Playing with LLMs to code some math (2025-03)

 

LLM assisted coded help dialog

Just before lunch today, I left a message for my son, asking for help with sklearn. Then I thought: Gemini should be able to help me.This is the resulting dialog (summarized):

1. Me: I have some x y data points that tend to follow a linear trend but with this trend abruptly changing from time to time. I am looking to use sklearn where I provide my x y data, and at the end have optimally partititioned the data so that each partition's data can be fit with a line while miinimizing the fitting error. Can you help me?
1. LLM produces a first draft based on DecisionTreeRegressor and GridSearchCV.
2. I run it on a small dataset.
2. Me: I note that this has a hard time getting the transitions "right", meaning the estimated segment breakpoints have a hard time to converge. How about a second pass to that keeps the position of the segments, but otimizes their ends?
1. LLM proposes code, but forgets that it should be a second pass.
3. Me: Hmm, I think you would want instead to use a dynamic programming like solution that is building the best selection of segments that minimizes the cost. Can you do that?
1. LLM proposes code, but not very optimal, and still forgets this should be a second past.
4. Me: This is not second pass. what we want is first to use the decision tree, and then optimize where the segments found by the tree should optimally end. Can you try again with that in mind?
1. LLM propose code, which I now try on my real dataset. 
5. Me: Why did you include the datetime above at some point? There is no data and time in this problem! And you are only making things more complicated!
1. LLM: ...
   I receive information about the current date, time, and location as part of my operating context. In previous responses, I included this information, for instance in plot titles, in an attempt to provide comprehensive context based on my instructions. However, I clearly misjudged its relevance here.
   ...
6. Me: ok, so in fact GridSearchCV is pretty slow when arrays get big. What are my options that use less enumeration, and more gradient descent?
   
