Gradient Descent and Optimizer Geometry: Momentum, Adam, and Loss Surfaces
AI Learning Project
Gradient Descent and Optimizer Geometry: Momentum, Adam, and Loss Surfaces
Hand-calculate early steps on a 2D quadratic, compare gradient descent, momentum, and Adam, and reproduce the loss-contour visualization.
AI Learning Project

Gradient Descent and Optimizer Geometry: Momentum, Adam, and Loss Surfaces

Reading info

Level: Intermediate Reading time: 13 min
  • Gradient Descent
  • Momentum
  • Adam
  • Loss Surface
Open knowledge map

An optimizer decides how parameters move along gradients. To understand gradient descent, momentum, and Adam, it is better to watch their paths on a loss surface than to memorize names.

This article uses a two-dimensional quadratic function. The start point and target are the same, but different optimizers take different routes because they treat curvature, history, and scale differently.

1. A Loss Surface You Can Differentiate By Hand

L(x, y) = 1/2 * (8x^2 + y^2) + 0.8xy
grad L = [8x + 0.8y, y + 0.8x]

The surface is steep in the x direction and flatter in the y direction. Neural-network losses often have this kind of uneven curvature.

Gradient descent, momentum, and Adam paths on a quadratic contour plot
Dense contour lines indicate fast loss change. Different optimizers trace different paths on the same surface.

2. Hand Calculate The First Gradient Descent Step

Starting from (2.2, -2.0), the gradient is:

grad = [8*2.2 + 0.8*(-2.0), -2.0 + 0.8*2.2]
     = [16.0, -0.24]

With learning rate 0.08:

x_new = 2.2  - 0.08 * 16.0  = 0.92
y_new = -2.0 - 0.08 * -0.24 = -1.9808

The lab output confirms it: step 1 for gradient descent is x=0.920000, y=-1.980800, and loss drops from 17.840000 to 3.889516.

3. What Momentum And Adam Change

Plain gradient descent only uses the current gradient. Momentum accumulates a velocity from previous gradients:

v_t = beta * v_{t-1} + grad_t
theta_t = theta_{t-1} - lr * v_t

Adam keeps first and second moments, using squared gradients to estimate per-coordinate scale:

m_t = beta1 * m_{t-1} + (1-beta1) * grad_t
v_t = beta2 * v_{t-1} + (1-beta2) * grad_t^2
theta_t = theta_{t-1} - lr * m_hat / (sqrt(v_hat) + eps)

4. Core Code

def grad(point):
    x, y = point
    return np.array([8.0 * x + 0.8 * y, y + 0.8 * x])

point = point - lr * grad(point)

Production optimizers add many details, but the central question remains: where should the parameters move next, given gradient, history, and scale?

5. What The Animation Shows

The animation compares gradient descent, momentum, and Adam on the same loss surface.

Watch whether the steep direction oscillates and whether the flatter direction progresses too slowly. Many training failures are optimizer-path problems rather than architecture problems.

6. Practical Notes

  • Plot training and validation loss before changing optimizers.
  • The learning rate usually matters more than the optimizer name.
  • Adam can still diverge when the learning rate is too high.
  • For noisy loss curves, try lowering the learning rate or adding warmup.

The next article moves into convolution and shows how local image operations become matrix computations.

Internal links

Related tutorials

Review foundation Backpropagation as a Computation Graph

Trace local gradients through ReLU and softmax cross-entropy in a two-layer MLP.

 Continue next Convolution and Receptive Field Math

Compute convolution output size, receptive fields, channel mixing, and im2col layout.

 Learning path AI Learning Path

Read AI foundations, machine learning, CNN work, traffic defense, and AI security in order.

 Resources Resource Library

Download companion code, datasets, diagrams, model files, and notes.

 Topic hub Student AI Projects

Connect handwritten digits, browser demos, and CNN projects into a portfolio path.

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FAQ

Who is this article for?

This article is for readers who want an intermediate-level guide to Gradient Descent and Optimizer Geometry. It takes about 13 min and focuses on Gradient Descent, Momentum, Adam, Loss Surface.

What should I read next?

The recommended next step is Convolution and Receptive Field Math, so the article connects into a longer learning route instead of ending as an isolated note.

Does this article include runnable code or companion resources?

Yes. Use the run notes, resource cards, and download links on the page to reproduce the example or inspect the companion files.

How does this article fit into the larger site?

It is connected to the article context block, learning routes, resources, and project timeline so readers can move from concept to implementation.

Article context

AI Learning Project

A practical route from AI concepts to machine learning workflow, evaluation, neural networks, Python practice, handwritten digits, a CIFAR-10 CNN, adversarial traffic-defense notes, and AI security.

