NL}$ being improved by a factor of $\sim 3$ upon their combination. (abridged)Comment: 30 pages, 4 figures, 4 tables. To appear on JCA

Robust pricing and hedging of double no-touch options

Double no-touch options, contracts which pay out a fixed amount provided an underlying asset remains within a given interval, are commonly traded, particularly in FX markets. In this work, we establish model-free bounds on the price of these options based on the prices of more liquidly traded options (call and digital call options). Key steps are the construction of super- and sub-hedging strategies to establish the bounds, and the use of Skorokhod embedding techniques to show the bounds are the best possible. In addition to establishing rigorous bounds, we consider carefully what is meant by arbitrage in settings where there is no {\it a priori} known probability measure. We discuss two natural extensions of the notion of arbitrage, weak arbitrage and weak free lunch with vanishing risk, which are needed to establish equivalence between the lack of arbitrage and the existence of a market model.Comment: 32 pages, 5 figure

Optimal Power Flow with Step-Voltage Regulators in Multi-Phase Distribution Networks

This paper develops a branch-flow based optimal power flow (OPF) problem for multi-phase distribution networks that allows for tap selection of wye, closed-delta, and open-delta step-voltage regulators (SVRs). SVRs are assumed ideal and their taps are represented by continuous decision variables. To tackle the non-linearity, the branch-flow semidefinite programming framework of traditional OPF is expanded to accommodate SVR edges. Three types of non-convexity are addressed: (a) rank-1 constraints on non-SVR edges, (b) nonlinear equality constraints on SVR power flows and taps, and (c) trilinear equalities on SVR voltages and taps. Leveraging a practical phase-separation assumption on the SVR secondary voltage, novel McCormick relaxations are provided for (c) and certain rank-1 constraints of (a), while dropping the rest. A linear relaxation based on conservation of power is used in place of (b). Numerical simulations on standard distribution test feeders corroborate the merits of the proposed convex formulation.Comment: This manuscript has been submitted to IEEE Transactions on Power System

Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law

In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spectral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate period. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed "metastability" of the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou