function regardless of the original problem; iii) G can be −∞ for some α and β Theorem 1. Lower Bounds Property: If α ⪰ 0, then G(α, β ) ≤ p∗ where p∗ is the optimal value of the (original) primal problem duality d∗ ≤ p∗ always holds for all optimization problems, and can be used to find non-trivial lower bounds. The duality is said to be strong if d∗ = p∗. In this case, we can optimize the original problem constructed by composing these rules. In SVM, Mercer’s condition can be translated to another way to check whether K is a valid kernel. The kernel function K also defines the so-called ker- nel matrix over

� G is concave, can be −∞ for some α, β Feng Li (SDU) SVM December 28, 2021 19 / 82 The Lower Bounds Property If α ⪰ 0, then G(α, β ) ≤ p∗, where p∗ is the optimal value of the primal problem Proof: 2021 21 / 82 Weak Duality Weak duality: d∗ ≤ p∗ Always holds Can be used to find nontrivial lower bounds for difficult problems Optimal duality gap: p∗ − d∗ Feng Li (SDU) SVM December 28, 2021 22 / 82 Duality V.s. Strong Duality Weak duality: d∗ ≤ p∗ Always holds Can be used to find nontrivial lower bounds for difficult problems Strong duality: d∗ = p∗ Does not hold in general (Usually) holds for convex

handwritten samples in multiple inputs. Suppose, we have confusion in one input then we need to check again other inputs to recognize the correct context which takes the decision from the past. Workflow used to evaluate the prediction of the algorithm / Model (once the machine learn) and to cross check the efficiency of the learning process.  Compile the model: Compile the algorithm / model, the prediction with actual result of the test data.  Freeze, Modify or choose new algorithm: Check whether the evaluation of the model is successful. If yes, save the algorithm for future prediction

vsphere, kubernetes pip install "skypilot-nightly[aws,gcp]" 随后，您需要用如下命令确认是否能使用云： sky check For more information, check the official document and see if you have set up your cloud accounts correctly. Alternatively transformers.Trainer, output_dir: str, bias="none" ): """Collects the state dict and dump to disk.""" # check if zero3 mode enabled if deepspeed.is_deepspeed_zero3_enabled(): state_dict = trainer.model_wrapped issue that there␣ �→are problems with # loading the checkpoint when using LoRA with DeepSpeed. # Check this issue https://github.com/huggingface/peft/issues/746 for more␣ �→information. if ( list(pathlib

(Contd.) What if x ∈ {1, 2, · · · , u} and y ∈ {1, 2, · · · , k}? Can we get the same results? Check it yourself! Feng Li (SDU) GDA, NB and EM September 27, 2023 71 / 122 Classification by Naive Bayes y) �v t=1 count(i)(t) p(y) = �m i=1 1(y(i) = y) m where count(i)(t) = ni � j=1 1(x(i) j = t) Check them by yourselves! What if y = 1, 2, · · · , k? Feng Li (SDU) GDA, NB and EM September 27, 2023 get a solution Σj = �m i=1 ω(i) j (x(i) − µj)(x(i) − µj)T �m i=1 ω(i) j where j = 1, · · · , k Check the derivations by yourself! Feng Li (SDU) GDA, NB and EM September 27, 2023 112 / 122 Naive Bayes

2-D tensor e.g. black&white images 3-D tensor e.g. RGB images Tensors – Shape of Tensors ● Check with .shape() (5, ) (3, 5) (4, 5, 3) Note: dim in PyTorch == axis in NumPy dim 0 dim 0 dim 1 dim appropriate devices. ● CPU x = x.to(‘cpu’) ● GPU x = x.to(‘cuda’) Tensors – Device (GPU) ● Check if your computer has NVIDIA GPU torch.cuda.is_available() ● Multiple GPUs: specify ‘cuda:0’, ‘cuda:1’

denote string ▪ One – hot ▪ [0, 1, 0, 0, …] ▪ Embedding ▪ Word2vec ▪ glove Data type Type check Dimension 0 / rank 0 Loss Dim 0 Dim 1 / rank 1 Bias Linear Input Dim 1 Dim 2 Linear Input

择视频h最w关键帧作i该视频的首图。 • 效果a • r工评测w酷原始f封面图算法（s评测可对比出w劣l分的数据_a 算法w，占比)%.%% 内容理解——视频智能封面图 内容理解——总结 • ����������check��QU���������7�3� ��������NDCG ��1%���� • �������������� • 测试集a 语kr工标注gTQuPG VTuVh • 目前最高:215

whatever device (cuda or cpu) • Fallback to cpu if gpu is unavailable: • torch.cuda.is_available() • Check cpu/gpu tensor OR numpy array ? • type(t) or t.type() • returns • numpy.ndarray • torch.Tensor •

But since in this example we have only one feature, being able to plot this gives a nice sanity-check on our result. (3) Finally, we’d like to make some predictions using the learned hypothesis. Use