Level: Intermediate Reading time: 13 min
  • Gradient Descent
  • Momentum
  • Adam
  • Loss Surface

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Continue: Convolution and Receptive Field Math
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Other language version 梯度下降与优化器几何：Momentum、Adam 和 loss surface 轨迹
Share summary Gradient Descent and Optimizer Geometry

Compare gradient descent, momentum, and Adam on a visible quadratic loss surface.

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Project timeline

Published posts

  1. AI Basics Learning Roadmap Separate AI, machine learning, and deep learning before going into implementation details.
  2. Machine Learning Workflow Follow the practical path from data and features to training, prediction, and evaluation.
  3. Model Training and Evaluation Understand loss, overfitting, train/test splits, accuracy, recall, and F1.
  4. Neural Network Basics Move from perceptrons to activation, forward propagation, backpropagation, and training loops.
  5. Matrix Calculus for Neural Networks Derive dL/dW for y = Wx + b and verify it with finite differences.
  6. Backpropagation as a Computation Graph Trace local gradients through ReLU and softmax cross-entropy in a two-layer MLP.
  7. Gradient Descent and Optimizer Geometry Compare gradient descent, momentum, and Adam on a visible quadratic loss surface.
  8. Convolution and Receptive Field Math Compute convolution output size, receptive fields, channel mixing, and im2col layout.
  9. Transformer Attention Math Hand-calculate Q/K/V scores, softmax weights, masks, multi-head structure, and KV cache.
  10. NLP Basics: Understanding Bag of Words and TF-IDF An introduction to the most fundamental text representation methods in NLP: Bag of Words (BoW) and TF-IDF.
  11. RNN Basics: Handling Sequential Data with Memory Understand the core concepts of Recurrent Neural Networks (RNN), the role of hidden states, and their application in NLP.
  12. Transformer Self-Attention Read Q/K/V, scaled dot-product attention, multi-head attention, and positional encoding before exploring LLM internals.
  13. Python AI Mini Practice Run a small scikit-learn classification task and read the experiment output.
  14. Handwritten Digit Dataset Basics Read train.csv, test.csv, labels, and the flattened 28 by 28 pixel layout before training the classifier.
  15. Handwritten Digit Softmax in C Follow the C implementation from logits and softmax probabilities to confusion matrices and submission export.
  16. Handwritten Digit Playground Notes See how the offline classifier was adapted into a browser demo with drawing input and probability output.
  17. CIFAR-10 Tiny CNN Tutorial in C Build and train a small convolutional neural network for CIFAR-10 image classification, then read its loss and accuracy output.
  18. Building a Tiny CIFAR-10 CNN in C: Convolution, Pooling, and Backpropagation A source-based walkthrough of cifar10_tiny_cnn.c, covering CIFAR-10 binary input, 3x3 convolution, ReLU, max pooling, fully connected logits, softmax, backpropagation, and local commands.
  19. High-Entropy Traffic Defense Notes Study encrypted metadata leaks, entropy, traffic classifiers, and a defensive Python chaffing prototype.
  20. AI Security Threat Modeling Build a defense map with NIST adversarial ML, MITRE ATLAS, and OWASP LLM risks.
  21. Adversarial Examples and Robust Evaluation Evaluate clean and perturbed accuracy with an FGSM-style digits experiment.
  22. Data Poisoning and Backdoor Defense Study poison rate, trigger behavior, attack success rate, and training pipeline controls.
  23. Model Privacy and Extraction Defense Measure membership inference signal and surrogate fidelity against a local toy model.
  24. LLM, RAG, and Agent Security Separate instructions from data and enforce tool permissions against indirect prompt injection.

