经济代写|博弈论代写Game Theory代考|ECON6025

2023年3月24日

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经济代写|博弈论代写Game Theory代考|Formal Methods and Verification

Trinity includes formal methods for verifying deep neural networks. An initial prototype for improving trust of deep learning models is publicly available in an open-source tool, Trinity. ${ }^1$ At an abstract level, a deep neural network computes a function from a set of inputs to some set of outputs. The formal verification addresses the following: Given a neural network (NN), and constraints (assumptions) which define a set of inputs to the network, provide a tight over-approximation (guarantee) of the output set. This serves as one of the main primitives in verification of neural networks. Sherlock uses mixed-integer linear programming (MILP) solver to address this verification problem but it does not merely compile the verification into an MILP problem. Sherlock first uses sound piecewise linearization of the nonlinear activation function to define an encoding of the neural network semantics into mixed-integer constraints involving real-valued variables and binary variables that arise from the (piecewise) linearized activation functions. Such an encoding into MILP is a standard approach to handling piecewise linear functions. As such, the input constraints $\phi(\mathbf{x})$ are added to the MILP and next, the output variable is separately maximized and minimized to infer the corresponding guarantee that holds on the output. This enables us to infer an assume-guarantee contract on the overall deep neural network. Sherlock augments this simple use of MILP solving with a local search that exploits the local continuity and differentiability properties of the function represented by the network. These properties are not exploited by MILP solvers which typically use a branch-and-cut approach. On the other hand, local search alone may get “stuck” in local minima. Sherlock handles local minima by using the MILP solver to search for a solution that is “better” than the current local minimum or conclude that no such solution exists. Thus, by alternating between inexpensive local search iterations and relatively expensive MILP solver calls, Sherlock can exploit local properties of the neural network function but at the same time avoid the problem of local minima, and thus, solve the verification of deep neural networks more efficiently.

Deep neural networks are very common in applications such as image classification and autonomous control. In image classification networks, since each image is a point in the high dimensional pixel space, a polyhedral set may be used to represent all possible bounded perturbations to a given image. If, for such a set, we can guarantee that the output of the classification remains unaltered, then we have proved that the neural network is robust to bounded pixel noise. Besides image classification, neural networks are increasingly used in the control of autonomous systems, such as self-driving cars, unmanned aerial vehicles, and other robotic systems. A typical approach to verify these systems involves a reachability computation to estimate the forward reachable set as time evolves. Using this, it is possible to prove that, no matter what the initial condition of a system is, it always reaches a target region in finite time. For instance, we wish to prove that, an autonomous car whose inputs are provided by a neural network controller’s feedback, will remain within a fixed lateral distance from the center of the road (desired trajectory), while remaining under the speed limit.

经济代写|博弈论代写Game Theory代考|Top-down Analysis by Synthesis

Machine learning models are typically bottom-up inferring intent from the effect which makes the learned models fragile and nonresilient. Consider the component of a car’s vision system concerned with detecting traffic lanes. A basic method will look for more-or-less straight lines painted on the road, and a bottom-up approach will perform this process afresh as each image frame is processed. But this is inefficientthe traffic lanes in the current image frame are likely to be similar to those in the previous few frames and we should surely use this to seed the searchand it is fragilemissing or scuffed lane markers might cause lanes to go undetected where they could have been extrapolated from previous images. A better approach builds a model of the road and its traffic lanes and uses this to seed the search for lanes in the current image by predicting their location.

There will be some uncertainty in the model and its projection of lanes, so what is sent to the vision system will be a best guess, or perhaps a probability distribution over several such estimates. The vision system will use this to seed its search for lanes in the current image and will send back the difference or “error” between the prediction and its current observation. The error signal is used to refine the model in a way that aims to minimize future prediction errors and thereby bring it closer to reality.

This is an example of “analysis by synthesis” meaning that we formulate hypotheses (i.e., candidate world models) and favor those whose predictions match the input data. In practical applications, we need to consider the level of the “predictions” concerned: do we use the world model to synthesize the raw data (e.g., pixels) that we predict the sensor will detect, or do we target some higher level of its local processing (e.g., objects)? This is a topic for experimental research, but we would expect a level of representation comparable to that produced by an object detector to be suitable.

博弈论代考

经济代写|博弈论代写Game Theory代考|Formal Methods and Verification

Trinity 包括用于验证深度神经网络的正式方法。用于提高深度学习模型信任度的初始原型在开源工具 Trinity 中公开可用。1在抽象层面上，深度神经网络计算从一组输入到一组输出的函数。形式验证解决以下问题：给定一个神经网络 (NN) 和定义一组网络输入的约束（假设），提供输出集的紧密过度近似（保证）。这是神经网络验证的主要原语之一。Sherlock 使用混合整数线性规划 (MILP) 求解器来解决此验证问题，但它不仅仅将验证编译为 MILP 问题。Sherlock 首先使用非线性激活函数的合理分段线性化，将神经网络语义编码定义为混合整数约束，涉及由（分段）线性化激活函数产生的实值变量和二元变量。这种对 MILP 的编码是处理分段线性函数的标准方法。因此，输入约束ϕ(x)被添加到 MILP，接下来，输出变量分别最大化和最小化以推断对输出保持的相应保证。这使我们能够在整个深度神经网络上推断出一个假设保证合同。Sherlock 通过利用网络表示的函数的局部连续性和可微性属性的局部搜索增强了 MILP 求解的这种简单使用。这些属性未被通常使用分支切割方法的 MILP 求解器利用。另一方面，单独的局部搜索可能会“陷入”局部最小值。Sherlock 通过使用 MILP 求解器搜索比当前局部最小值“更好”的解决方案或断定不存在这样的解决方案来处理局部最小值。因此，

有限元方法代写

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。