College Level Neural Nets [I] - Basic Nets: Math & Practice!

Explore how neural networks build simple linear classifiers using two features, height and weight, to separate classes with a decision boundary and learn from labeled training data.

Examine how insufficiently representative training data shapes a decision boundary, leading to overfitting, and how linear and nonlinear classifiers, including the perceptron, underpin neural networks.

Derive the perceptron as a linear classifier for two features x1 and x2, and compute its decision boundary line x2 = 3/2 x1 from training data. Use F(X) to classify.

Visualize how the perceptron equation defines a decision boundary as the intersection of a plane with the input plane, explaining why points on one side yield positive values.

Demonstrates that the weight vector is perpendicular to the decision boundary via the dot product, with f(x)=w^T x and positive and negative regions on each side.

Explore how the perceptron weight vector remains perpendicular to a shifted decision boundary, even when the origin is not on the boundary, using w^T x and the intercept form.

Visualize how adding a bias term augments the perceptron weight vector and shifts the decision boundary from the x1-x2 plane to the x3=1 plane, while maintaining the same classification.

Show how a linear classifier uses f(x)=W^T x + B and a bias term, with sign or unit step activation to label data as class 1 or -1.

Explore graphical representations of neural networks, from a single neuron with input x and weights w to multi-neuron layouts with bias terms, activation functions, and output y.

Explore supervised, unsupervised, and reinforcement learning, including classification and regression, clustering, data compression, and generative models, with agent learning and rewards.

design a single neuron to classify two-dimensional training points by graphically selecting a decision boundary and deriving weights and bias for a perceptron.

Explore how multi-layer perceptrons solve not linearly separable data by combining simple perceptrons into layers, using intermediate activations y1 and y2 and a final classifier.

Design a graphical multi-layer perceptron to solve a not linearly separable classification problem by using three hidden-layer perceptrons, computing line equations, and a second-layer perceptron to fuse outputs.

Explore how a hidden layer in a multi-layer perceptron transforms height and weight into intermediate features, making the data linearly separable for final classification.

Explore how a multi-layer neural network builds progressively complex, hierarchical features from four inputs - height, weight, heart condition, and lung condition - to improve classification with deeper layers.

Generate new features, such as x3 equals x1 x2, to convert the xor problem from nonlinear to linearly separable in higher dimensions, enabling a single perceptron to classify data.

Explore multi-class classification with three perceptrons, one per class. Compare threshold activation with sigmoid and 10 H activation to model confidence by distance to decision boundaries.

Transform the neuron equations into a weight matrix times an input vector plus a bias, highlighting vectorized notation for fast matrix operations on GPUs.

Learn how the perceptron learning rule updates weights to separate linearly separable data by adjusting a random hyperplane when misclassifications occur.

Update the weight vector when a point is misclassified using the perceptron learning rule, by adding or subtracting eta times x0 toward the correct class.

Prove the perceptron convergence theorem using the perceptron learning rule. Assume data is linearly separable and converges in a finite number of iterations.

Prove a part of the perceptron convergence inequality by showing w_{N+1}^2 >= (N−t)^2 alpha^2 when t of the n examples are correctly classified.

The video presents the final inequality in the perceptron convergence theorem by expanding norms, summing bounds, and bounding growth with a positive beta.

Identify the three main problems of the threshold perceptron: non-unique predictions in multi-class regions, convergence only for linearly separable data, and training challenges in multi-layer networks, which backpropagation later solved.

Replace the threshold activation with the sigmoid function to enable smooth optimization, derive its derivative sigma(z)(1-sigma(z)), and discuss beta-controlled slope and probability interpretation.

Derive the gradient descent algorithm for minimizing a two-parameter cost function with weights W1 and W2. Show iterative updates W^{k+1}=W^k-eta grad C to guarantee negative change toward a local minimum.

Learn how gradient descent updates weights using the derivative of the cost function to move toward its minimum, and how learning rate size affects overshoot or slow progress.

Update weights using gradient descent across a multi-output network, and explore a numerical derivative method before backpropagation. Apply mini-batch and online gradient descent to train large datasets efficiently.

Explore how gradient descent converges for the function E(w)= w1^2 + 10 w2^2 by testing learning-rate ranges that satisfy |1-2 eta|<1 and |1-20 eta|<1, revealing convergence versus oscillation.

This lecture demonstrates that a multi-layer perceptron with linear activations equals a single-layer network, since successive linear mappings compose into one equivalent weight matrix.

Derivation of back propagation presents an efficient method to compute the partial derivative of the cost with respect to network weights, enabling practical gradient descent and scalable neural network training.

