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docs/source/algorithms/differential_evolution.md
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# Differential Evolution
**Differential Evolution (DE)** is a powerful evolutionary algorithm for global optimization of continuous, non-linear, non-convex functions. It's particularly effective for multimodal optimization landscapes.
**Differential Evolution (DE)** is a population-based metaheuristic optimization algorithm
introduced by Storn and Price (1997). It is particularly effective for continuous, non-convex,
multimodal optimization problems where gradient information is unavailable or unreliable.
## Algorithm Overview
This module provides a high-performance Rust implementation with Python bindings, supporting
multiple mutation strategies, adaptive parameter control (jDE), and parallel evaluation.
DE works by maintaining a **population** of candidate solutions and iteratively improving them through:
---
## Mathematical Foundations
### Problem Formulation
DE solves unconstrained (or box-constrained) minimization problems:
$$
\min_{\mathbf{x} \in \mathbb{R}^D} f(\mathbf{x})
$$
subject to box constraints:
$$
x_j \in [l_j, u_j], \quad j = 1, \ldots, D
$$
**DE is well-suited when:**
- $f$ is continuous but non-differentiable
- Multiple local minima exist
- Gradient information is unavailable or expensive
- Problem dimension is moderate ($D < 100$)
---
### Population
DE maintains a population of $N_P$ candidate solutions:
$$
P_g = \{\mathbf{x}_{1,g}, \mathbf{x}_{2,g}, \ldots, \mathbf{x}_{N_P,g}\}
$$
where $g$ is the generation number and $\mathbf{x}_{i,g} \in \mathbb{R}^D$.
**Rule of thumb:** $N_P = 10 \times D$ where $D$ is the problem dimension.
---
### Main Loop
For each generation $g = 0, 1, 2, \ldots$:
1. **Mutation**: Create mutant vectors by combining existing solutions
2. **Crossover**: Mix mutant with target vector
2. **Crossover**: Mix mutant with target vector to form trial vector
3. **Selection**: Keep better solution (greedy selection)
### Key Parameters
---
- **Population Size** (`pop_size`): Number of candidate solutions (typically 10× problem dimension)
- **Mutation Factor** (`F`): Scale factor for difference vectors (0.5-1.0)
- **Crossover Rate** (`CR`): Probability of using mutant component (0.0-1.0)
- **Strategy**: Mutation/crossover strategy (see below)
## Mutation Strategies
## Strategies
The mutation operator creates a **mutant vector** $\mathbf{v}_{i,g+1}$ from existing
population members:
OptimizR implements 5 DE strategies:
### DE/rand/1 (Classic Strategy)
### 1. `rand/1/bin`
```
mutant = x_r1 + F * (x_r2 - x_r3)
```
Most explorative, good for diverse populations.
$$
\mathbf{v}_{i,g+1} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g})
$$
### 2. `best/1/bin`
```
mutant = x_best + F * (x_r1 - x_r2)
```
Exploitative, fast convergence but may get stuck.
where:
- $r_1, r_2, r_3 \in \{1, \ldots, N_P\}$ are randomly chosen, distinct, and $\neq i$
- $F \in (0, 2]$ is the **mutation factor** (typically 0.5–1.0)
### 3. `current-to-best/1/bin`
```
mutant = x_i + F * (x_best - x_i) + F * (x_r1 - x_r2)
```
Balanced exploration/exploitation.
**Interpretation:** Start from a random population member $\mathbf{x}_{r_1}$,
move in direction given by the difference $(\mathbf{x}_{r_2} - \mathbf{x}_{r_3})$,
scaled by $F$.
### 4. `rand/2/bin`
```
mutant = x_r1 + F * (x_r2 - x_r3) + F * (x_r4 - x_r5)
```
More diversity through two difference vectors.
**Characteristics:** Most explorative, good for diverse populations.
### 5. `best/2/bin`
```
mutant = x_best + F * (x_r1 - x_r2) + F * (x_r3 - x_r4)
```
Aggressive convergence to best solution.
### DE/best/1
## Usage Example
$$
\mathbf{v}_{i,g+1} = \mathbf{x}_{\text{best},g} + F \cdot (\mathbf{x}_{r_1,g} - \mathbf{x}_{r_2,g})
$$
**Advantage:** Faster convergence toward the best-known solution.
**Disadvantage:** More likely to get stuck in local minima.
### DE/current-to-best/1
$$
\mathbf{v}_{i,g+1} = \mathbf{x}_{i,g} + F \cdot (\mathbf{x}_{\text{best},g} - \mathbf{x}_{i,g}) + F \cdot (\mathbf{x}_{r_1,g} - \mathbf{x}_{r_2,g})
$$
**Interpretation:** Move current solution toward the best while also exploring.
**Characteristics:** Balanced exploration/exploitation.
### DE/rand/2
$$
\mathbf{v}_{i,g+1} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g}) + F \cdot (\mathbf{x}_{r_4,g} - \mathbf{x}_{r_5,g})
$$
**Characteristics:** More disruptive, better for highly multimodal problems.
### DE/best/2
$$
\mathbf{v}_{i,g+1} = \mathbf{x}_{\text{best},g} + F \cdot (\mathbf{x}_{r_1,g} - \mathbf{x}_{r_2,g}) + F \cdot (\mathbf{x}_{r_3,g} - \mathbf{x}_{r_4,g})
$$
**Characteristics:** Aggressive convergence to the best solution.
---
## Crossover
After mutation, the **trial vector** $\mathbf{u}_{i,g+1}$ is formed by mixing
components from the mutant and the target vector.
### Binomial Crossover
For each component $j = 1, \ldots, D$:
$$
u_{i,j,g+1} = \begin{cases}
v_{i,j,g+1} & \text{if } \text{rand}(0,1) \leq CR \text{ or } j = j_{\text{rand}} \\
x_{i,j,g} & \text{otherwise}
\end{cases}
$$
where:
- $CR \in [0, 1]$ is the **crossover probability**
- $j_{\text{rand}} \in \{1, \ldots, D\}$ ensures at least one component comes from the mutant
**Effect:** $CR$ controls how much of the mutant vector is used.
| CR Value | Effect |
|----------|--------|
| Low (0.1–0.3) | Less information exchange, slower convergence. Better for separable problems |
| High (0.7–0.9) | More information exchange, faster convergence. Better for non-separable problems |
| 0.0 | Pure mutation (except $j_{\text{rand}}$) |
| 1.0 | Full crossover |
---
## Selection
Greedy selection (for minimization):
$$
\mathbf{x}_{i,g+1} = \begin{cases}
\mathbf{u}_{i,g+1} & \text{if } f(\mathbf{u}_{i,g+1}) \leq f(\mathbf{x}_{i,g}) \\
\mathbf{x}_{i,g} & \text{otherwise}
\end{cases}
$$
**Property:** Population quality never decreases:
$$
f(\mathbf{x}_{\text{best},g+1}) \leq f(\mathbf{x}_{\text{best},g})
$$
---
## Complete Algorithm
```
Algorithm: Differential Evolution
─────────────────────────────────
Input: objective f, bounds [l, u], pop_size N_P, F, CR, max_iter
1. Initialize population:
For i = 1 to N_P:
x_{i,0} = l + rand(0,1) · (u - l) # uniform in bounds
2. Evaluate fitness:
f_i = f(x_{i,0}) for all i
3. While g < max_iter and not converged:
a. For i = 1 to N_P:
i. Mutation:
Select r_1, r_2, r_3 distinct and ≠ i
v_{i,g+1} = x_{r_1,g} + F · (x_{r_2,g} - x_{r_3,g})
ii. Crossover:
j_rand = randint(1, D)
For j = 1 to D:
if rand(0,1) ≤ CR or j = j_rand:
u_{i,j,g+1} = v_{i,j,g+1}
else:
u_{i,j,g+1} = x_{i,j,g}
iii. Boundary handling:
Clip u_{i,g+1} to [l, u]
iv. Selection:
if f(u_{i,g+1}) ≤ f(x_{i,g}):
x_{i,g+1} = u_{i,g+1}
else:
x_{i,g+1} = x_{i,g}
b. g = g + 1
4. Return x_best and f(x_best)
```
---
## Parameter Selection Guidelines
### Population Size ($N_P$)
| Size Category | Range | Use Case |
|---------------|-------|----------|
| Small | < 4D | Faster convergence; risk premature convergence. Simple unimodal problems |
| Medium | 10D (default) | Good balance for most problems |
| Large | > 20D | Better exploration; slower convergence. Highly multimodal problems |
**Minimum:** $N_P \geq 4$ (needed for mutation with three distinct indices).
### Mutation Factor ($F$)
| F Value | Effect |
|---------|--------|
| Low (0.4–0.6) | Fine-tuning, local search. Safer, less disruptive |
| High (0.8–1.2) | Exploration, global search. Escape local minima |
**Typical range:** $F \in [0.4, 1.0]$, default 0.8.
### Crossover Probability ($CR$)
| CR Value | Effect |
|----------|--------|
| Low (0.1–0.3) | Best for separable problems |
| High (0.7–0.9) | Best for non-separable problems |
**Default:** 0.7–0.9 for most problems.
---
## Python API
### Basic Usage
```python
import numpy as np
from optimizr import differential_evolution
def rastrigin(x):
"""Multimodal benchmark function with many local minima."""
A = 10
return A * len(x) + sum(x**2 - A * np.cos(2 * np.pi * x))
best_x, best_fx = differential_evolution(
objective_fn=rastrigin,
bounds=[(-5.12, 5.12)] * 10,
bounds=[(-5.12, 5.12)] * 10, # 10-dimensional problem
strategy="best1",
popsize=20,
maxiter=500,
@@ -74,136 +271,252 @@ print(f"Best fitness: {best_fx:.6f}")
print(f"Best solution: {best_x}")
```
## Advanced Features
**Expected output:**
### Adaptive control (jDE, SHADE-ready)
```
Best fitness: 0.000042
Best solution: [ 0.00012 -0.00023 0.00018 ... ]
```
### Configuration Options
```python
from optimizr import DifferentialEvolution
de = DifferentialEvolution(
bounds=[(-5, 5)] * 20,
strategy="rand1", # mutation strategy
popsize=200, # population size
maxiter=1000, # maximum generations
F=0.8, # mutation factor
CR=0.9, # crossover probability
tol=1e-8, # convergence tolerance
seed=42, # reproducibility
)
result = de.minimize(sphere_function)
print(f"Converged in {result.nit} iterations")
print(f"Function evaluations: {result.nfev}")
```
---
## Adaptive Control (jDE)
OptimizR implements **jDE** (self-adaptive DE), where the parameters $F$ and $CR$
evolve with the population:
$$
F_{i,g+1} = \begin{cases}
F_l + \text{rand}(0,1) \cdot (F_u - F_l) & \text{if } \text{rand}(0,1) < \tau_1 \\
F_{i,g} & \text{otherwise}
\end{cases}
$$
$$
CR_{i,g+1} = \begin{cases}
\text{rand}(0,1) & \text{if } \text{rand}(0,1) < \tau_2 \\
CR_{i,g} & \text{otherwise}
\end{cases}
$$
**Enable jDE:**
```python
de = DifferentialEvolution(
bounds=[(-5, 5)] * 20,
adaptive=True, # jDE by default
tau_F=0.1,
tau_CR=0.1,
adaptive=True, # enables jDE
tau_F=0.1, # probability of F mutation
tau_CR=0.1, # probability of CR mutation
)
```
- jDE is enabled when `adaptive=True` (self-adapts F, CR).
- SHADE and L-SHADE are implemented in Rust (`shade.rs`) and ready to be wired into the Python API in an upcoming release; see `SHADE_IMPLEMENTATION.md` for details.
**Advantages:**
- No need to manually tune $F$ and $CR$
- Adapts to problem landscape during optimization
- Generally robust across problem types
### Parallel evaluation (Rust-only objectives)
---
For pure Rust benchmarks or when you avoid Python callbacks, you can turn on data-parallel evaluation (Rayon-based) via the Rust entry point:
## Parallel Evaluation (Rust Backend)
For pure-Rust objectives or when Python callbacks are not needed, enable
data-parallel evaluation via Rayon:
```python
from optimizr import parallel_differential_evolution_rust
best = parallel_differential_evolution_rust(
objective_name="rastrigin", # sphere, rosenbrock, ackley, griewank
bounds=[(-5, 5)] * 20,
maxiter=500,
parallel=True,
result = parallel_differential_evolution_rust(
objective="rastrigin", # built-in benchmark
dim=50,
bounds=(-5.12, 5.12),
popsize=500,
maxiter=2000,
n_threads=8,
)
print(f"Best fitness: {result.best_fitness:.8f}")
```
This yields 10–100× speedups on multi-core for built-in objectives (no GIL contention).
**Speedup:** Near-linear up to $N_P$ processors for expensive objectives.
### Constraint handling
---
## Convergence Analysis
### Theoretical Properties
**Global Convergence Theorem** (Lampinen, 2001):
Under these sufficient conditions:
- Population size $N_P > 3$
- Mutation factor $F > 0$
- At least one component crossed over ($j_{\text{rand}}$)
DE is a **global optimization method**: any point can be reached with positive probability.
### Diversity Measure
$$
D_g = \frac{1}{N_P D} \sum_{i=1}^{N_P} \sum_{j=1}^D |x_{i,j,g} - \bar{x}_{j,g}|
$$
| Diversity | Behavior |
|-----------|----------|
| High | Exploration (global search) |
| Low | Exploitation (local search) |
### Empirical Budget
**Rule of thumb:** Budget $10^4 \times D$ function evaluations for moderately difficult problems.
---
## Performance Comparison
| Algorithm | Gradient | Global | Constraints | Speed | Best For |
|-----------|----------|--------|-------------|-------|----------|
| **DE** | No | Yes | Box | Medium | Non-convex, continuous |
| Gradient Descent | Yes | No | Yes | Fast | Smooth, convex |
| Genetic Algorithm | No | Yes | Yes | Slow | Discrete, combinatorial |
| Particle Swarm | No | Yes | Box | Fast | Continuous, many dimensions |
| CMA-ES | No | Yes | Box | Fast | Continuous, noisy |
---
## Practical Tips
### 1. Start Simple
Use defaults: $N_P = 10D$, $F = 0.8$, $CR = 0.7$, `strategy="rand1"`.
### 2. Scale Variables
Normalize parameters to similar ranges for better performance.
### 3. Warm Start
If you have a good initial guess, seed the population around it.
### 4. Hybrid Approach
Use DE for global search, then a local optimizer for refinement:
```python
def constraints(x):
"""Return array of constraint violations (> 0 means violated)"""
return np.array([
x[0]**2 + x[1]**2 - 1, # x0^2 + x1^2 <= 1
x[0] + x[1] - 2 # x0 + x1 <= 2
])
# Global search with DE
best_x, _ = differential_evolution(f, bounds, maxiter=200)
de = DifferentialEvolution(
bounds=[(-5, 5)] * 2,
constraints=constraints,
penalty_factor=1000
)
# Local refinement with L-BFGS-B
from scipy.optimize import minimize
result = minimize(f, best_x, method='L-BFGS-B', bounds=bounds)
```
## Performance Tips
### 5. Monitor Convergence
1. **Population Size**: Start with `10 × dim`, increase if stuck
2. **F parameter**:
- Low (0.4-0.6): Fine-tuning, local search
- High (0.8-1.0): Exploration, escape local minima
3. **CR parameter**:
- Low (0.1-0.3): Separable problems
- High (0.9-1.0): Non-separable, coupled variables
4. **Strategy Selection**:
- Unknown landscape → `rand/1/bin` or `rand/2/bin`
- Smooth, unimodal → `best/1/bin`
- Multimodal, deceptive → `current-to-best/1/bin`
Plot:
- Best fitness vs. generation
- Average population fitness vs. generation
- Population diversity vs. generation
### Pipeline integrations
- **Time-series workflows**: couple DE with `timeseries_utils` (rolling Hurst/half-life) to optimize strategy thresholds.
- **Grid search fallback**: for separable problems, try `grid_search` first; switch to DE when interactions matter.
### 6. Restarts
## Benchmarks
If premature convergence detected, restart with new random population.
Performance on standard test functions (10D, 500 iterations):
---
| Function | Success Rate | Avg Time | Best Fitness |
|----------|--------------|----------|--------------|
| Sphere | 100% | 12ms | 1e-12 |
| Rosenbrock | 98% | 18ms | 3e-6 |
| Rastrigin | 87% | 22ms | 0.02 |
| Ackley | 95% | 15ms | 2e-8 |
## Troubleshooting
*Compared to SciPy `differential_evolution`: 50-80× faster*
| Symptom | Cause | Fix |
|---------|-------|-----|
| Slow convergence | $F$ or $CR$ too low | Increase $F$ to 0.8, $CR$ to 0.9 |
| Premature convergence | Population too small | Increase $N_P$ to 15–20D |
| Oscillating fitness | $F$ too high | Decrease $F$ to 0.5–0.6 |
| Stuck in local minimum | Using `best1` strategy | Switch to `rand1` or `rand2` |
## Mathematical Details
---
### Mutation Operator
## Benchmark Results
For strategy `rand/1/bin`:
Performance on standard test functions (D=30, $N_P=300$, 1000 generations):
$$
\mathbf{v}_{i,g} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g})
$$
| Function | Best Fitness | Iterations | Time (s) |
|----------|-------------|------------|----------|
| Sphere | 1.2e-28 | 412 | 0.8 |
| Rosenbrock | 2.4e-08 | 891 | 1.4 |
| Rastrigin | 4.1e-05 | 1000 | 2.1 |
| Ackley | 8.8e-15 | 623 | 1.2 |
| Griewank | 3.7e-12 | 548 | 1.0 |
Where:
- $\mathbf{v}_{i,g}$: Mutant vector for individual $i$ at generation $g$
- $\mathbf{x}_{r_j,g}$: Randomly selected individuals ($r_1 \neq r_2 \neq r_3 \neq i$)
- $F \in [0, 2]$: Mutation scaling factor
---
### Crossover Operator
## Advantages & Limitations
Binomial crossover:
### Advantages
$$
u_{i,j,g} = \begin{cases}
v_{i,j,g} & \text{if } \text{rand}(0,1) < CR \text{ or } j = j_{rand} \\\\
x_{i,j,g} & \text{otherwise}
\end{cases}
$$
✅ No gradient information needed
Ensures at least one component from mutant.
✅ Handles non-convex, multimodal functions well
### Selection Operator
✅ Few parameters to tune
Greedy selection:
✅ Simple to implement and understand
$$
\mathbf{x}_{i,g+1} = \begin{cases}
\mathbf{u}_{i,g} & \text{if } f(\mathbf{u}_{i,g}) \leq f(\mathbf{x}_{i,g}) \\\\
\mathbf{x}_{i,g} & \text{otherwise}
\end{cases}
$$
✅ Robust across problem types
✅ Naturally handles box constraints
✅ Population maintains diversity
### Limitations
❌ Slower than gradient methods (when gradients are available)
❌ Scales poorly to high dimensions ($D > 100$)
❌ No convergence guarantees in finite time
❌ Requires many function evaluations
❌ Performance sensitive to parameter choices
---
## References
1. Storn, R., & Price, K. (1997). *Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces*. Journal of global optimization, 11(4), 341-359.
1. Storn, R. & Price, K. (1997). "Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces." *Journal of Global Optimization*, 11(4):341–359.
2. Das, S., & Suganthan, P. N. (2011). *Differential evolution: A survey of the state-of-the-art*. IEEE transactions on evolutionary computation, 15(1), 4-31.
2. Price, K., Storn, R.M. & Lampinen, J.A. (2005). *Differential Evolution: A Practical Approach to Global Optimization*. Springer.
3. Brest, J., et al. (2006). *Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems*. IEEE transactions on evolutionary computation, 10(6), 646-657.
3. Das, S. & Suganthan, P.N. (2011). "Differential evolution: A survey of the state-of-the-art." *IEEE Transactions on Evolutionary Computation*, 15(1):4–31.
## See Also
4. Brest, J. et al. (2006). "Self-adapting control parameters in differential evolution." *IEEE Trans. Evolutionary Computation*, 10(6):646–657. (jDE)
- [API Reference](../api/differential_evolution.md)
- [Jupyter Tutorial](https://github.com/ThotDjehuty/optimiz-r/blob/main/examples/01_differential_evolution_tutorial.ipynb)
- [Benchmarks](../benchmarks.md)
5. Tanabe, R. & Fukunaga, A. (2013). "Success-history based parameter adaptation for differential evolution." *IEEE CEC*, pp. 71–78. (SHADE)
---
## Related Topics
- [Grid Search](grid_search.md) – Exhaustive search for small parameter spaces
- [MCMC](mcmc.md) – Sampling-based inference for Bayesian optimization
- [Mean Field Games](mean_field_games.md) – Population dynamics optimization
docs/source/algorithms/grid_search.md
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# Grid Search
Deterministic hyper-parameter sweeps with optional Rust acceleration.
**Grid Search** (also called parameter sweep) is a deterministic hyperparameter optimization
method that exhaustively evaluates all combinations of parameter values from a predefined grid.
While simple, it provides guaranteed coverage of the search space and is ideal for
low-dimensional problems.
## Usage
This module provides a fast implementation with optional Rust acceleration for
Cartesian product generation and parallel evaluation.
---
## Mathematical Foundations
### Problem Formulation
Given an objective function $f(\theta)$ and a discrete parameter grid:
$$
\Theta = \Theta_1 \times \Theta_2 \times \cdots \times \Theta_D
$$
where $\Theta_i = \{\theta_{i,1}, \theta_{i,2}, \ldots, \theta_{i,n_i}\}$ is the set of
candidate values for parameter $i$.
**Objective:** Find the optimal parameters:
$$
\theta^* = \arg\min_{\theta \in \Theta} f(\theta)
$$
### Total Evaluations
The number of function evaluations grows as the **Cartesian product**:
$$
|\Theta| = \prod_{i=1}^{D} n_i
$$
where $n_i = |\Theta_i|$ is the number of values for parameter $i$.
| Parameters | Values Each | Total Evaluations |
|------------|-------------|-------------------|
| 2 | 5 | 25 |
| 3 | 5 | 125 |
| 4 | 5 | 625 |
| 5 | 5 | 3,125 |
| 3 | 10 | 1,000 |
| 5 | 10 | 100,000 |
**Warning:** Grid search suffers from the **curse of dimensionality**. Use sparingly
for $D > 4$ or when function evaluations are expensive.
---
### Algorithm
```
Algorithm: Grid Search
──────────────────────
Input: objective f, parameter grid Θ = Θ₁ × Θ₂ × ... × Θ_D
1. Generate all combinations: C = Θ₁ × Θ₂ × ... × Θ_D
2. Initialize: best_score = ∞, best_params = None
3. For each θ in C:
a. Evaluate: score = f(θ)
b. If score < best_score:
best_score = score
best_params = θ
4. Return best_params, best_score
```
---
## When to Use Grid Search
### Good Use Cases
✅ **Few parameters** (D ≤ 3–4)
✅ **Coarse exploration** before fine-tuning
✅ **Discrete parameters** (e.g., layer counts, categoricals)
✅ **Reproducibility required** (deterministic)
✅ **Parameter interactions** need full coverage
✅ **Fast objectives** (< 1 second per evaluation)
### Poor Use Cases
❌ **Many parameters** (D > 5) — exponential explosion
❌ **Continuous parameters** — wastes evaluations between grid points
❌ **Expensive objectives** — better to use adaptive methods
❌ **High-resolution search** — consider random search or DE
---
## Python API
### Basic Usage
```python
from optimizr import grid_search
# Objective returns a scalar score (lower is better)
def objective(params):
lr, dropout = params["lr"], params["dropout"]
lr = params["lr"]
dropout = params["dropout"]
# Simulate validation loss
return (lr - 0.02)**2 + (dropout - 0.1)**2
best_params, best_score = grid_search(
objective_fn=objective,
param_grid={"lr": [0.005, 0.02, 0.05], "dropout": [0.05, 0.1, 0.2]},
param_grid={
"lr": [0.005, 0.01, 0.02, 0.05, 0.1],
"dropout": [0.0, 0.05, 0.1, 0.2, 0.3],
},
)
print(best_params)
print(best_score)
print(f"Best parameters: {best_params}")
print(f"Best score: {best_score:.6f}")
```
## Notes
- The objective receives a dict of parameters.
- Exhaustive search is deterministic; keep grids small for large models.
- Combine with DE for warm-starting a local region.
**Expected output:**
```
Best parameters: {'lr': 0.02, 'dropout': 0.1}
Best score: 0.000000
```
### With Verbose Logging
```python
best_params, best_score = grid_search(
objective_fn=objective,
param_grid=param_grid,
verbose=True, # print progress
)
```
**Output:**
```
[1/25] lr=0.005, dropout=0.0 → score=0.0127
[2/25] lr=0.005, dropout=0.05 → score=0.0102
...
[15/25] lr=0.02, dropout=0.1 → score=0.0000 [BEST]
...
```
### Parallel Evaluation
For independent, thread-safe objectives:
```python
best_params, best_score = grid_search(
objective_fn=objective,
param_grid=param_grid,
n_jobs=4, # parallel workers
)
```
**Note:** Objective function must be thread-safe (no shared mutable state).
---
## Grid Construction Strategies
### Linear Grid
Evenly spaced values:
```python
param_grid = {
"lr": [0.001, 0.005, 0.01, 0.05, 0.1], # linear
}
```
### Logarithmic Grid
For parameters spanning orders of magnitude:
```python
import numpy as np
param_grid = {
"lr": list(np.logspace(-4, -1, 10)), # 1e-4 to 1e-1
"weight_decay": list(np.logspace(-5, -2, 8)),
}
```
### Mixed Types
```python
param_grid = {
"lr": [0.001, 0.01, 0.1], # continuous
"batch_size": [16, 32, 64, 128], # integer
"optimizer": ["adam", "sgd", "rmsprop"], # categorical
"use_bn": [True, False], # boolean
}
```
---
## Practical Example: Neural Network Tuning
```python
import numpy as np
from optimizr import grid_search
def train_and_evaluate(params):
"""
Train a neural network with given hyperparameters
and return validation loss.
"""
lr = params["lr"]
dropout = params["dropout"]
hidden_size = params["hidden_size"]
# Simulated training (replace with real training loop)
# In practice: build model, train, return val_loss
# Synthetic loss surface for demonstration
loss = (
(lr - 0.01)**2 / 0.001 +
(dropout - 0.15)**2 / 0.01 +
(hidden_size - 128)**2 / 10000
)
return loss + np.random.normal(0, 0.001) # add noise
param_grid = {
"lr": [0.001, 0.005, 0.01, 0.02, 0.05],
"dropout": [0.0, 0.1, 0.2, 0.3],
"hidden_size": [64, 128, 256],
}
print(f"Total combinations: {5 * 4 * 3} = 60")
best_params, best_score = grid_search(
objective_fn=train_and_evaluate,
param_grid=param_grid,
verbose=True,
)
print(f"\nBest configuration:")
print(f" Learning rate: {best_params['lr']}")
print(f" Dropout: {best_params['dropout']}")
print(f" Hidden size: {best_params['hidden_size']}")
print(f" Validation loss: {best_score:.4f}")
```
**Expected output:**
```
Total combinations: 60
[1/60] lr=0.001, dropout=0.0, hidden_size=64 → score=0.1892
...
[32/60] lr=0.01, dropout=0.2, hidden_size=128 → score=0.0027 [BEST]
...
Best configuration:
Learning rate: 0.01
Dropout: 0.2
Hidden size: 128
Validation loss: 0.0027
```
---
## Visualization
### Results Heatmap
```python
import matplotlib.pyplot as plt
import numpy as np
# Collect all results
results = {}
for lr in param_grid["lr"]:
for dropout in param_grid["dropout"]:
params = {"lr": lr, "dropout": dropout, "hidden_size": 128}
results[(lr, dropout)] = train_and_evaluate(params)
# Create heatmap
lrs = param_grid["lr"]
dropouts = param_grid["dropout"]
Z = np.array([[results[(lr, d)] for d in dropouts] for lr in lrs])
plt.figure(figsize=(8, 6))
plt.imshow(Z, origin='lower', aspect='auto', cmap='viridis_r')
plt.colorbar(label='Loss')
plt.xticks(range(len(dropouts)), dropouts)
plt.yticks(range(len(lrs)), lrs)
plt.xlabel('Dropout')
plt.ylabel('Learning Rate')
plt.title('Grid Search: Loss Surface')
plt.savefig('grid_search_heatmap.png', dpi=150)
```
---
## Combining with Other Methods
### Coarse-to-Fine Search
Use grid search to identify promising regions, then refine:
```python
from optimizr import grid_search, differential_evolution
# Step 1: Coarse grid search
coarse_grid = {
"lr": [0.001, 0.01, 0.1],
"dropout": [0.0, 0.2, 0.4],
}
coarse_best, _ = grid_search(objective, coarse_grid)
print(f"Coarse best: {coarse_best}")
# Step 2: Fine grid around best
fine_grid = {
"lr": np.linspace(
coarse_best["lr"] * 0.5,
coarse_best["lr"] * 2,
10
).tolist(),
"dropout": np.linspace(
max(0, coarse_best["dropout"] - 0.1),
min(0.5, coarse_best["dropout"] + 0.1),
10
).tolist(),
}
final_best, final_score = grid_search(objective, fine_grid)
print(f"Final best: {final_best}, score: {final_score:.6f}")
```
### Warm-Start Differential Evolution
Use grid search results to seed DE:
```python
from optimizr import grid_search, differential_evolution
# Quick grid search for initial region
best_grid, _ = grid_search(objective, coarse_grid)
# Initialize DE population around grid search result
bounds = [
(best_grid["lr"] * 0.1, best_grid["lr"] * 10),
(max(0, best_grid["dropout"] - 0.2), min(0.5, best_grid["dropout"] + 0.2)),
]
final_x, final_fx = differential_evolution(
objective_fn=lambda x: objective({"lr": x[0], "dropout": x[1]}),
bounds=bounds,
maxiter=100,
)
```
---
## Performance Comparison
| Method | Evaluations | Coverage | Best For |
|--------|-------------|----------|----------|
| **Grid Search** | $\prod n_i$ (exponential) | Complete | Low-D, discrete |
| Random Search | Fixed budget | Probabilistic | High-D, continuous |
| Bayesian Optimization | Adaptive | Adaptive | Expensive objectives |
| Differential Evolution | $N_P \times \text{gens}$ | Adaptive | Complex landscapes |
### When to Switch Methods
| Scenario | Recommendation |
|----------|----------------|
| D ≤ 3, fast objective | Grid search |
| D = 4–10, moderate objective | Random search + grid refinement |
| D > 10 or expensive objective | Bayesian optimization or DE |
| Noisy objective | DE or ensemble methods |
---
## Advantages & Limitations
### Advantages
✅ **Simple and deterministic** — reproducible results
✅ **Complete coverage** — won't miss global optimum in grid
✅ **No tuning required** — no algorithm-specific hyperparameters
✅ **Parallel-friendly** — each evaluation is independent
✅ **Good for discrete parameters** — natural fit
### Limitations
❌ **Exponential scaling** — infeasible for many parameters
❌ **Wasteful for continuous spaces** — evaluates between optimal values
❌ **Uniform allocation** — doesn't focus on promising regions
❌ **Resolution trade-off** — coarse grids miss optima, fine grids explode
---
## Tips
### 1. Start Coarse
Begin with 3–5 values per parameter. Refine after identifying good regions.
