What Can We Learn Privately?

Learning problems form an important category of computational tasks that generalizes many of the computations researchers apply to large real-life data sets. We ask: what concept classes can be learned privately, namely, by an algorithm whose output does not depend too heavily on any one input or specific training example? More precisely, we investigate learning algorithms that satisfy differential privacy, a notion that provides strong confidentiality guarantees in contexts where aggregate information is released about a database containing sensitive information about individuals. We demonstrate that, ignoring computational constraints, it is possible to privately agnostically learn any concept class using a sample size approximately logarithmic in the cardinality of the concept class. Therefore, almost anything learnable is learnable privately: specifically, if a concept class is learnable by a (non-private) algorithm with polynomial sample complexity and output size, then it can be learned privately using a polynomial number of samples. We also present a computationally efficient private PAC learner for the class of parity functions. Local (or randomized response) algorithms are a practical class of private algorithms that have received extensive investigation. We provide a precise characterization of local private learning algorithms. We show that a concept class is learnable by a local algorithm if and only if it is learnable in the statistical query (SQ) model. Finally, we present a separation between the power of interactive and noninteractive local learning algorithms.Comment: 35 pages, 2 figure

On the Learnability of Monotone Functions

A longstanding lacuna in the field of computational learning theory is the learnability of succinctly representable monotone Boolean functions, i.e., functions that preserve the given order of the input. This thesis makes significant progress towards understanding both the possibilities and the limitations of learning various classes of monotone functions by carefully considering the complexity measures used to evaluate them. We show that Boolean functions computed by polynomial-size monotone circuits are hard to learn assuming the existence of one-way functions. Having shown the hardness of learning general polynomial-size monotone circuits, we show that the class of Boolean functions computed by polynomial-size depth-3 monotone circuits are hard to learn using statistical queries. As a counterpoint, we give a statistical query learning algorithm that can learn random polynomial-size depth-2 monotone circuits (i.e., monotone DNF formulas). As a preliminary step towards a fully polynomial-time, proper learning algorithm for learning polynomial-size monotone decision trees, we also show the relationship between the average depth of a monotone decision tree, its average sensitivity, and its variance. Finally, we return to monotone DNF formulas, and we show that they are teachable (a different model of learning) in the average case. We also show that non-monotone DNF formulas, juntas, and sparse GF2 formulas are teachable in the average case

ErmineJ: Tool for functional analysis of gene expression data sets

BACKGROUND: It is common for the results of a microarray study to be analyzed in the context of biologically-motivated groups of genes such as pathways or Gene Ontology categories. The most common method for such analysis uses the hypergeometric distribution (or a related technique) to look for "over-representation" of groups among genes selected as being differentially expressed or otherwise of interest based on a gene-by-gene analysis. However, this method suffers from some limitations, and biologist-friendly tools that implement alternatives have not been reported. RESULTS: We introduce ErmineJ, a multiplatform user-friendly stand-alone software tool for the analysis of functionally-relevant sets of genes in the context of microarray gene expression data. ErmineJ implements multiple algorithms for gene set analysis, including over-representation and resampling-based methods that focus on gene scores or correlation of gene expression profiles. In addition to a graphical user interface, ErmineJ has a command line interface and an application programming interface that can be used to automate analyses. The graphical user interface includes tools for creating and modifying gene sets, visualizing the Gene Ontology as a table or tree, and visualizing gene expression data. ErmineJ comes with a complete user manual, and is open-source software licensed under the Gnu Public License. CONCLUSION: The availability of multiple analysis algorithms, together with a rich feature set and simple graphical interface, should make ErmineJ a useful addition to the biologist's informatics toolbox. ErmineJ is available from