Derive back propagation in a general, layer-agnostic way using the chain rule, compute z values, and obtain final-layer weight derivatives as delta times previous activations.

Derive the derivative of the cost with respect to weights using the chain rule, propagate deltas backward through layers, and update weights by gradient descent.

Vectorize backpropagation by transforming delta calculations into a gradient vector and applying the derivative of the cost with respect to activations using sigma prime z, replacing loops for efficiency.

Demonstrates vectorized backpropagation for two neurons, deriving delta updates via the weight matrix transposed and sigma dash z, and expressing z as W a_{l-1} plus bias.

Explore vectorization of backpropagation by deriving the weight matrix gradients via outer products of delta vectors and activations, and apply the final backpropagation equations.

Leverage batch vectorization to perform backpropagation across multiple examples in parallel. Represent inputs as matrices and update weights using matrix-based activations and deltas.

Learn how regression models fit data, why overfitting and underfitting occur, and how regularization and polynomial features help generalize to unseen data.

use regularization to prevent overfitting by penalizing large weights in the cost function with a lambda term, and tune it via cross validation across training, validation, and test sets.

Learn about L2 and L1 regularization, data augmentation, and data mutation to increase training data and variability, reduce overfitting, and improve model generalization.

The video derives how L2 regularization introduces a lambda over n term, causing weight decay in gradient descent. It then shows L1 updates using a sign-based subgradient.

Compare L1 and L2 regularization, showing how L2 shrinks large weights more while L1 promotes sparsity, using intuitive, numerical, and graphical analyses.

Explore the intuition of dropout regularization in neural networks, where randomly deactivating neurons during training reduces overfitting and at test time all neurons contribute to decisions.

Explore dropout and inverted dropout in neural nets, choosing keep probabilities, applying dropout to input layers, and training versus testing behavior with scaling to maintain consistent activations.

Use cross-validation with training, validation, and test splits to detect overfitting or underfitting, tuning hyperparameters and applying regularization to improve generalization while tracking accuracy.

Explain why accuracy fails with imbalanced classes, use confusion matrix to illustrate true/false positives and negatives, and discuss class-weighted loss and resampling to improve minority-class performance.

Define precision, recall, and the F1 score for binary and multiclass classification, explaining true positives, false positives, and false negatives. Show how the F1 score balances precision and recall.

The lecture compares the F1 score with the simple average and explains why F1 better handles imbalanced precision and recall, with concrete cases showing why averages can mislead.

Learn how changing thresholds affects precision and recall using sigmoid probabilities, with examples, and compare models via the precision–recall and ROC curves.

Explore the receiver operating characteristics curve (ROC) and AUC as metrics to compare classifiers, focusing on true positive rate and false positive rate, threshold selection, and area under the curve.

Introduce momentum to gradient descent by adding a velocity term and memory, letting updates accumulate in a consistent direction to accelerate training toward the global minimum.

Explain how momentum speeds gradient descent in high-dimensional nets, reduces oscillations on elongated cost surfaces, and helps avoid saddle points, while showing integration with L2 regularization and Adam.

Explore adaptive gradient methods like Adagrad and RMSProp, which assign per-weight learning rates by accumulating squared gradients, with RMSProp adding alpha decay to stabilize memory and momentum.

Explore the Adam algorithm, its combination of momentum and RMSProp, and how learning rate decay shapes training by balancing velocity and gradient-driven updates.

Explain the vanishing gradient problem in backpropagation with sigmoid activation, showing near-zero gradients from large z, and outline remedies: switch activation, adjust cost function, weight initialization, and input normalization.

Center each feature by shifting its mean to zero. Normalize each feature on its own to achieve unit variance, while keeping the features independent and speeding training.

Learn how input centering enables mixed weight updates and faster learning, and how input normalization equalizes feature scales to smooth the cost surface and ensure consistent training and testing.

Initialize weights randomly to break symmetry; identical weights cause identical activations across neurons and slow learning, even to the point of no learning with zero or fixed values.

Explore weight initialization to control the z value and combat vanishing gradients. Use a zero-mean, unit-variance normal distribution and scale by 1 over sqrt(n) for the previous layer size.

Explore activation functions beyond sigmoid, such as tanh and relu variants, to mitigate vanishing gradients and saturation; learn how initialization, learning rate, and leaky relu keep neurons active.

Estimate probability density function parameters with maximum likelihood, using Gaussian examples and likelihood functions. Learn how to maximize the log-likelihood to identify mu and sigma.

Compute maximum likelihood estimates for a one-dimensional Gaussian to determine mu and sigma. Identify mu_hat as the sample mean and sigma_hat^2 as the mean squared deviation.