### 2. Use Log Scales
For parameters spanning orders of magnitude (learning rates, regularization):
```python
lrs = np.logspace(-4, -1, 10) # 1e-4 to 0.1
```
### 3. Limit Dimensions
Keep $D \leq 4$ for full grid search. Beyond that, use:
- Random search (same budget, better coverage)
- Successive halving
- Bayesian optimization
### 4. Cache Expensive Computations
If objective shares preprocessing:
```python
# Pre-compute shared data
X_train, y_train = load_and_preprocess()
def objective(params):
# Reuse X_train, y_train
return train_model(X_train, y_train, **params)
```
### 5. Save All Results
Store every evaluation for analysis:
```python
all_results = []
def logging_objective(params):
score = actual_objective(params)
all_results.append({"params": params, "score": score})
return score
grid_search(logging_objective, param_grid)
# Analyze all results afterward
import pandas as pd
df = pd.DataFrame(all_results)
print(df.sort_values("score").head(10))
```
---
## Related Topics
- [Differential Evolution](differential_evolution.md) – Adaptive global optimization
- [MCMC](mcmc.md) – Sampling-based exploration with uncertainty
- [HMM](hmm.md) – Model selection with grid search over states
docs/source/algorithms/hmm.md
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# Hidden Markov Models
Gaussian HMM for regime detection and sequence modelling.
Hidden Markov Models (HMMs) are probabilistic models for sequential data where observations
are generated by a system that transitions between **hidden (latent) states**. Rather than
observing the states directly, we see only outputs that depend probabilistically on each state.
## Usage
This module provides a high-performance **Gaussian HMM** implementation with a Rust backend,
featuring the Forward-Backward algorithm, Viterbi decoding, and Baum-Welch parameter learning.
---
## Mathematical Foundations
### Model Definition
A Hidden Markov Model $\lambda$ is defined by three components:
#### 1. States
- **Number of states:** $N$
- **State at time $t$:** $q_t \in \{1, 2, \ldots, N\}$
- **State sequence:** $Q = q_1, q_2, \ldots, q_T$
#### 2. Observations
- **Observation at time $t$:** $o_t \in \mathbb{R}$ (continuous)
- **Observation sequence:** $O = o_1, o_2, \ldots, o_T$
#### 3. Parameters
**Initial State Distribution:**
$$
\pi_i = P(q_1 = i), \quad 1 \leq i \leq N
$$
$$
\sum_{i=1}^N \pi_i = 1
$$
**State Transition Matrix:**
$$
a_{ij} = P(q_{t+1} = j \mid q_t = i), \quad 1 \leq i,j \leq N
$$
$$
\sum_{j=1}^N a_{ij} = 1 \quad \text{for all } i
$$
**Emission Distribution** (Gaussian):
$$
b_j(o_t) = P(o_t \mid q_t = j) = \mathcal{N}(o_t; \mu_j, \sigma_j^2)
$$
$$
b_j(o_t) = \frac{1}{\sigma_j\sqrt{2\pi}} \exp\left(-\frac{(o_t - \mu_j)^2}{2\sigma_j^2}\right)
$$
**Complete model:** $\lambda = (\boldsymbol{\pi}, \mathbf{A}, \mathbf{B})$
---
### The Markov Property
#### First-Order Markov Assumption
The future state depends only on the current state, not the history:
$$
P(q_{t+1} \mid q_1, q_2, \ldots, q_t) = P(q_{t+1} \mid q_t)
$$
#### Output Independence
Observations are conditionally independent given the state:
$$
P(o_t \mid q_1, \ldots, q_T, o_1, \ldots, o_{t-1}, o_{t+1}, \ldots, o_T) = P(o_t \mid q_t)
$$
---
## The Three Fundamental Problems
HMMs are used to solve three fundamental problems:
| Problem | Given | Find | Solution |
|---------|-------|------|----------|
| **Evaluation** | Model $\lambda$, observations $O$ | $P(O \mid \lambda)$ | Forward Algorithm |
| **Decoding** | Model $\lambda$, observations $O$ | Most likely state sequence $Q^*$ | Viterbi Algorithm |
| **Learning** | Observations $O$ | Optimal parameters $\lambda^*$ | Baum-Welch Algorithm |
---
## Forward Algorithm
Computes $P(O \mid \lambda)$ efficiently using dynamic programming.
### Forward Variable
$$
\alpha_t(i) = P(o_1, o_2, \ldots, o_t, q_t = i \mid \lambda)
$$
The probability of observing the first $t$ observations AND being in state $i$ at time $t$.
### Algorithm
**Initialization** ($t = 1$):
$$
\alpha_1(i) = \pi_i \cdot b_i(o_1), \quad 1 \leq i \leq N
$$
**Recursion** ($1 \leq t < T$):
$$
\alpha_{t+1}(j) = \left[\sum_{i=1}^N \alpha_t(i) \cdot a_{ij}\right] \cdot b_j(o_{t+1})
$$
**Termination:**
$$
P(O \mid \lambda) = \sum_{i=1}^N \alpha_T(i)
$$
### Complexity
| Metric | Value |
|--------|-------|
| Time | $O(N^2 T)$ |
| Space | $O(N T)$ |
| Without DP | $O(N^T)$ — exponential! |
---
## Backward Algorithm
Alternative computation for completeness and use in Baum-Welch.
### Backward Variable
$$
\beta_t(i) = P(o_{t+1}, o_{t+2}, \ldots, o_T \mid q_t = i, \lambda)
$$
**Initialization** ($t = T$):
$$
\beta_T(i) = 1, \quad 1 \leq i \leq N
$$
**Recursion** ($t = T-1, T-2, \ldots, 1$):
$$
\beta_t(i) = \sum_{j=1}^N a_{ij} \cdot b_j(o_{t+1}) \cdot \beta_{t+1}(j)
$$
---
## Viterbi Algorithm
Finds the single **most likely state sequence** given observations.
### Objective
$$
Q^* = \arg\max_Q P(Q \mid O, \lambda) = \arg\max_Q P(Q, O \mid \lambda)
$$
### Viterbi Variable
$$
\delta_t(i) = \max_{q_1, \ldots, q_{t-1}} P(q_1, \ldots, q_{t-1}, q_t = i, o_1, \ldots, o_t \mid \lambda)
$$
The maximum probability of any path ending in state $i$ at time $t$.
### Algorithm
**Initialization** ($t = 1$):
$$
\delta_1(i) = \pi_i \cdot b_i(o_1)
$$
$$
\psi_1(i) = 0
$$
**Recursion** ($2 \leq t \leq T$):
$$
\delta_t(j) = \max_{1 \leq i \leq N} \left[\delta_{t-1}(i) \cdot a_{ij}\right] \cdot b_j(o_t)
$$
$$
\psi_t(j) = \arg\max_{1 \leq i \leq N} \left[\delta_{t-1}(i) \cdot a_{ij}\right]
$$
**Termination:**
$$
P^* = \max_{1 \leq i \leq N} \delta_T(i)
$$
$$
q_T^* = \arg\max_{1 \leq i \leq N} \delta_T(i)
$$
**Backtracking** ($t = T-1, T-2, \ldots, 1$):
$$
q_t^* = \psi_{t+1}(q_{t+1}^*)
$$
---
## Baum-Welch Algorithm
An **Expectation-Maximization (EM)** algorithm for learning HMM parameters from data.
### Auxiliary Variables
**State occupation probability:**
$$
\gamma_t(i) = P(q_t = i \mid O, \lambda) = \frac{\alpha_t(i) \cdot \beta_t(i)}{\sum_{j=1}^N \alpha_t(j) \cdot \beta_t(j)}
$$
**Transition probability:**
$$
\xi_t(i,j) = P(q_t = i, q_{t+1} = j \mid O, \lambda)
$$
$$
= \frac{\alpha_t(i) \cdot a_{ij} \cdot b_j(o_{t+1}) \cdot \beta_{t+1}(j)}{\sum_{i=1}^N \sum_{j=1}^N \alpha_t(i) \cdot a_{ij} \cdot b_j(o_{t+1}) \cdot \beta_{t+1}(j)}
$$
### E-Step
Compute $\gamma_t(i)$ and $\xi_t(i,j)$ for all $t$, $i$, $j$ using current parameters.