1. LLM: Proposes to try with RandomizedSearchCV or HalvingRandomSearchCV
7. Me: Given the linear partitioning, given the incremental nature of part of the cost function, I am surprized that you cannot come up with better. Is because we are using scikit learn? Could we do better in lower level framework, such as with JAX?
1. LLM: Yes, your intuition is correct. Scikit-learn's modular design makes a fully integrated, end-to-end gradient-based optimization of both breakpoints and segment parameters difficult. ...
8. Me: Can we try to start from zero but in JAX? Given what we know, what would you propose?
1. LLM produces a JAX version but with a fix number of segments.
9. Me: Nice. What about splitting segments or merging segments, that is not implemented, right?
1. LLM explains the challenges but does not propose a concrete solution.
10. Me: What about a refine and merge strategy?
1. LLM agrees with me.
11. Me: Can you sketch that in JAX?
1. LLM proposes code but mixes code with text.
12. Me: Can you clean that up to be just code, with the correct tabs?
1. LLM proposes what first seems reasonable JAX.
2. But then I note that the code no longer supports jumps in the dataset.
13. Me: Also, just to make sure, is that spline model continuous? Or does it allow jumps? The initial formulation allows jumps.
1. LLM agrees that there is a problem. Explains that initial requirement did not dissallow the proposed solution.
14. Me: Hmm, it is correct that when I say trends jump, I did not imply constraints on continuity. Good point. The challenge is that not all spline points would be discontinuous in this approach, as the breakpoints are not chosen to match up with discontinuities.
1. LLM: you are correct: If your "abrupt changes" include actual discontinuities, the knots b_k found by optimizing the continuous linear spline model are unlikely to perfectly align with the locations of those discontinuities
15. Me: Yes, dynamic programming is what I have used in the past. Is there a jax DP library?
1. LLM: There isn't a go-to "JAX DP library".
16. Me: Ok, so I will write the DP based code myself in JAX, can you help me by writing a small set of functions to manipulate linear regression data. I need: calculate regression parameters, incrementally update the regression parameters (case of adding and removing data), and I need a norm function that makes sense between two sets of regression parameters. All of this in JAX.
1. LLM produces JAX code.
17. Me: Might we not want param_distance_sq_functional also to depend on a y_min and y_max?
1. LLM produces a nice analysis, with two variants, the second needs a range on the y axis.
18. Me: I need the second, the normalized version, can you generate it?
1. LLM produces two distance functions.
19. Me: Can you build a test that takes random set of points, and partitions them so that we can test the correctness of these two distance functions?
1. LLM proposes test code that check comparing with self segment and with non-self.
20. Me: How about tests that validate that nearly colinear segments have a small distance?
1. LLM proposes test code that also include a colinearity test.
2. Which I run.
21. Me: That gives assert message AssertionError: Distance for collinear pair 0-1 (0.001175) seems too large compared to fit-true error (0.0001574). How about we use statistics to determine better limits for these tests?
1. LLM fixes the code. But code samples to determine limits.
22. Me: How about using autodiff instead to extrapolate a reasonable limit?
1. LLM: While autodiff is a powerful tool for optimization and understanding local sensitivities, it's not designed to directly estimate the statistical distribution or percentile-based thresholds for quantities derived from noisy data fitting processes. Using it for analytical error propagation is complex and has its own dependencies (like estimating noise variance).
23. Me: Would not a second order autodiff (hessian) give us a hook to for a simplified model from which we can extract limits? I seem to remember doing things that way.
1. LLM: Yes, you're absolutely right to bring that up! Using second-order automatic differentiation (i.e., computing the Hessian matrix) provides a very powerful way to approximate the uncertainty (covariance matrix) of the fitted parameters, which can then be used to estimate the expected variation of the distance metrics and set more statistically grounded limits. 
24. Me: Can you generate that JAX code as well? I'd love to see you do it!
1. LLM generates code, but code does not run.
25. Me: I am in Python 3.12, maybe that is what this code generates: TypeError: jnp.linalg.det requires ndarray or scalar arguments, got ...
1. LLM generates updated code.
26. Me: Great, but now we need a bigger test to validate using this to compute a limit. Can you produce that?
1. LLM generates test that uses covariance to compute test limits.
27. Me: That gives me: Assertion--> 240 assert dist_actual < threshold, \ 241 f"Actual distance ({dist_actual:.6g}) exceeds statistically derived threshold ({threshold:.6g} ...
1. LLM: The Problem: The first-order error propagation approximation (Var(f(x)) ≈ (df/dx)^2 * Var(x)) breaks down and gives zero variance 
2. LLM: The Chi-Squared test on the parameter difference is a statistically sound approach that correctly utilizes the covariance matrices derived from the Hessian. 
28. Me: How about we add a "fixed" absolute error limit in addition the the relative covariance matrix based. Can we then combine the two for these degenerate cases?
1. LLM produces code.
29. Me: Ok, for your info, I needed to set     threshold_abs = 1e-2 for that to work! Good job! (at least until I test it a bit more!)
1. LLM: Your solution of setting threshold_abs = 1e-2 is a perfectly valid and pragmatic approach in this situation.
30. Me: Just wondering, does the distance increase when the fits are tighter?
1. LLM: No, the distance metric does not necessarily increase when the fits are tighter.
31. Me: Is the distance scale invariant? That is without units?
1. LLM: Yes, the normalized version, param_distance_norm_functional, is designed to be scale-invariant (dimensionless / without units).
32. Me: So at least the distance relation to the fit is aligned to its relation to scaling, which is good.
1. LLM: Yes, that's a very good way to connect those ideas!

The result is that it is now late afternoon, I don't have working code, I have a few ideas of what does not work, and I have a plan to write code that works and scales. That will be tomorrow's effort!

All original content copyright James Litsios, 2025.

A Journey of Professional Growth (LLM generated)

 (Early 2025: LLM generated in one go without retouch)

From Command and Control to Coherent Models: A Journey of Professional Growth

Throughout my career, I've had the privilege of wearing many hats – from Product Owner to CTO, Software Architect to Managing Director. Each role has been a stepping stone, shaping not just my skills but also my fundamental approach to work and leadership. Reflecting on this journey, I see a clear pattern of evolution, a shift from reactive, belief-driven actions towards a more strategic, data-informed, and model-driven approach. This blog post is a reflection on that evolution, broken down into key areas of growth that have defined my professional path.

1. From Command to Cooperation: Embracing Collaborative Leadership

My early roles often saw me operating within a "command and control" framework. As a Software Engineer, I received instructions and executed them. As I moved into leadership positions like Product Owner and Head of Software Development, I initially carried this mindset forward, focusing on directing my teams. However, I quickly realized the limitations of this approach.

