Maximum Margin Planning

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转载最新推荐文章于 2024-07-06 18:32:46 发布 · 363 阅读

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原文链接：http://www.cnblogs.com/justin_s/archive/2011/06/05/2073279.html

本文探讨了如何自动化地将感知特征映射到成本，不再局限于与期望点的距离，而是扩展到与示例数据点集的距离。通过引入二次规划形式和有效优化方法，论文提出了在非可微目标函数上的收敛解决方案，并提供了伪代码以辅助理解。

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[source] ICML

[year] 2006

to automate the mapping from perception features to costs

不再只是与一个点（期望）间的距离，而是与示例数据点集间的距离

文章先提出Quadratic Programming Formulation

再Efficient Optimization，有伪代码，取subgradient 方向，即使在non-differentiable目标函数上依然收敛。

From wiki:

Subgradient methods are iterative methods for solving convex minimization problems. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient methods are convergent when applied even to a non-differentiable objective function. When the objective function is differentiable, subgradient methods for unconstrained problems use the same search direction as the method of steepest descent.

Subgradient methods are slower than Newton's method when applied to minimize twice continuously differentiable convex functions. However, Newton's method fails to converge on problems that have non-differentiable kinks.

The subgradient

The concepts of subderivative and subdifferential can be generalized to functions of several variables. If f:U→ R is a real-valued convex function defined on a convex open setin the Euclidean space Rⁿ, a vector v in that space is called a subgradient at a point x₀ in U if for any x in U one has

$f(x)-f(x_0)\ge v\cdot (x-x_0)$

where the dot denotes the dot product. The set of all subgradients at x₀ is called the subdifferential at x₀ and is denoted ∂f(x₀). The subdifferential is always a nonempty convex compact set.

These concepts generalize further to convex functions f:U→ R on a convex set in a locally convex space V. A functional v^∗ in the dual space V^∗ is called subgradient at x₀ in Uif

$f(x)-f(x_0)\ge v^*(x-x_0).$

The set of all subgradients at x₀ is called the subdifferential at x₀ and is again denoted ∂f(x₀). The subdifferential is always a convex closed set. It can be an empty set; consider for example an unbounded operator, which is convex, but has no subgradient. If f is continuous, the subdifferential is nonempty.

次导数和次微分的概念可以推广到多元函数。如果f:U→ R是一个实变量凸函数，定义在欧几里得空间Rⁿ内的凸集，则该空间内的向量v称为函数在点x₀的次梯度，如果对于所有U内的x，都有：

$f(x)-f(x_0)\ge v\cdot (x-x_0)$

所有次梯度的集合称为次微分，记为∂f(x₀)。次微分总是非空的凸紧集。

转载于:https://www.cnblogs.com/justin_s/archive/2011/06/05/2073279.html