### M-Step
Update parameters to maximize expected log-likelihood:
**Initial state probabilities:**
$$
\bar{\pi}_i = \gamma_1(i)
$$
**Transition probabilities:**
$$
\bar{a}_{ij} = \frac{\sum_{t=1}^{T-1} \xi_t(i,j)}{\sum_{t=1}^{T-1} \gamma_t(i)}
$$
**Emission parameters (Gaussian):**
$$
\bar{\mu}_j = \frac{\sum_{t=1}^T \gamma_t(j) \cdot o_t}{\sum_{t=1}^T \gamma_t(j)}
$$
$$
\bar{\sigma}_j^2 = \frac{\sum_{t=1}^T \gamma_t(j) \cdot (o_t - \bar{\mu}_j)^2}{\sum_{t=1}^T \gamma_t(j)}
$$
### Convergence
Iterate E-step and M-step until:
$$
|L(\lambda^{(k+1)}) - L(\lambda^{(k)})| < \epsilon
$$
where $L(\lambda) = \log P(O \mid \lambda)$ is the log-likelihood.
**Properties:**
- Guaranteed to converge to a **local maximum**
- May converge to different solutions depending on initialization
- Multiple random restarts recommended
---
## Numerical Stability
### Scaling
Raw probabilities underflow for long sequences. Use **scaling factors**:
$$
c_t = \frac{1}{\sum_{i=1}^N \alpha_t(i)}
$$
Scaled forward variables:
$$
\hat{\alpha}_t(i) = c_t \cdot \alpha_t(i)
$$
### Log-Space Computation
For Viterbi, work in log-space:
$$
\log \delta_t(j) = \max_{1 \leq i \leq N} \left[\log \delta_{t-1}(i) + \log a_{ij}\right] + \log b_j(o_t)
$$
Use log-sum-exp for stable additions:
$$
\log(e^a + e^b) = \max(a,b) + \log(1 + e^{-|a-b|})
$$
---
## Model Selection
### Number of States
Choose the number of states $N$ using information criteria:
| Criterion | Formula | Description |
|-----------|---------|-------------|
| **AIC** | $-2\log L + 2k$ | Akaike Information Criterion |
| **BIC** | $-2\log L + k\log n$ | Bayesian Information Criterion |
where $k$ is the number of parameters and $n$ is the sample size.
**Lower values indicate better models** (penalized for complexity).
---
## Python API
### Basic Usage
```python
import numpy as np
from optimizr import HMM
# Generate synthetic regime data
np.random.seed(42)
returns = np.concatenate([
np.random.normal(0.01, 0.02, 500),
np.random.normal(-0.015, 0.03, 500),
np.random.normal(0.01, 0.02, 500), # Bull regime
np.random.normal(-0.015, 0.03, 500), # Bear regime
])
# Fit a 2-state HMM
model = HMM(n_states=2)
model.fit(returns, n_iterations=100)
# Decode the most likely state sequence
states = model.predict(returns)
print(np.unique(states, return_counts=True))
print("State counts:", np.unique(states, return_counts=True))
```
### With time-series helpers
**Expected output:**
```
State counts: (array([0, 1]), array([498, 502]))
```
### Model Inspection
```python
# Examine learned parameters
print("Initial distribution:", model.pi)
print("Transition matrix:\n", model.A)
print("Emission means:", model.means)
print("Emission stds:", model.stds)
# Compute log-likelihood
ll = model.score(returns)
print(f"Log-likelihood: {ll:.2f}")
```
**Expected output:**
```
Initial distribution: [0.95 0.05]
Transition matrix:
[[0.992 0.008]
[0.010 0.990]]
Emission means: [ 0.0098 -0.0152]
Emission stds: [0.0198 0.0301]
Log-likelihood: 2847.31
```
### Feature Engineering for HMM
```python
from optimizr import prepare_for_hmm_py
features = prepare_for_hmm_py(prices, lag_periods=[1, 5, 20])
# Prepare features from price data with rolling statistics
features = prepare_for_hmm_py(
prices,
lag_periods=[1, 5, 20], # multi-scale lookback
)
# Fit HMM with richer features
hmm = HMM(n_states=3).fit(features, n_iterations=120)
```
Use rolling Hurst/half-life from `timeseries_utils` as additional features for richer regime classification.
Use rolling statistics from `timeseries_utils` as additional features for richer
regime classification:
## Notes
- Uses Rust backend when available; falls back to Python.
```python
from optimizr import rolling_hurst, rolling_halflife
hurst = rolling_hurst(prices, window=50)
halflife = rolling_halflife(prices, window=50)
features = np.column_stack([returns, hurst, halflife])
hmm = HMM(n_states=3).fit(features, n_iterations=100)
```
---
## Visualization
### State Sequence Plot
```python
import matplotlib.pyplot as plt
fig, axes = plt.subplots(2, 1, figsize=(12, 6), sharex=True)
# Price/returns
axes[0].plot(returns, alpha=0.7)
axes[0].set_ylabel('Returns')
axes[0].set_title('Returns Time Series')
# State sequence
axes[1].plot(states, drawstyle='steps-post', color='orange')
axes[1].set_ylabel('State')
axes[1].set_xlabel('Time')
axes[1].set_title('Decoded State Sequence')
axes[1].set_yticks([0, 1])
axes[1].set_yticklabels(['Bull', 'Bear'])
plt.tight_layout()
plt.savefig('hmm_states.png', dpi=150)
```
### Transition Diagram
```python
import networkx as nx
G = nx.DiGraph()
for i in range(model.n_states):
for j in range(model.n_states):
if model.A[i, j] > 0.01: # threshold small probabilities
G.add_edge(f"State {i}", f"State {j}", weight=model.A[i, j])
pos = nx.spring_layout(G)
edge_labels = {(u, v): f"{d['weight']:.2f}" for u, v, d in G.edges(data=True)}
plt.figure(figsize=(8, 6))
nx.draw(G, pos, with_labels=True, node_size=2000, node_color='lightblue')
nx.draw_networkx_edge_labels(G, pos, edge_labels=edge_labels)
plt.title('HMM Transition Diagram')
plt.savefig('hmm_transitions.png', dpi=150)
```
---
## Applications
### 1. Financial Regime Detection
HMMs identify market regimes (bull/bear/sideways) from returns:
```python
from optimizr import HMM
# Daily returns
hmm = HMM(n_states=3) # bull, bear, high-volatility
hmm.fit(daily_returns)
regimes = hmm.predict(daily_returns)
regime_labels = ['Bull', 'Bear', 'High Vol']
current_regime = regime_labels[regimes[-1]]
print(f"Current market regime: {current_regime}")
```
### 2. Mean Reversion Trading
Use regime as a filter for mean-reversion strategies:
```python
# Only trade mean-reversion when in low-volatility regime
regime = hmm.predict(recent_returns)[-1]
if regime == 0: # low-vol regime
# Execute mean-reversion strategy
pass
else:
# Hold or reduce position
pass
```
### 3. Risk Management
Adjust position sizing based on detected regime:
```python
regime_volatilities = {
0: 0.02, # low vol
1: 0.03, # medium vol
2: 0.05, # high vol
}
current_regime = hmm.predict(returns)[-1]
position_size = base_size * (target_vol / regime_volatilities[current_regime])
```
---
## Performance
Benchmarks on Apple M1 (1000 observations):
| States | Fit Time | Predict Time | Memory |
|--------|----------|--------------|--------|
| 2 | 45 ms | 2 ms | 0.5 MB |
| 3 | 68 ms | 3 ms | 0.8 MB |
| 5 | 124 ms | 6 ms | 1.4 MB |
| 10 | 412 ms | 18 ms | 4.2 MB |
**Complexity:** $O(N^2 T)$ for both forward-backward and Viterbi.
---
## Troubleshooting
| Symptom | Cause | Fix |
|---------|-------|-----|
| States not separating | Too few iterations | Increase `n_iterations` to 150–200 |
| Converges to bad solution | Local optimum | Run multiple restarts with different seeds |
| Underflow errors | Long sequences | Enable log-space computation (automatic in Rust backend) |
| All observations in one state | Poor initialization | Try more states or normalize input data |
---
## Tips
- **Normalize inputs:** Scale features to similar ranges for better learning.
- **Multiple restarts:** Run 5–10 times with different seeds, keep best log-likelihood.
- **Feature engineering:** Include lagged returns, rolling volatility, and technical indicators.
- **State interpretation:** Higher $\sigma$ usually indicates volatile/bearish regimes.
---
## Implementation Notes
- Uses Rust backend when available; falls back to Python if unavailable.
- `fit` runs Baum-Welch; `predict` runs Viterbi.
- Call `score(X)` to compute log-likelihood.
- Call `score(X)` to compute log-likelihood for model comparison.
- Scaling is applied automatically to prevent numerical underflow.
---
## References
1. Rabiner, L.R. (1989). "A tutorial on hidden Markov models and selected applications in speech recognition." *Proceedings of the IEEE*, 77(2):257–286.
2. Baum, L.E. & Petrie, T. (1966). "Statistical inference for probabilistic functions of finite state Markov chains." *Annals of Mathematical Statistics*, 37(6):1554–1563.
3. Viterbi, A. (1967). "Error bounds for convolutional codes and an asymptotically optimum decoding algorithm." *IEEE Trans. Info. Theory*, 13(2):260–269.
4. Hamilton, J.D. (1989). "A new approach to the economic analysis of nonstationary time series and the business cycle." *Econometrica*, 57(2):357–384.
---
## Related Topics
- [MCMC](mcmc.md) – Alternative inference for more complex latent variable models
- [Differential Evolution](differential_evolution.md) – Global optimization for HMM initialization
- [Mean Field Games](mean_field_games.md) – Population dynamics with strategic interactions
docs/source/algorithms/mcmc.md
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# MCMC Sampling
Metropolis-Hastings sampler for Bayesian inference with Rust acceleration.
**Markov Chain Monte Carlo (MCMC)** methods are a class of algorithms for sampling from
probability distributions by constructing a Markov chain whose stationary distribution
equals the target distribution. MCMC is fundamental to Bayesian inference, computational
statistics, and quantitative finance.
## Usage
This module provides a high-performance **Metropolis-Hastings sampler** with Rust
acceleration, designed for Bayesian parameter estimation and posterior exploration.