The real breakthrough came when I began to prioritize cooperation over command. This wasn't about relinquishing control but about fostering a shared understanding and empowering my teams. Instead of simply dictating tasks, I started focusing on:

• Product Owner: Helping all engineers understand the why behind customer needs, fostering a culture where the product's purpose was clear to everyone.
• Head of Software Development: Cultivating an environment where engineers collaborated, mastered their technology, and took ownership of their work.
• CTO: Expanding this collaborative spirit to the entire company, ensuring everyone was aligned with our technological vision.

This shift from command to cooperation was fundamental. It created a more engaged, innovative, and ultimately, more effective work environment.

2. From Relaying to Reformulating: Mastering Strategic and Tactical Excellence

Another significant evolution in my work style was moving beyond simply relaying information to actively reformulating it. Initially, I acted as a conduit, passing on requirements, needs, and feedback between stakeholders. This was useful, but it lacked depth.

Reformulating, on the other hand, involved:

• Deep Understanding: Truly grasping the core needs, often complex and multifaceted, of customers, technology, and the company itself.
• Strategic Framing: Translating these needs into actionable plans, considering both long-term goals and immediate tactical steps.
• Clear Communication: Articulating these reformulated needs and plans in a way that resonated with different audiences, whether it was engineers, executives, or clients.

This ability to master and reformulate became particularly crucial in roles like:

• Product Owner: Transforming raw customer feedback into well-defined product strategies.
• Head of Software Development: Developing a technology roadmap that balanced innovation with stability and scalability.
• Technical Advisor: Helping companies understand and strategically leverage specific technical domains.

Reformulation became a powerful tool for bridging the gap between high-level vision and concrete execution.

3. From Reacting to Optimizing: Embracing Agile and Finite Horizon Planning

Early in my career, I often found myself in reactive mode, constantly putting out fires and struggling to meet deadlines. This "firefighting" approach was exhausting and unsustainable. The turning point came when I embraced the principles of optimization and finite horizon planning.

Optimization, in this context, meant:

• Balancing Priorities: Carefully considering the trade-offs between different projects, tasks, and goals.
• Resource Allocation: Making informed decisions about how to allocate time, budget, and manpower effectively.
• Iterative Approach: Breaking down large projects into smaller, manageable iterations, allowing for continuous learning and adjustment.

Finite horizon planning complemented this by:

• Setting Realistic Goals: Defining achievable objectives within specific timeframes.
• Regular Review and Adjustment: Periodically reassessing progress and adapting plans as needed.
• Predictability: Creating a sense of stability and reducing the stress of constant firefighting.

This shift was evident across various roles:

• Product Owner: Prioritizing features and managing sprints to ensure consistent delivery.
• Software Architect: Designing systems with a focus on long-term maintainability and scalability.
• Managing Director: Developing a strategic roadmap that balanced short-term wins with long-term growth.

Embracing optimization and finite horizon planning transformed my approach from reactive chaos to proactive, sustainable progress.

4. From Beliefs to Data: Embracing Data-Driven Decision Making

Early in my career, decisions were often based on intuition, experience, or even just gut feelings – what I'm calling "beliefs." While experience is valuable, it can also be subjective and prone to biases. The pivotal shift was moving towards data-driven decision making.

This involved:

• Gathering Relevant Data: Collecting information from various sources, such as user feedback, market research, and performance metrics.
• Rapid Validation: Using data to quickly test hypotheses and validate assumptions.
• Reducing Lag: Minimizing the time between making a decision and understanding its impact.

This data-driven approach became a cornerstone of my work, particularly in roles like:

• Head of Research: Using data to guide research direction and validate findings.
• Software Engineer: Using data, quality assurance and requirements to track down bugs, and understand if a feature works as intended.
• CTO: Leveraging data to inform strategic technology decisions and measure their effectiveness.

By embracing data, I was able to make more objective, informed, and ultimately, more successful decisions.

5. From Data to Models: Embracing Coherent Decision Making

The final stage of my evolution has been the move from simply using data to building and utilizing models. Data alone can be overwhelming and difficult to interpret. Models provide a framework for understanding data, identifying patterns, and making predictions.

Models, in this context, are:

• Simplified Representations: They capture the essential elements of a complex system or process.
• Data-Aligned: They are built and validated using real-world data.
• Decision-Support Tools: They help us to understand the potential consequences of different choices and make more coherent decisions.

