holds with equality, ω∗ actually minimizes L(ω, α∗, β ∗) over ω. 2.2.3 Karush-Kuhn-Tucker (KKT) Conditions We assume that the objective function and the inequality constraint functions are differentiable domains of the primal problem and the dual problem, respectively, we have the primal feasibility conditions (18)∼(18) and the dual feasibility condition (20) holds gi(ω∗) ≤ 0, ∀i = 1, · · · , k (18) hj(ω∗) hj(ω∗) = 0, ∀j = 1, · · · , l (19) α∗ i ≥ 0, ∀i = 1, · · · , k (20) (21) All these conditions (16)∼(20) are so-called Karush-Kuhn-Tucker (KKT) condi- tions. For any optimization problem with differentiable

December 28, 2021 15 / 82 Convex Optimization Review Optimization Problem Lagrangian Duality KKT Conditions Convex Optimization S. Boyd and L. Vandenberghe, 2004. Convex Optimization. Cambridge university Karush-Kuhn-Tucker (KKT) Conditions Let ω∗ and (α∗, β ∗) by any primal and dual optimal points wither zero duality gap (i.e., the strong duality holds), the following conditions should be satisfied Stationarity: problems Strong duality: d∗ = p∗ Does not hold in general (Usually) holds for convex problems Conditions that guarantee strong duality in convex problems are called constraint qualifications Feng Li

where xj ∈ {0, 1} for ∀j ∈ [n], we have vj = 2 for ∀j. Note that, p(y) satisfies the following two conditions p(y) ≥ 0, ∀y ∈ [k] k � y=1 p(y) = k � y=1 �m i=1 1(y(i) = y) + 1 m + k = �k y=1 �m i=1 vector given that the data sample is labeled by y. Also, p(t | y) should respect the following conditions: i) p(t | y) ≥ 0, and ii) �v t=1 p(t | y) = 1. We also define p(y) = P(Y = y) for ∀y ∈ [k]. We its label being z(i) ∈ Ω (i.e., Qi(z(i)) = P(Z(i) = z(i))). Qi(z(i)) should satisfy the following conditions: � z(i)∈Ω Qi(z(i)) = 1, Qi(z(i)) ≥ 0, ∀z(i) ∈ Ω Also, suppose φ(Z(i)) is a function of random

Bayes’ theorem (or Bayes’ rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event P(A | B) = P(B | A)P(A) P(B) In the Bayesian interpretation Feng Li (SDU) GDA, NB and EM September 27, 2023 83 / 122 Convex Functions (Contd.) First-order conditions: Suppose f is differentiable (i.e., its gradient ∇f exists at each point in domf , which is open) Feng Li (SDU) GDA, NB and EM September 27, 2023 84 / 122 Convex Functions (Contd.) Second-order conditions: Assume f is twice differentiable (i.t., its Hes- sian matrix or second derivative ∇2f exists at

learning_rate (Float) {'default': 0.0001, 'conditions': [], 'min_value': 0.0001, 'max_value': 0.01, 'step': None, 'sampling': 'log'} dropout_rate (Float) {'default': 0.1, 'conditions': [], 'min_value': 0.1, 'max_value':

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Linear Regression September 13, 2023 15 / 31 GD Algorithm (Contd.) Stopping criterion (i.e., conditions to convergence) the gradient has its magnitude less than or equal to a predefined thresh- old

underexposed photos, and contains a small number of underexposed images that cover limited lighting conditions. Our Dataset Quantitative Comparison: Our Dataset Method PSNR SSIM HDRNet 26.33 0.743 DPE

57 Supervised Regression Problems Predict tomorrow’s stock market price given current market conditions and other possible side information Predict the age of a viewer watching a given video on YouTube

-th residual block for a network with five blocks and the final probability ( ). Under these conditions, the expected network depth during training reduces to . By expected network depth we informally