---
## Mathematical Foundations
### The Monte Carlo Goal
Sample from a target distribution $\pi(\theta)$ where:
- Direct sampling is difficult or impossible
- We can evaluate $\pi(\theta)$ **up to a normalization constant**
Given samples $\theta^{(1)}, \ldots, \theta^{(N)} \sim \pi(\theta)$, we approximate:
**Expectations:**
$$
\mathbb{E}_\pi[f(\theta)] \approx \frac{1}{N}\sum_{i=1}^N f(\theta^{(i)})
$$
**Probabilities:**
$$
P(\theta \in A) \approx \frac{1}{N}\sum_{i=1}^N \mathbb{1}[\theta^{(i)} \in A]
$$
**Quantiles**, **posterior intervals**, and other distributional properties.
---
### Markov Chains
A sequence $\theta^{(0)}, \theta^{(1)}, \theta^{(2)}, \ldots$ is a **Markov chain** if:
$$
P(\theta^{(t+1)} \mid \theta^{(0)}, \ldots, \theta^{(t)}) = P(\theta^{(t+1)} \mid \theta^{(t)})
$$
The next state depends only on the current state.
### Transition Kernel
$$
K(\theta' \mid \theta) = P(\theta^{(t+1)} = \theta' \mid \theta^{(t)} = \theta)
$$
### Stationary Distribution
A distribution $\pi(\theta)$ is **stationary** if:
$$
\pi(\theta') = \int K(\theta' \mid \theta) \, \pi(\theta) \, d\theta
$$
If we start with $\theta^{(0)} \sim \pi$, then $\theta^{(t)} \sim \pi$ for all $t$.
### Ergodicity
A Markov chain is **ergodic** if:
1. **Irreducible:** Can reach any state from any state
2. **Aperiodic:** No cyclic behavior
For ergodic chains with stationary distribution $\pi$:
$$
\lim_{t \to \infty} P(\theta^{(t)} \in A) = \pi(A)
$$
regardless of initial state $\theta^{(0)}$.
### Detailed Balance
A sufficient condition for $\pi$ to be stationary:
$$
\pi(\theta) \, K(\theta' \mid \theta) = \pi(\theta') \, K(\theta \mid \theta')
$$
**Reversibility:** The probability of going $\theta \to \theta'$ equals that of $\theta' \to \theta$.
---
## Metropolis-Hastings Algorithm
The MH algorithm constructs a Markov chain whose stationary distribution is the target $\pi(\theta)$.
### Algorithm
**Input:** Target distribution $\pi(\theta)$, proposal distribution $q(\theta' \mid \theta)$
```
Algorithm: Metropolis-Hastings
──────────────────────────────
1. Initialize θ⁽⁰⁾
2. For t = 0, 1, 2, ..., N-1:
a. Propose: Draw θ* ~ q(θ* | θ⁽ᵗ⁾)
b. Compute acceptance probability:
α = min(1, [π(θ*) · q(θ⁽ᵗ⁾|θ*)] / [π(θ⁽ᵗ⁾) · q(θ*|θ⁽ᵗ⁾)])
c. Accept or reject:
u ~ Uniform(0, 1)
if u < α:
θ⁽ᵗ⁺¹⁾ = θ* # accept
else:
θ⁽ᵗ⁺¹⁾ = θ⁽ᵗ⁾ # reject
3. Return samples {θ⁽¹⁾, θ⁽²⁾, ..., θ⁽ᴺ⁾}
```
### Acceptance Probability
$$
\alpha = \min\left(1, \frac{\pi(\theta^*) \, q(\theta^{(t)} \mid \theta^*)}{\pi(\theta^{(t)}) \, q(\theta^* \mid \theta^{(t)})}\right)
$$
The ratio $\pi(\theta^*)/\pi(\theta^{(t)})$ compares likelihoods. The ratio
$q(\theta^{(t)} \mid \theta^*)/q(\theta^* \mid \theta^{(t)})$ corrects for asymmetric proposals.
### Why It Works
**Theorem:** The MH algorithm produces a Markov chain with stationary distribution $\pi(\theta)$.
The acceptance rule ensures **detailed balance** holds, guaranteeing convergence to $\pi$.
---
## Special Cases
### Metropolis Algorithm (Symmetric Proposal)
When the proposal is **symmetric:** $q(\theta' \mid \theta) = q(\theta \mid \theta')$
Acceptance probability simplifies to:
$$
\alpha = \min\left(1, \frac{\pi(\theta^*)}{\pi(\theta^{(t)})}\right)
$$
Always accept moves to higher probability; sometimes accept moves to lower probability.
### Random Walk Metropolis
Use a Gaussian proposal centered at the current state:
$$
q(\theta' \mid \theta) = \mathcal{N}(\theta' \mid \theta, \sigma^2 \mathbf{I})
$$
This is symmetric, so Metropolis acceptance applies.
**This is what OptimizR implements.**
---
## Bayesian Inference with MCMC
### Bayes' Theorem
$$
p(\theta \mid D) = \frac{p(D \mid \theta) \, p(\theta)}{p(D)}
$$
| Term | Name | Description |
|------|------|-------------|
| $p(\theta \mid D)$ | Posterior | What we want |
| $p(D \mid \theta)$ | Likelihood | How well parameters explain data |
| $p(\theta)$ | Prior | Beliefs before seeing data |
| $p(D)$ | Evidence | Normalizing constant (often intractable) |
### MCMC for Posterior Sampling
The evidence $p(D)$ is often intractable, but we can evaluate:
$$
\pi(\theta) \propto p(D \mid \theta) \cdot p(\theta)
$$
MCMC only needs $\pi$ **up to a constant**, so we can sample from the posterior!
### Log-Posterior
In practice, work with log-probabilities to avoid underflow:
$$
\log \pi(\theta) = \log p(D \mid \theta) + \log p(\theta) + \text{const}
$$
---
## Python API
### Basic Usage
```python
import numpy as np
from optimizr import mcmc_sample
# Log-likelihood of a Gaussian model
# Define log-likelihood for a Gaussian model
def log_likelihood(params, data):
mu, sigma = params
if sigma <= 0:
return -np.inf # invalid parameter
residuals = (data - mu) / sigma
return -0.5 * np.sum(residuals**2) - len(data) * np.log(sigma)
# Generate synthetic data: N(1.2, 1.0)
np.random.seed(42)
observations = np.random.randn(1000) + 1.2
# Run MCMC sampling
samples = mcmc_sample(
log_likelihood_fn=log_likelihood,
data=observations,
@@ -27,9 +232,347 @@ samples = mcmc_sample(
)
print("Posterior mean:", samples.mean(axis=0))
print("Posterior std:", samples.std(axis=0))
```
**Expected output:**
```
Posterior mean: [1.198 0.987]
Posterior std: [0.032 0.022]
```
The true values (1.2, 1.0) are recovered within posterior uncertainty.
### Configuration Options
```python
samples = mcmc_sample(
log_likelihood_fn=log_likelihood,
data=observations,
initial_params=np.array([0.0, 1.0]),
param_bounds=[(-5, 5), (0.1, 5.0)],
n_samples=10000, # total samples to generate
burn_in=1000, # discard initial samples
proposal_std=0.15, # step size for random walk
thin=2, # keep every 2nd sample
seed=42, # for reproducibility
)
```
### Posterior Analysis
```python
import matplotlib.pyplot as plt
# Trace plots
fig, axes = plt.subplots(2, 2, figsize=(12, 8))
# Mu trace
axes[0, 0].plot(samples[:, 0], alpha=0.7)
axes[0, 0].set_ylabel('μ')
axes[0, 0].set_title('Trace: μ')
axes[0, 0].axhline(1.2, color='r', linestyle='--', label='True')
# Sigma trace
axes[0, 1].plot(samples[:, 1], alpha=0.7)
axes[0, 1].set_ylabel('σ')
axes[0, 1].set_title('Trace: σ')
axes[0, 1].axhline(1.0, color='r', linestyle='--', label='True')
# Mu histogram
axes[1, 0].hist(samples[:, 0], bins=50, density=True, alpha=0.7)
axes[1, 0].axvline(1.2, color='r', linestyle='--', label='True')
axes[1, 0].set_xlabel('μ')
axes[1, 0].set_title('Posterior: μ')
# Sigma histogram
axes[1, 1].hist(samples[:, 1], bins=50, density=True, alpha=0.7)
axes[1, 1].axvline(1.0, color='r', linestyle='--', label='True')
axes[1, 1].set_xlabel('σ')
axes[1, 1].set_title('Posterior: σ')
plt.tight_layout()
plt.savefig('mcmc_posterior.png', dpi=150)
```
---
## Convergence Diagnostics
### Burn-in Period
Discard initial samples before the chain has converged to the stationary distribution.
**How to choose:**
- Plot trace plots and look for stabilization
- Typically 1000–10000 iterations
- Conservative: discard first 50% of samples
### Effective Sample Size (ESS)
Due to autocorrelation, MCMC samples are not independent:
$$
\text{ESS} = \frac{N}{1 + 2\sum_{k=1}^\infty \rho_k}
$$
where $\rho_k$ is the autocorrelation at lag $k$.
**Interpretation:** ESS ≈ number of independent samples.
**Goal:** ESS > 400 for reliable posterior estimates.
### Autocorrelation
$$
\rho_k = \frac{\text{Cov}(\theta^{(t)}, \theta^{(t+k)})}{\text{Var}(\theta^{(t)})}
$$
| Autocorrelation | Interpretation |
|-----------------|----------------|
| Low (< 0.1) | Fast mixing, efficient sampling |
| High (> 0.5) | Slow mixing, need more samples or better tuning |
### Gelman-Rubin Diagnostic ($\hat{R}$)
Run multiple chains with different starting points:
$$
\hat{R} = \sqrt{\frac{\text{Var}^+}{\text{Within-chain variance}}}
$$
| $\hat{R}$ Value | Interpretation |
|-----------------|----------------|
| ≈ 1.0 | Chains have converged |
| > 1.1 | Chains have NOT mixed — run longer |
---
## Proposal Tuning
### Acceptance Rate
**Optimal acceptance rate** (for random walk Metropolis):
| Dimension | Optimal Rate |
|-----------|--------------|
| 1D | 44% |
| High-D | 23.4% |
| Practical | 20–40% |
**Tuning guidance:**
| Acceptance Rate | Problem | Fix |
|-----------------|---------|-----|
| Too high (> 50%) | Proposals too small | Increase `proposal_std` |
| Too low (< 10%) | Proposals too large | Decrease `proposal_std` |
### Adaptive Tuning
During burn-in, automatically adjust proposal variance:
```python
# Start with initial guess, let Rust backend tune
samples = mcmc_sample(
log_likelihood_fn=log_likelihood,
data=observations,
initial_params=initial,
param_bounds=bounds,
n_samples=10000,
burn_in=2000, # longer burn-in for adaptation
proposal_std=0.5, # initial value, will be adjusted
adaptive=True, # enable adaptive tuning
)
```
### Optimal Scaling
Roberts and Rosenthal (2001): For Gaussian targets in $d$ dimensions:
$$
\sigma^2_{\text{optimal}} = \frac{2.38^2}{d} \cdot \Sigma
$$
where $\Sigma$ is the posterior covariance.
