Lower bounds to randomized algorithms for graph properties

AbstractFor any property P on n-vertex graphs, let C(P) be the minimum number of edges needed to be examined by any decision tree algorithm for determining P. In 1975 Rivest and Vuillemin settled the Aanderra-Rosenberg Conjecture, proving that C(P)=Ω(n2) for every nontrivial monotone graph property P. An intriguing open question is whether the theorem remains true when randomized algorithms are allowed. In this paper we show that Ω(n(log n)112 edges need to be examined by any randomized algorithm for determining any nontrivial monotone graph property

Quantum replication at the Heisenberg limit

No process in nature can perfectly clone an arbitrary quantum state. But is it possible to engineer processes that replicate quantum information with vanishingly small error? Here we demonstrate the possibility of probabilistic super-replication phenomena where N equally prepared quantum clocks are transformed into a much larger number of M nearly perfect replicas, with an error that rapidly vanishes whenever M is small compared to the square of N. The quadratic replication rate is the ultimate limit imposed by Quantum Mechanics to the proliferation of information and is fundamentally linked with the Heisenberg limit of quantum metrology.Comment: 9 + 16 pages, 2 figures, published versio

Credible, Truthful, and Two-Round (Optimal) Auctions via Cryptographic Commitments

We consider the sale of a single item to multiple buyers by a revenue-maximizing seller. Recent work of Akbarpour and Li formalizes \emph{credibility} as an auction desideratum, and prove that the only optimal, credible, strategyproof auction is the ascending price auction with reserves (Akbarpour and Li, 2019). In contrast, when buyers' valuations are MHR, we show that the mild additional assumption of a cryptographically secure commitment scheme suffices for a simple \emph{two-round} auction which is optimal, strategyproof, and credible (even when the number of bidders is only known by the auctioneer). We extend our analysis to the case when buyer valuations are $\alpha$-strongly regular for any $\alpha > 0$, up to arbitrary $\varepsilon$ in credibility. Interestingly, we also prove that this construction cannot be extended to regular distributions, nor can the $\varepsilon$ be removed with multiple bidders

When do Models Generalize? A Perspective from Data-Algorithm Compatibility

One of the major open problems in machine learning is to characterize generalization in the overparameterized regime, where most traditional generalization bounds become inconsistent (Nagarajan and Kolter, 2019). In many scenarios, their failure can be attributed to obscuring the crucial interplay between the training algorithm and the underlying data distribution. To address this issue, we propose a concept named compatibility, which quantitatively characterizes generalization in a both data-relevant and algorithm-relevant manner. By considering the entire training trajectory and focusing on early-stopping iterates, compatibility exploits the data and the algorithm information and is therefore a more suitable notion for generalization. We validate this by theoretically studying compatibility under the setting of solving overparameterized linear regression with gradient descent. Specifically, we perform a data-dependent trajectory analysis and derive a sufficient condition for compatibility in such a setting. Our theoretical results demonstrate that in the sense of compatibility, generalization holds with significantly weaker restrictions on the problem instance than the previous last iterate analysis

PrivacyFL: A simulator for privacy-preserving and secure federated learning

Federated learning is a technique that enables distributed clients to collaboratively learn a shared machine learning model while keeping their training data localized. This reduces data privacy risks, however, privacy concerns still exist since it is possible to leak information about the training dataset from the trained model's weights or parameters. Setting up a federated learning environment, especially with security and privacy guarantees, is a time-consuming process with numerous configurations and parameters that can be manipulated. In order to help clients ensure that collaboration is feasible and to check that it improves their model accuracy, a real-world simulator for privacy-preserving and secure federated learning is required. In this paper, we introduce PrivacyFL, which is an extensible, easily configurable and scalable simulator for federated learning environments. Its key features include latency simulation, robustness to client departure, support for both centralized and decentralized learning, and configurable privacy and security mechanisms based on differential privacy and secure multiparty computation. In this paper, we motivate our research, describe the architecture of the simulator and associated protocols, and discuss its evaluation in numerous scenarios that highlight its wide range of functionality and its advantages. Our paper addresses a significant real-world problem: checking the feasibility of participating in a federated learning environment under a variety of circumstances. It also has a strong practical impact because organizations such as hospitals, banks, and research institutes, which have large amounts of sensitive data and would like to collaborate, would greatly benefit from having a system that enables them to do so in a privacy-preserving and secure manner

On Computational Power of Quantum Read-Once Branching Programs

In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting technique, we show that any Boolean function with linear polynomial presentation can be computed by a quantum read-once branching program using a relatively small (usually logarithmic in the size of input) number of qubits. Then we show that the described class of Boolean functions is closed under the polynomial projections.Comment: In Proceedings HPC 2010, arXiv:1103.226