Add hill climbing visualization blog post and Princeton PhD news

Author: Yixun-Hu

_posts/2025-10-22-probability-for-ml.md

@@ -83,3 +83,8 @@ The more coin flips per experiment and the more experiments you run, the more cl
 
 Try adjusting the parameters in the visualization to build your intuition about how the CLT works!
 
+## More Resources to learn
+1. [Diffusion Models from MIT courses]()
+2. [Optimization theory]()
+3. [Probability theory for machine learning]()
+4. 

_posts/2025-10-23-hill-climbing-search.md

@@ -0,0 +1,149 @@
+---
+layout: post
+title: Hill Climbing Search for Predicate Discovery
+date: 2025-10-23 10:00:00
+description: An interactive visualization of the hill climbing algorithm used in grammar search invention for task planning
+tags: algorithms optimization machine-learning
+categories: tutorial
+related_posts: false
+---
+
+Hill climbing is a fundamental optimization algorithm that iteratively improves a solution by making locally optimal choices. In this post, we'll explore how it's used to discover predicates for task planning systems.
+
+## What is Hill Climbing?
+
+Hill climbing is a **local search algorithm** that starts from an initial solution and repeatedly moves to a better neighboring solution until no improvement can be found. Think of it as climbing a hill in dense fog - you can only see your immediate surroundings, so you keep taking steps upward until you reach a peak.
+
+### Key Characteristics:
+
+- **Greedy approach**: Always chooses the best immediate option
+- **Local search**: Only explores nearby solutions
+- **No backtracking**: Cannot undo previous decisions
+- **Fast convergence**: Typically finds solutions quickly
+
+## Application: Predicate Discovery for Planning
+
+In the **Predicators** system for neuro-symbolic learning, hill climbing is used to discover the optimal set of logical predicates that describe a planning domain. 
+
+### The Problem:
+
+Given a large grammar of 121 possible predicates, find the subset that:
+- Best describes the planning domain
+- Minimizes the heuristic score (planning difficulty)
+- Enables efficient task completion
+
+### The Solution:
+
+Start with an empty set and iteratively add predicates that provide the most improvement, until no better predicates can be found.
+
+## Interactive Demonstration
+
+Below is a real example from the Predicators paper, showing how hill climbing discovered 5 optimal predicates in just 5 steps, reducing the heuristic from 535,231 to 293!
+
+<div class="row mt-3">
+    <div class="col-sm mt-3 mt-md-0">
+        <iframe src="/assets/html/hill-climbing-visualization.html" width="100%" height="1200" frameborder="0" style="border: 1px solid #ddd; border-radius: 8px;"></iframe>
+    </div>
+</div>
+
+## Key Observations
+
+### Massive Early Improvement
+The first predicate reduced the heuristic by **99.7%** (from 535,231 to 13,659)! This shows that even basic logical conditions can dramatically improve planning efficiency.
+
+### Diminishing Returns
+Later steps show smaller improvements:
+- Step 1: -99.7%
+- Step 2: -1.5%
+- Step 3: -97.0%
+- Step 4: -15.4%
+- Step 5: -13.0%
+
+This is typical of hill climbing - early steps make big improvements, later steps fine-tune.
+
+### Fast Convergence
+Only **5 iterations** were needed to find the optimal set from 121 candidates. The algorithm evaluated 726 predicates total but efficiently pruned the search space.
+
+## The Mathematics
+
+The hill climbing algorithm can be formalized as:
+
+$$
+s_{t+1} = \arg\max_{s' \in N(s_t)} f(s')
+$$
+
+where:
+- $$s_t$$ is the current state at time $$t$$
+- $$N(s_t)$$ is the set of neighbor states
+- $$f(s')$$ is the evaluation function (heuristic)
+- We terminate when $$f(s_{t+1}) \leq f(s_t)$$ for all neighbors
+
+In this application:
+- **State**: A set of predicates
+- **Neighbors**: States with one additional predicate
+- **Evaluation**: Heuristic score from learned operators
+
+## Advantages and Limitations
+
+### ✅ Advantages:
+1. **Simple to implement** - straightforward algorithm
+2. **Fast execution** - finds solutions quickly
+3. **Low memory** - only stores current state
+4. **Works well** - effective for many problems
+
+### ⚠️ Limitations:
+1. **Local optima** - may get stuck at peaks that aren't global maxima
+2. **No backtracking** - cannot undo bad early decisions
+3. **Order dependent** - different starting points yield different results
+4. **Plateau problem** - can get stuck on flat regions
+
+## Extensions and Variations
+
+To address limitations, several variants exist:
+
+### Random-Restart Hill Climbing
+Run hill climbing multiple times with different starting points and choose the best result.
+
+### Simulated Annealing
+Occasionally accept worse solutions to escape local optima, with probability decreasing over time.
+
+### Tabu Search
+Maintain a list of recently visited states to avoid cycling.
+
+### Beam Search
+Keep track of multiple candidate solutions simultaneously.
+
+## Real-World Performance
+
+The predicates discovered by this hill climbing search enabled the system to:
+- ✅ Solve **50/50 test tasks** (100% success rate)
+- ⏱️ Average planning time: **0.0023 seconds**
+- 🎯 Found optimal solution in **5 steps**
+
+This demonstrates that simple algorithms can be highly effective when applied to the right problem structure!
+
+## Try It Yourself!
+
+Use the interactive visualization above to:
+1. **Step through** the algorithm iteration by iteration
+2. **Auto-play** to watch the full search process
+3. **See the predicates** being added at each step
+4. **Track the heuristic** improvement on the chart
+
+Use keyboard shortcuts:
+- `→` Next step
+- `←` Previous step  
+- `Space` Play/Pause
+- `R` Reset
+
+## Conclusion
+
+Hill climbing is a powerful yet simple optimization technique that works remarkably well for many real-world problems. While it has limitations, understanding when and how to apply it is a crucial skill in AI and machine learning.
+
+The predicate discovery example shows how hill climbing can efficiently search through large spaces to find compact, effective solutions - a key principle in neuro-symbolic AI systems.
+
+## References
+
+- [Predicators: Neuro-Symbolic Learning for Task Planning](https://arxiv.org/abs/2210.00649)
+- Silver, T., Hariprasad, V., Shuttleworth, R. S., Kumar, N., Lozano-Pérez, T., & Kaelbling, L. P. (2022)
+