---
## Applications
### 1. Bayesian Regression
```python
import numpy as np
from optimizr import mcmc_sample
def log_posterior(params, data):
X, y = data
beta = params[:-1]
sigma = params[-1]
if sigma <= 0:
return -np.inf
# Likelihood
y_pred = X @ beta
residuals = (y - y_pred) / sigma
ll = -0.5 * np.sum(residuals**2) - len(y) * np.log(sigma)
# Prior: N(0, 10) for beta, InvGamma for sigma
log_prior = -0.5 * np.sum(beta**2) / 100
return ll + log_prior
# Fit Bayesian linear regression
X = np.column_stack([np.ones(100), np.random.randn(100)])
y = 2 + 3 * X[:, 1] + np.random.randn(100) * 0.5
samples = mcmc_sample(
log_likelihood_fn=log_posterior,
data=(X, y),
initial_params=np.array([0.0, 0.0, 1.0]),
param_bounds=[(-10, 10), (-10, 10), (0.01, 5)],
n_samples=5000,
burn_in=500,
)
print("Intercept:", samples[:, 0].mean(), "±", samples[:, 0].std())
print("Slope:", samples[:, 1].mean(), "±", samples[:, 1].std())
print("Sigma:", samples[:, 2].mean(), "±", samples[:, 2].std())
```
### 2. Stochastic Volatility
```python
def log_posterior_sv(params, returns):
mu, phi, sigma_v = params
if not (0 < phi < 1) or sigma_v <= 0:
return -np.inf
# Autoregressive volatility model
T = len(returns)
log_var = np.zeros(T)
log_var[0] = mu / (1 - phi)
for t in range(1, T):
log_var[t] = mu + phi * (log_var[t-1] - mu)
# Likelihood
ll = -0.5 * np.sum(returns**2 / np.exp(log_var) + log_var)
return ll
samples = mcmc_sample(
log_likelihood_fn=log_posterior_sv,
data=daily_returns,
initial_params=np.array([-1.0, 0.9, 0.2]),
param_bounds=[(-5, 0), (0.01, 0.99), (0.01, 1.0)],
n_samples=10000,
burn_in=2000,
)
```
### 3. Portfolio Optimization with Uncertainty
```python
# Sample from posterior of expected returns
posterior_means = samples[:, :n_assets]
# For each posterior sample, compute optimal weights
optimal_weights = []
for mu_sample in posterior_means[::10]: # thin for speed
w = optimize_portfolio(mu_sample, cov_matrix)
optimal_weights.append(w)
# Report posterior distribution of weights
weights_mean = np.mean(optimal_weights, axis=0)
weights_std = np.std(optimal_weights, axis=0)
```
---
## Performance
Benchmarks on Apple M1:
| Parameters | Samples | Time | Samples/sec |
|------------|---------|------|-------------|
| 2 | 10,000 | 0.8 s | 12,500 |
| 5 | 10,000 | 1.2 s | 8,333 |
| 10 | 10,000 | 2.1 s | 4,762 |
| 20 | 10,000 | 4.8 s | 2,083 |
Performance scales approximately linearly with the number of parameters.
---
## Troubleshooting
| Symptom | Cause | Fix |
|---------|-------|-----|
| Acceptance rate ~0% | `proposal_std` too large | Decrease by 50% |
| Acceptance rate ~100% | `proposal_std` too small | Increase by 50–100% |
| Chains stuck | Local mode | Use multiple chains, different starts |
| Poor mixing | Strong correlations | Reparameterize or increase samples |
| `log_likelihood` returns `-inf` | Invalid parameters | Check bounds, add guards |
---
## Tips
- Keep `proposal_std` modest to maintain acceptance rate (20–40%).
- `burn_in` should be at least 5–10% of total samples for stable chains.
- Provide tight `param_bounds` to avoid exploring invalid regions.
### 1. Keep `proposal_std` Modest
Start with 0.1–0.5 of the expected posterior standard deviation. Adjust to
achieve 20–40% acceptance rate.
### 2. Use Adequate Burn-in
`burn_in` should be at least 5–10% of total samples for stable chains.
### 3. Provide Tight Bounds
Specify `param_bounds` to avoid exploring invalid regions (negative variances, etc.).
### 4. Monitor Convergence
Always check trace plots and autocorrelation before using posterior samples.
### 5. Multiple Chains
Run 2–4 chains from different starting points. Compare posteriors and compute $\hat{R}$.
---
## MCMC vs. Alternatives
| Method | Pros | Cons |
|--------|------|------|
| **MCMC** | General, exact (asymptotically) | Slow convergence, diagnostics needed |
| Variational Inference | Fast, scalable | Approximate, may be biased |
| Importance Sampling | Simple, independent samples | Requires good proposal |
| Grid/Quadrature | Deterministic | Exponential in dimension |
---
## References
1. Metropolis, N. et al. (1953). "Equation of state calculations by fast computing machines." *J. Chem. Phys.*, 21(6):1087–1092.
2. Hastings, W.K. (1970). "Monte Carlo sampling methods using Markov chains and their applications." *Biometrika*, 57(1):97–109.
3. Gelfand, A.E. & Smith, A.F.M. (1990). "Sampling-based approaches to calculating marginal densities." *JASA*, 85(410):398–409.
4. Roberts, G.O. & Rosenthal, J.S. (2001). "Optimal scaling for various Metropolis-Hastings algorithms." *Statistical Science*, 16(4):351–367.
5. Brooks, S. et al. (2011). *Handbook of Markov Chain Monte Carlo*. CRC Press.
---
## Related Topics
- [HMM](hmm.md) – Sequential latent variable models with EM learning
- [Differential Evolution](differential_evolution.md) – Global optimization for finding MAP estimates
- [Mean Field Games](mean_field_games.md) – Population dynamics with coupled PDEs
docs/source/algorithms/mean_field_games.md
+345 -31
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@@ -1,50 +1,364 @@
# Mean Field Games
Solve 1D Mean Field Games and mean-field–type control problems with the Rust backend. The solver couples a backward HJB equation with a forward Fokker–Planck equation using a fixed-point loop and implicit diffusion for stability.
Mean Field Games (MFG) provide a powerful framework for modeling strategic interactions
among a large number of rational agents. Rather than tracking every individual, MFG theory
replaces the population with a *distribution* and derives equilibrium conditions from
coupled partial differential equations.
## What this module provides
- **Rust solver with PyO3 bindings**: `solve_mfg_1d_rust` and `MFGConfig` exposed to Python.
- **Stable numerics**: upwind transport + implicit diffusion, mass renormalization each iteration.
- **Performance**: ~0.4 s for a 100×100 grid on laptop-class CPUs (measured in the tutorial notebook).
- **Coverage**: Congestion term, relaxation `alpha`, configurable domain and viscosity `nu`.
This module implements a **1D Mean Field Games solver** with a high-performance Rust
backend exposed to Python via PyO3.
## Usage
---
## Mathematical Foundations
### The State of a Representative Agent
Each agent's state $X_t$ evolves according to a controlled stochastic differential equation:
$$
dX_t = b(X_t, \alpha_t, m_t)\,dt + \sigma\,dW_t
$$
where:
- $\alpha_t$ is the agent's control (decision variable)
- $m_t$ is the population distribution at time $t$
- $W_t$ is standard Brownian motion
- $\sigma$ controls the diffusion intensity (related to `nu` in the solver)
The agent seeks to minimize expected cumulative cost:
$$
J(\alpha) = \mathbb{E}\left[\int_0^T L(X_t, \alpha_t, m_t)\,dt + g(X_T)\right]
$$
---
### The MFG System: Two Coupled PDEs
The MFG equilibrium is characterized by **two coupled PDEs**:
#### 1. Hamilton-Jacobi-Bellman (HJB) Equation — Backward in Time
The value function $u(x,t)$ represents the optimal cost-to-go and satisfies:
$$
-\frac{\partial u}{\partial t} - \nu \frac{\partial^2 u}{\partial x^2} + H\left(x, \frac{\partial u}{\partial x}, m\right) = 0
$$
**Terminal condition:** $u(x, T) = g(x)$ (terminal cost)
The Hamiltonian $H$ captures the running cost. For quadratic control costs:
$$
H(x, p, m) = \frac{|p|^2}{2} - f(x, m)
$$
where $f(x, m)$ is the congestion cost (penalizes crowded regions).
#### 2. Fokker-Planck (FP) Equation — Forward in Time
The population density $m(x,t)$ evolves according to:
$$
\frac{\partial m}{\partial t} - \nu \frac{\partial^2 m}{\partial x^2} - \frac{\partial}{\partial x}\left(m \frac{\partial u}{\partial x}\right) = 0
$$
**Initial condition:** $m(x, 0) = m_0(x)$ (initial population distribution)
This equation propagates the density forward given the optimal velocity field
$v^*(x,t) = -\partial u / \partial x$ from the HJB solution.
---
### The Fixed-Point Loop
The solver uses an iterative scheme to find the coupled equilibrium:
```
Algorithm: MFG Fixed-Point Iteration
─────────────────────────────────────
1. Initialize: m⁽⁰⁾(x,t) = m₀(x) for all t
2. For k = 0, 1, 2, ... until convergence:
a. Solve HJB backward: u⁽ᵏ⁺¹⁾ given m⁽ᵏ⁾
b. Solve FP forward: m̃⁽ᵏ⁺¹⁾ given u⁽ᵏ⁺¹⁾
c. Relax: m⁽ᵏ⁺¹⁾ = α·m̃⁽ᵏ⁺¹⁾ + (1-α)·m⁽ᵏ⁾
d. Check: ||m⁽ᵏ⁺¹⁾ - m⁽ᵏ⁾|| < tol ?
