Worst-Case Violation of Sampled Convex Programs for Optimization with Uncertainty

2011 ◽  
Vol 152 (1) ◽  
pp. 171-197 ◽  
Author(s):  
Takafumi Kanamori ◽  
Akiko Takeda
Author(s):  
Sven Gowal ◽  
Krishnamurthy Dvijotham ◽  
Robert Stanforth ◽  
Timothy Mann ◽  
Pushmeet Kohli

This paper addressed the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (e.g., robustness to bounded norm adversarial perturbations). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime, i.e., it can be stopped at any time and a valid bound on the maximum violation can be obtained. Finally, we highlight how this approach can be used to train models that are amenable to verification.


Author(s):  
J.D. Geller ◽  
C.R. Herrington

The minimum magnification for which an image can be acquired is determined by the design and implementation of the electron optical column and the scanning and display electronics. It is also a function of the working distance and, possibly, the accelerating voltage. For secondary and backscattered electron images there are usually no other limiting factors. However, for x-ray maps there are further considerations. The energy-dispersive x-ray spectrometers (EDS) have a much larger solid angle of detection that for WDS. They also do not suffer from Bragg’s Law focusing effects which limit the angular range and focusing distance from the diffracting crystal. In practical terms EDS maps can be acquired at the lowest magnification of the SEM, assuming the collimator does not cutoff the x-ray signal. For WDS the focusing properties of the crystal limits the angular range of acceptance of the incident x-radiation. The range is dependent upon the 2d spacing of the crystal, with the acceptance angle increasing with 2d spacing. The natural line width of the x-ray also plays a role. For the metal layered crystals used to diffract soft x-rays, such as Be - O, the minimum magnification is approximately 100X. In the worst case, for the LEF crystal which diffracts Ti - Zn, ˜1000X is the minimum.


2008 ◽  
Author(s):  
Sonia Savelli ◽  
Susan Joslyn ◽  
Limor Nadav-Greenberg ◽  
Queena Chen

Author(s):  
Akira YAMAWAKI ◽  
Hiroshi KAMABE ◽  
Shan LU
Keyword(s):  

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