Feng Li (SDU) GDA, NB and EM September 27, 2023 57 / 122 Lagrange Multiplier Maximize f (x, y) subject to g(x, y) = 0 f (x, y) is maximized at point (x0, y0) where they have common tangent line such (SDU) GDA, NB and EM September 27, 2023 58 / 122 Lagrange Multiplier (Contd.) Maximize f (x, y, z) subject to g(x, y, z) = 0 r(t) = (x(t), y(t), z(t)) be an arbitrary parameterized curve which lies on the log p(x(i)) = m � i=1 log k � y=1 � �p(y) n � j=1 pj(x(i) j | y) � � Maximizing ℓ(θ) subject to the following constraints p(y) ≥ 0 for ∀y ∈ {1, · · · , k}, and �k y=1 p(y) = 1 For ∀y ∈ {1

of textual data, the words have a certain affinity to each other. They are also attracted to the subject matter. We term this affinity or attraction as attention. Words like run, kick or throw have higher sequence. That makes sense because typically21 the sentences in both the languages begin with a subject. The attention matrix contains scores for each pair of elements from the two sequences. It takes

use as little data for training is critical when the user-data might be sensitive to handling / subject to various restrictions such as the General Data Protection Regulation (GDPR) law6 in Europe. Hence

constraints. The aim of the above optimiza- tion problem is to minimizing the objective function f(ω) subject to the inequal- ity constraints g1(ω), · · · , gk(ω) and the equality constraints h1(ω), · · · ,

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将尝试找到一个函数f ∗ F，这是我们在F中的最佳选择。例如，给定一个具有X特性和y标签的数据集，我们可 以尝试通过解决以下优化问题来找到它： f ∗ F := argmin f L(X, y, f) subject to f ∈ F. (7.6.1) 那么，怎样得到更近似真正f ∗的函数呢？唯一合理的可能性是，我们需要设计一个更强大的架构F′。换句话 说，我们预计f ∗ F′比f ∗ F“更近似”。然而，如果F 凸优化的一个很好的特性是能够让我们有效地处理约束（constraints）。即它使我们能够解决以下形式的约 束优化（constrained optimization）问题： minimize x f(x) subject to ci(x) ≤ 0 for all i ∈ {1, . . . , N}. (11.2.16) 这里f是目标函数，ci是约束函数。例如第一个约束c1(x) = ∥x∥2 − 1，则参数x被限制为单位球。如果第二个