Published resources

  1. Python AI practice code guide The article includes a runnable scikit-learn classification script.
  2. digit_softmax_classifier.c The C source for the handwritten digit softmax classifier.
  3. train.csv.zip Compressed handwritten digit training set with 42000 labeled samples.
  4. test.csv.zip Compressed handwritten digit test set with 28000 unlabeled samples.
  5. sample_submission.csv The official submission format example for checking the final output columns.
  6. submission.csv The prediction file generated by the current C project.
  7. digit-playground-model.json The compact softmax demo model and sample set used by the browser playground.
  8. digit-sample-grid.svg A small handwritten digit preview grid extracted from the training set.
  9. Handwritten digit project bundle Contains the source file, compressed datasets, submission files, browser model, and preview grid.
  10. cifar10_tiny_cnn.c source Single-file C tiny CNN with CIFAR-10 loading, convolution, pooling, softmax, and backpropagation.
  11. model_weights.bin sample weights Model weights generated by one local small-sample run.
  12. test_predictions.csv sample predictions Sample test prediction output from the CIFAR-10 tiny CNN.
  13. CNN project explanation PDF Companion explanation material for the CNN project.
  14. Virtual Mirror redacted code skeleton A redacted mld_chaffing_v2.py control-flow skeleton with secrets, node topology, and target lists removed.
  15. Virtual Mirror stress-test template A redacted CSV template for CPU, memory, peak threads, pulse rate, latency, and error measurements.
  16. Virtual Mirror classifier-evaluation template A CSV template for TP, FN, FP, TN, accuracy, precision, recall, F1, ROC-AUC, entropy, and JS divergence.
  17. Virtual Mirror resource notes Notes explaining why the public resources include only redacted code, test templates, and architecture context.
  18. AI Security Lab README Setup, safety boundaries, and quick-run commands for the AI Security series.
  19. AI Security Lab full bundle Includes safe toy scripts, result CSVs, risk register, attack-defense matrix, and architecture diagram.
  20. AI security risk register CSV risk register template for AI threat modeling and release review.
  21. AI attack-defense matrix Maps attack surface, toy demo, metric, and defensive control into one CSV table.
  22. AI Security Lab architecture diagram Shows threat modeling, robustness, data integrity, model privacy, and RAG guardrails.
  23. FGSM digits robustness script FGSM-style perturbation and accuracy-drop experiment for a local digits classifier.
  24. Data poisoning and backdoor toy script Demonstrates poison rate, trigger behavior, and attack success rate on digits.
  25. Model privacy and extraction toy script Outputs membership AUC, target accuracy, surrogate fidelity, and surrogate accuracy.
  26. RAG prompt injection guard toy script Uses a deterministic toy agent to demonstrate external-data demotion and tool-policy blocking.
  27. Deep Learning Math Lab README Setup commands, script entry points, generated outputs, and figure notes for the math series.
  28. Deep learning math full lab bundle Bundles NumPy scripts, CSV outputs, formula diagrams, loss contours, convolution figures, and attention heatmaps.
  29. Gradient check results CSV Stores MSE analytic gradients, finite-difference gradients, and error norms.
  30. Optimizer path CSV Step-by-step coordinates and loss for gradient descent, momentum, and Adam on a 2D quadratic.
  31. Attention weights CSV Scores, softmax weights, and context vectors for a three-token scaled dot-product attention example.
  32. Deep learning math figure set Includes matrix shapes, computation graphs, loss contours, convolution scans, and attention heatmaps.
  33. Deep learning math interactive visualizer Browser modules for gradient checking, optimizer paths, convolution output size, and attention heatmaps.
  34. Deep Learning topic share card A 1200x630 SVG card for sharing the Deep Learning / CNN topic hub.
  35. Machine Learning From Scratch share card A 1200x630 SVG card for the K-means, Iris, and ML workflow topic hub.
  36. Student AI Projects share card A 1200x630 SVG card for handwritten digits, C classifiers, and browser demos.
  37. CNN convolution scan animation An 8-second Remotion animation showing how a 3x3 convolution kernel scans an input and builds a feature map.

Current route

  1. AI Basics Learning Roadmap Learning path step
  2. Machine Learning Workflow Learning path step
  3. Model Training and Evaluation Learning path step
  4. Neural Network Basics Learning path step
  5. Matrix Calculus for Neural Networks Learning path step
  6. Backpropagation as a Computation Graph Learning path step
  7. Gradient Descent and Optimizer Geometry Learning path step
  8. Convolution and Receptive Field Math Learning path step
  9. Transformer Attention Math Learning path step
  10. Transformer Self-Attention Learning path step
  11. LLM Visualizer Learning path step
  12. Python AI Mini Practice Learning path step
  13. Handwritten Digit Dataset Basics Learning path step
  14. Handwritten Digit Softmax in C Learning path step
  15. Handwritten Digit Playground Notes Learning path step
  16. CIFAR-10 Tiny CNN Tutorial in C Learning path step
  17. High-Entropy Traffic Defense Notes Learning path step
  18. AI Security Threat Modeling Learning path step
  19. Adversarial Examples and Robust Evaluation Learning path step
  20. Data Poisoning and Backdoor Defense Learning path step
  21. Model Privacy and Extraction Defense Learning path step
  22. LLM, RAG, and Agent Security Learning path step

Next notes

  1. Add more image-classification and error-analysis cases
  2. Turn common metrics into a quick reference
  3. Add more AI security defense experiment notes
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