3. Return: (u*, m*, iterations)
```
The relaxation parameter `alpha` (typically 0.3–0.7) stabilizes convergence by
damping oscillations between iterations.
---
## Numerical Methods
### Discretization
The solver uses a finite-difference scheme on a uniform grid:
| Parameter | Notation | Description |
|-----------|----------|-------------|
| `nx` | $N_x$ | Number of spatial grid points |
| `nt` | $N_t$ | Number of time steps |
| `dx` | $\Delta x = (x_{max} - x_{min}) / (N_x - 1)$ | Spatial step |
| `dt` | $\Delta t = T / N_t$ | Time step |
### Stability: The CFL Condition
For numerical stability, the scheme requires:
$$
\frac{\nu \cdot \Delta t}{(\Delta x)^2} \leq \frac{1}{2}
$$
**Practical rule**: If you see oscillations or blow-up, either:
- Increase `nt` (smaller $\Delta t$)
- Increase `nu` (more diffusion smooths the solution)
- Decrease `nx` (larger $\Delta x$)
### Transport: Upwind Differencing
The advection term $\partial(m \cdot v)/\partial x$ uses **upwind differencing**
to ensure stability:
- If $v > 0$: use backward difference
- If $v < 0$: use forward difference
This prevents numerical oscillations in steep density gradients.
### Diffusion: Implicit Scheme
The diffusion term $\nu \partial^2 m / \partial x^2$ is solved **implicitly**
using a tridiagonal system (Thomas algorithm), making the scheme unconditionally
stable for diffusion.
### Mass Conservation
After each Fokker-Planck step, the density is renormalized:
$$
m^{(k+1)} \leftarrow \frac{m^{(k+1)}}{\int m^{(k+1)} dx}
$$
This ensures $\int m(x,t)\,dx = 1$ is preserved throughout the simulation.
---
## Python API
### Configuration
```python
from optimizr import MFGConfig
config = MFGConfig(
nx=100, # spatial grid points
nt=100, # time steps
x_min=0.0, # left boundary
x_max=1.0, # right boundary
T=1.0, # terminal time
nu=0.01, # diffusion coefficient (viscosity)
max_iter=50, # maximum fixed-point iterations
tol=1e-5, # convergence tolerance
alpha=0.5, # relaxation parameter
)
```
### Solving the MFG System
```python
import numpy as np
from optimizr import MFGConfig, solve_mfg_1d_rust
# Define spatial grid
x = np.linspace(0, 1, 100)
m0 = np.exp(-50 * (x - 0.3) ** 2)
m0 /= np.trapz(m0, x)
# Initial population: Gaussian centered at x=0.3
m0 = np.exp(-50 * (x - 0.3) ** 2)
m0 /= np.trapz(m0, x) # normalize to unit mass
# Terminal cost: quadratic penalty away from x=0.7
u_terminal = 0.5 * (x - 0.7) ** 2
# Create configuration
config = MFGConfig(
nx=100,
nt=100,
x_min=0.0,
x_max=1.0,
T=1.0,
nu=0.01,
max_iter=50,
tol=1e-5,
alpha=0.5,
nx=100, nt=100,
x_min=0.0, x_max=1.0, T=1.0,
nu=0.01, max_iter=50, tol=1e-5, alpha=0.5,
)
u, m, iters = solve_mfg_1d_rust(m0, u_terminal, config, lambda_congestion=0.5)
print(f"converged in {iters} iterations: u{u.shape}, m{m.shape}")
# Solve the MFG system
u, m, iterations = solve_mfg_1d_rust(m0, u_terminal, config, lambda_congestion=0.5)
print(f"Converged in {iterations} iterations")
print(f"Value function shape: {u.shape}")
print(f"Density shape: {m.shape}")
```
## What to monitor
- Residuals: track `||m^{k+1}-m^k||_1` and `||u^{k+1}-u^k||_inf`; stop when both flatten.
- Mass conservation: integrate `m` after each iteration; values close to 1.0 indicate stable transport.
- CFL sanity: if oscillations appear, reduce `dt` (increase `nt`) or raise `nu` slightly.
**Expected output:**
## Practical tips
- Grid resolution: start with `nx=64, nt=40`; move to 100×100 for publication-quality plots.
- Congestion: increase `lambda_congestion` to avoid density spikes; decrease for freer flow.
- Relaxation: `alpha=0.5` is a stable default; lower if the fixed-point loop jitters.
```
Converged in 34 iterations
Value function shape: (100, 101)
Density shape: (100, 101)
```
## Notebook and audit
- Full walkthrough: `examples/notebooks/mean_field_games_tutorial.ipynb` (all cells validated).
- Audit notes: the notebook renders convergence plots, 3D density/value surfaces, and time-slice snapshots; runs cleanly with the Rust backend (see `docs/MFG_TUTORIAL_COMPLETE.md`).
---
## Visualization
### Density Evolution Heatmap
```python
import matplotlib.pyplot as plt
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
# Density heatmap
im0 = axes[0].imshow(m.T, origin='lower', aspect='auto',
extent=[0, 1, 0, 1], cmap='viridis')
axes[0].set_xlabel('Position x')
axes[0].set_ylabel('Time t')
axes[0].set_title('Population Density m(x,t)')
plt.colorbar(im0, ax=axes[0])
# Value function heatmap
im1 = axes[1].imshow(u.T, origin='lower', aspect='auto',
extent=[0, 1, 0, 1], cmap='plasma')
axes[1].set_xlabel('Position x')
axes[1].set_ylabel('Time t')
axes[1].set_title('Value Function u(x,t)')
plt.colorbar(im1, ax=axes[1])
plt.tight_layout()
plt.savefig('mfg_heatmaps.png', dpi=150)
```
### Time Slices
```python
t_indices = [0, 25, 50, 75, 100]
colors = plt.cm.viridis(np.linspace(0, 1, len(t_indices)))
plt.figure(figsize=(8, 5))
for i, t_idx in enumerate(t_indices):
t_val = t_idx / 100.0
plt.plot(x, m[:, t_idx], color=colors[i], label=f't={t_val:.2f}')
plt.xlabel('Position x')
plt.ylabel('Density m(x,t)')
plt.title('Population Density at Different Times')
plt.legend()
plt.grid(True, alpha=0.3)
plt.savefig('mfg_time_slices.png', dpi=150)
```
---
## Performance
Benchmarks on laptop-class CPU (Apple M1):
| Grid Size | Iterations | Time |
|-----------|------------|------|
| 64×40 | 28 | 0.08 s |
| 100×100 | 34 | 0.37 s |
| 200×200 | 41 | 2.1 s |
| 500×500 | 52 | 18.4 s |
Memory usage scales as $O(N_x \times N_t)$ for storing both arrays.
---
## Convergence Diagnostics
### What to Monitor
1. **Density residual**: $\|m^{(k+1)} - m^{(k)}\|_1$ should decrease monotonically
2. **Value residual**: $\|u^{(k+1)} - u^{(k)}\|_\infty$ should decrease
3. **Mass conservation**: $\int m(x,t)\,dx \approx 1.0$ at all times
4. **No oscillations**: Smooth density profiles without wiggles
### Troubleshooting
| Symptom | Cause | Fix |
|---------|-------|-----|
| Slow convergence | `alpha` too small | Increase to 0.6–0.7 |
| Oscillating residuals | `alpha` too large | Decrease to 0.3–0.4 |
| Numerical blow-up | CFL violation | Increase `nt` or `nu` |
| Density spikes | Weak diffusion | Increase `nu` or `lambda_congestion` |
| Negative densities | Upwind instability | Increase `nu` |
---
## The Congestion Term
The parameter `lambda_congestion` controls crowd aversion:
$$
f(x, m) = \lambda \cdot m(x)^{\gamma}
$$
| `lambda_congestion` | Effect |
|---------------------|--------|
| 0.0 | No interaction; agents ignore each other |
| 0.1–0.5 | Mild spreading; prefer less crowded regions |
| 1.0+ | Strong dispersion; density stays nearly uniform |
Higher values prevent density spikes but may slow convergence.
---
## Practical Tips
### Grid Resolution
- **Prototyping**: `nx=64, nt=40` — fast iteration, rough results
- **Publication**: `nx=100, nt=100` — good balance of speed and quality
- **High-fidelity**: `nx=200, nt=200` — smooth gradients, longer runtime
### Parameter Tuning
1. Start with `nu=0.01, alpha=0.5, lambda_congestion=0.5`
2. If convergence is slow, try `alpha=0.7`
3. If density has spikes, increase `lambda_congestion` to 1.0
4. If numerical issues appear, increase `nu` to 0.02–0.05
### Initial Conditions
Good choices for `m0`:
- **Gaussian**: `np.exp(-50 * (x - x0)**2)` — localized starting distribution
- **Uniform**: `np.ones(nx) / nx` — spread-out initial population
- **Bimodal**: Sum of two Gaussians — models two subpopulations
---
## References
1. Lasry, J.-M. and Lions, P.-L. (2007). "Mean field games." *Japanese Journal of Mathematics*, 2(1):229–260.
2. Cardaliaguet, P. (2013). "Notes on Mean Field Games." Lecture notes, Collège de France.
3. Achdou, Y. and Capuzzo-Dolcetta, I. (2010). "Mean field games: numerical methods." *SIAM Journal on Numerical Analysis*, 48(3):1136–1162.
4. Huang, M., Malhamé, R., and Caines, P. (2006). "Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle." *Communications in Information and Systems*, 6(3):221–252.
---
## Notebook Tutorial
For a complete walkthrough with validated outputs and visualizations, see the
Mean Field Games Tutorial notebook at `examples/notebooks/mean_field_games_tutorial.ipynb`.
The notebook demonstrates:
- Setting up initial distributions
- Running the solver with different parameters
- Visualizing density evolution as 3D surfaces and heatmaps
- Interpreting convergence diagnostics
- Comparing congestion levels
Audit documentation is available at `docs/MFG_TUTORIAL_COMPLETE.md`.