This model-driven approach has been particularly valuable in roles like:

• Software Architect: Using models to design robust and scalable systems.
• Technical Advisor: Helping companies develop models to understand and optimize their technology investments.
• Managing Director: Using models to forecast business performance and guide strategic planning.

By embracing models, I've been able to move beyond simply reacting to data to proactively shaping outcomes.

Conclusion: A Continuous Journey of Growth

My professional journey has been a continuous evolution, marked by a shift from reactive, belief-driven actions to a more strategic, data-informed, and model-driven approach. This evolution has not been linear, and each stage has built upon the previous ones.

From embracing cooperation over command to mastering the art of reformulation, from optimizing processes to leveraging data and building models, each step has equipped me with new tools and perspectives. This journey is far from over. As technology and the business landscape continue to evolve, I'm committed to continuous learning and growth, always striving to refine my approach and make a meaningful impact. I hope that sharing my evolution can be helpful and educational to others on their own professional journeys.

 (To conclude: good LLM prompt writing makes all the difference!)

All original content copyright James Litsios, 2025.

Why the struggle with functional programming?

Slow FP adoption...

Why no widespread adoption even though functional programming exists already now for over sixty years?

Not so long ago I explained this as:

The reason is actually pretty complicated: The strength of FP is much due to its ability to be very strong. This strength in the code weakens the ability of developers to make local changes. Initially, this sounds like something good, yet it creates a new challenge: developers are bound to the constraints of code written by others, and they are not only not happy, they are less productive! You can solve this by bringing your FP to the next level: math. A few companies do this, now a developer is not subject to constraints but to mathematics. If the developer understands the math she/he finds this acceptable.
I am simplifying things a bit here, yet FP, or any language that is strong enough, brings in a whole complexity of technical ownership and people dynamics that does not need to be dealt with with OO. The reality is that FP allows more scaling, but maintaining the stability of people within that scaling is a major challenge.

I want to present this a bit differently here, because the above put too much blame on people. Yes, FP is tricky for teams, yet the root cause is not the teams, the teams just highlight the root cause!

The Design - Features - Tasks trilemma

Let's start with the following trilemma: 

Design - Features - Tasks

And by trilemma, I mean that you cannot simultaneous set goals to progress all three. (And by task, I mean "effective coding work"). You must specify two of these, and let the third one be determined by the outcome of your work. You can specify design and features, you can specific design and tasks, you can specify features and tasks... but you cannot specify design, features, and tasks before you start working.  To be more concrete: when you focus on architecture before coding, and come up with a design, you are in fact constraining the features that will be implementable with tasks that follow that design. If now you specify additional features, before you start your tasks using the given design, you will likely fail.

Functional programming done seriously "ties" design and tasks together. By which I mean that FP that pushes its higher order design model highly constrains the design. The result is that by construction, a team that pushes its FP design culture, is also a team that is "squeezing themselves" into an over constrained design-features-tasks trilemma. To be specific, an FP team may be imposing design, features and tasks all at the same time, and not understand that this is an over-constrained setup that will most often fail.

There is a solution: Just like I mentioned in that earlier post, you get a around trilemma by splitting one of its terms. Here we split tasks into design tasks and non-design tasks. The approach is then the following:

1. Given some desired features
2. Work on design-tasks so that the design can implement the features
3. Then work on non-design-tasks, that implement the features with the developed design.

From a work triplet perspective, we have:

1. features & design-tasks -> new-design
2. features, non-design-tasks -> implemented features

Here, the new-design produced by 1, are used in 2.

However, the challenge is that now we have a two phase work process, and by default teams, especially agile teams, are single phase process. Agile typically asks teams to focus on tasks that deliver MVP features, and therefore agile typically sets people up to fail, as there is no new feature in good FP without new design, and... and teams often struggles to juggle the three needs to progress design, features and tasks within a same agile effort.

TDD helps

Test driven development (TDD) is one way to escape the limits of a one phase agile process. The idea is the following: 

1. Given some design features
2. Develop tests for these features, AND extend the design to cover the features.
3. Then work on non-design-tasks, that implement the tasks, and pass the tests.

However...

Yet there is still a challenge: design changes most often depend on strong opinions. FP depends on design changes. FP depends on strong opinions. Getting teams to gel around strong opinions is often not trivial. And that why I wrote the statement shared above.

All original content copyright James Litsios, 2024.