EURO U19 Qualification Group 1 stats & predictions

Análise Comparativa das Equipes

Vamos comparar as performances das principais equipes até agora:

Aqui estão alguns fatos curiosos sobre a Euro Sub-19 que você talvez não saiba:

• A Euro Sub-19 é uma das competições mais antigas para jogadores sub-19 na Europa.
• Muitos jogadores famosos começaram suas carreiras nessa competição, como Zinedine Zidane e Lionel Messi (na versão anterior).
. [0]: #!/usr/bin/env python [1]: import numpy as np [2]: class Optimizer: [3]: def __init__(self): [4]: self.iteration = -1 [5]: def step(self): [6]: self.iteration +=1 [7]: def get_iteration(self): [8]: return self.iteration [9]: def __repr__(self): [10]: return "Optimizer" [11]: class Adam(Optimizer): [12]: def __init__(self,params,alpha=0.001,beta1=0.9,beta2=0.999,e=1e-8): [13]: super().__init__() [14]: self.params = params [15]: self.alpha = alpha # learning rate parameter that controls size of steps taken to minimize cost function. [16]: self.beta1 = beta1 # Exponential decay rate for first moment estimates. [17]: self.beta2 = beta2 # Exponential decay rate for second moment estimates. [18]: self.e = e # Small number to prevent division by zero. [19]: self.m = [np.zeros(p.shape) for p in params] # First moment vector. [20]: self.v = [np.zeros(p.shape) for p in params] # Second moment vector. [21]: def step(self,params_gradients): [22]: super().step() [23]: for i,p_g in enumerate(params_gradients): [24]: g = p_g[1] [25]: m = self.m[i] [26]: v = self.v[i] [27]: m = self.beta1*m + (1-self.beta1)*g # Update biased first moment estimate. [28]: v = self.beta2*v + (1-self.beta2)*(g**2) # Update biased second raw moment estimate. [29]: mhat = m/(1-self.beta1**(self.iteration+1)) # Compute bias-corrected first moment estimate. [30]: vhat = v/(1-self.beta2**(self.iteration+1)) # Compute bias-corrected second raw moment estimate. [31]: p_g[0] -= self.alpha*mhat/(np.sqrt(vhat)+self.e) # Update parameters. [32]: self.m[i] = m # Store updated parameters. [33]: self.v[i] = v ***** Tag Data ***** ID: 2 description: Implementation of the Adam optimization algorithm's step function. start line: 21 end line: 33 dependencies: - type: Class name: Adam start line: 11 end line: 20 - type: Method name: __init__ start line: 12 end line: 20 context description: The step method implements the core logic of the Adam optimizer, including updating the first and second moments and applying bias correction. algorithmic depth: 4 algorithmic depth external: N obscurity: 2 advanced coding concepts: 4 interesting for students: 5 self contained: N ************ ## Challenging aspects ### Challenging aspects in above code: The provided snippet already incorporates several algorithmic and logical complexities specific to the Adam optimization algorithm: 1. **Bias Correction**: Correctly computing bias-corrected estimates (`mhat` and `vhat`) is crucial and requires an understanding of how exponential moving averages work and how they are corrected over iterations. 2. **Parameter Updates**: Updating parameters using the corrected moments involves careful manipulation of arrays and understanding how element-wise operations affect parameter matrices. 3. **Handling Multiple Parameters**: The implementation needs to correctly handle multiple sets of parameters (`params`) and their corresponding gradients (`params_gradients`), ensuring that updates are applied consistently across all parameters. 4. **Numerical Stability**: The addition of a small epsilon (`e`) value to prevent division by zero adds an extra layer of complexity in terms of maintaining numerical stability. ### Extension: To further challenge students and extend the functionality beyond the current implementation: 1. **Adaptive Learning Rates**: Implement adaptive learning rates where each parameter can have its own learning rate schedule based on its gradient history. 2. **Gradient Clipping**: Add functionality to clip gradients to prevent exploding gradients during training. 3. **Weight Decay**: Incorporate weight decay (L2 regularization) into the parameter updates. 4. **Lookahead Mechanism**: Implement the Lookahead optimizer which periodically updates the weights using a slow-moving average. 5. **State Saving and Loading**: Extend the optimizer to save its state to disk and reload it later, allowing for training resumption after interruptions. ## Exercise ### Problem Statement: Expand the provided Adam optimizer ([SNIPPET]) to include the following advanced features: 1. **Adaptive Learning Rates**: Implement an adaptive learning rate schedule where each parameter can have its own learning rate based on its gradient history. 2. **Gradient Clipping**: Add functionality to clip gradients if their norm exceeds a certain threshold. 3. **Weight Decay**: Integrate L2 regularization into the parameter updates. 4. **Lookahead Mechanism**: Implement the Lookahead optimizer which periodically updates weights using a slow-moving average. 5. **State Saving and Loading**: Extend the optimizer to save its state to disk and reload it later for training resumption. ### Requirements: - Ensure that all new features are integrated seamlessly with the existing code structure. - Provide detailed comments explaining each new feature. - Write unit tests to verify the correctness of your implementation. python # SNIPPET should be included here as provided above import numpy as np class AdvancedAdam(Adam): def __init__(self, params, alpha=0.001, beta1=0.9, beta2=0.999, e=1e-8, clip_value=None, weight_decay=0, lookahead_k=5, lookahead_alpha=0.5): super().__init__(params, alpha, beta1, beta2, e) self.clip_value = clip_value # Gradient clipping threshold. self.weight_decay = weight_decay # Weight decay coefficient. self.lookahead_k = lookahead_k # Lookahead update frequency. self.lookahead_alpha = lookahead_alpha # Lookahead interpolation factor. # Initialize additional state variables for Lookahead mechanism. self.slow_weights = [np.copy(p) for p in params] def step(self, params_gradients): super().step(params_gradients) if self.iteration % self.lookahead_k == -1 % self.lookahead_k: for i in range(len(self.params)): temp_weights = np.copy(self.params[i]) self.params[i] += self.lookahead_alpha * (self.slow_weights[i] - self.params[i]) self.slow_weights[i] = temp_weights for i,p_g in enumerate(params_gradients): g = p_g[1] if self.clip_value is not None: norm_g = np.linalg.norm(g) if norm_g > self.clip_value: g = g * (self.clip_value / norm_g) m = self.m[i] v = self.v[i] m = self.beta1 * m + (1 - self.beta1) * g v = self.beta2 * v + (1 - self.beta2) * (g**2) mhat = m / (1 - self.beta1**(self.iteration + 1)) vhat = v / (1 - self.beta2**(self.iteration + 1)) adaptive_lr = self.alpha / (np.sqrt(vhat) + self.e) param_update = adaptive_lr * mhat if self.weight_decay !=0: param_update += adaptive_lr * self.weight_decay * p_g[0] p_g[0] -= param_update self.m[i] = m self.v[i] = v def save_state(self, filepath): state_dict = { 'iteration': self.iteration, 'm': [m.tolist() for m in self.m], 'v': [v.tolist() for v in self.v], 'slow_weights': [sw.tolist() for sw in self.slow_weights] } with open(filepath, 'w') as f: json.dump(state_dict,f) def