The persistent shadow of the supermassive black hole of M 87

In April 2019, the Event Horizon Telescope (EHT) Collaboration reported the first-ever event-horizon-scale images of a black hole, resolving the central compact radio source in the giant elliptical galaxy M 87. These images reveal a ring with a southerly brightness distribution and a diameter of ∼42 μas, consistent with the predicted size and shape of a shadow produced by the gravitationally lensed emission around a supermassive black hole. These results were obtained as part of the April 2017 EHT observation campaign, using a global very long baseline interferometric radio array operating at a wavelength of 1.3 mm. Here, we present results based on the second EHT observing campaign, taking place in April 2018 with an improved array, wider frequency coverage, and increased bandwidth. In particular, the additional baselines provided by the Greenland telescope improved the coverage of the array. Multiyear EHT observations provide independent snapshots of the horizon-scale emission, allowing us to confirm the persistence, size, and shape of the black hole shadow, and constrain the intrinsic structural variability of the accretion flow. We have confirmed the presence of an asymmetric ring structure, brighter in the southwest, with a median diameter of 43.3−3.1+1.5 μas. The diameter of the 2018 ring is remarkably consistent with the diameter obtained from the previous 2017 observations. On the other hand, the position angle of the brightness asymmetry in 2018 is shifted by about 30° relative to 2017. The perennial persistence of the ring and its diameter robustly support the interpretation that the ring is formed by lensed emission surrounding a Kerr black hole with a mass ∼6.5 × 109 M⊙. The significant change in the ring brightness asymmetry implies a spin axis that is more consistent with the position angle of the large-scale jet

First Sagittarius A* Event Horizon Telescope results. II. EHT and multiwavelength observations, data processing, and calibration

We present Event Horizon Telescope (EHT) 1.3 mm measurements of the radio source located at the position of the supermassive black hole Sagittarius A* (Sgr A*), collected during the 2017 April 5–11 campaign. The observations were carried out with eight facilities at six locations across the globe. Novel calibration methods are employed to account for Sgr A*'s flux variability. The majority of the 1.3 mm emission arises from horizon scales, where intrinsic structural source variability is detected on timescales of minutes to hours. The effects of interstellar scattering on the image and its variability are found to be subdominant to intrinsic source structure. The calibrated visibility amplitudes, particularly the locations of the visibility minima, are broadly consistent with a blurred ring with a diameter of ∼50 μas, as determined in later works in this series. Contemporaneous multiwavelength monitoring of Sgr A* was performed at 22, 43, and 86 GHz and at near-infrared and X-ray wavelengths. Several X-ray flares from Sgr A* are detected by Chandra, one at low significance jointly with Swift on 2017 April 7 and the other at higher significance jointly with NuSTAR on 2017 April 11. The brighter April 11 flare is not observed simultaneously by the EHT but is followed by a significant increase in millimeter flux variability immediately after the X-ray outburst, indicating a likely connection in the emission physics near the event horizon. We compare Sgr A*'s broadband flux during the EHT campaign to its historical spectral energy distribution and find that both the quiescent emission and flare emission are consistent with its long-term behavior.http://iopscience.iop.org/2041-8205Physic

First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration

We present Event Horizon Telescope (EHT) 1.3 mm measurements of the radio source located at the position of the supermassive black hole Sagittarius A* (Sgr A*), collected during the 2017 April 5–11 campaign. The observations were carried out with eight facilities at six locations across the globe. Novel calibration methods are employed to account for Sgr A*'s flux variability. The majority of the 1.3 mm emission arises from horizon scales, where intrinsic structural source variability is detected on timescales of minutes to hours. The effects of interstellar scattering on the image and its variability are found to be subdominant to intrinsic source structure. The calibrated visibility amplitudes, particularly the locations of the visibility minima, are broadly consistent with a blurred ring with a diameter of ∼50 μas, as determined in later works in this series. Contemporaneous multiwavelength monitoring of Sgr A* was performed at 22, 43, and 86 GHz and at near-infrared and X-ray wavelengths. Several X-ray flares from Sgr A* are detected by Chandra, one at low significance jointly with Swift on 2017 April 7 and the other at higher significance jointly with NuSTAR on 2017 April 11. The brighter April 11 flare is not observed simultaneously by the EHT but is followed by a significant increase in millimeter flux variability immediately after the X-ray outburst, indicating a likely connection in the emission physics near the event horizon. We compare Sgr A*’s broadband flux during the EHT campaign to its historical spectral energy distribution and find that both the quiescent emission and flare emission are consistent with its long-term behavior