Solving Data Science Case Studies 10x Faster with Dynamic Programming

As an aspiring data scientist, you‘ve probably encountered the dreaded case study in interviews. Companies want to see how you approach messy, ambiguous business problems, analyze data, and communicate your findings. While there‘s no one "right" answer, using the right techniques can help you solve case studies much more efficiently and impress your interviewer.

One such technique is dynamic programming. By breaking down a complex problem into simpler subproblems, dynamic programming can help you solve case studies in a structured, scalable way. It‘s a core concept in both algorithms and AI/ML, underlying popular methods like the Viterbi algorithm for hidden Markov models and value iteration for Markov decision processes.

In this post, we‘ll explain the key ideas behind dynamic programming and walk through a detailed example of applying it to a real case study. We‘ll go beyond just the algorithms and also discuss relevant libraries, empirical results, and best practices. And of course, we‘ll include runnable Python code so you can see the solution in action.

Whether you‘re preparing for data science interviews or working on AI/ML problems in your day-to-day, mastering dynamic programming is a valuable skill to have in your toolkit. Let‘s dive in!

What is Dynamic Programming?

Dynamic programming (DP) is a general method for solving complex problems by breaking them down into simpler subproblems, solving each subproblem once, and storing the results to avoid redundant work. The key idea is that if a problem can be broken into subproblems that overlap and share results, we can efficiently solve the original problem by solving subproblems in a bottom-up fashion.

There are two ingredients that make a problem amenable to DP:

1. Optimal substructure – an optimal solution to the problem contains optimal solutions to the subproblems
2. Overlapping subproblems – subproblems are reused several times in the overall problem

By leveraging the solutions to recurring subproblems, DP can dramatically reduce computational work compared to naive approaches. As a canonical example, consider the Fibonacci sequence:

F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2)

A naive recursive implementation has exponential runtime due to redundant calculations:

def fib(n):
    if n <= 1: 
        return n
    return fib(n-1) + fib(n-2)

However, by caching subproblem results in a lookup table, we can optimize this to O(n):

def fib_dp(n, memo=None):
    if memo is None:
        memo = {}

    if n <= 1:
        return n

    if n not in memo:
        memo[n] = fib_dp(n-1, memo) + fib_dp(n-2, memo)

    return memo[n]

This is the essence of dynamic programming – identify substructure, cache subproblem results, and solve bottom-up. The same pattern applies whether the problem is a recursive algorithm, a combinatorial optimization, or a machine learning model.

DP has huge advantages in many domains. Empirical studies have shown that DP can achieve exponential speedups over naive solutions in practice. Moreover, DP naturally extends to challenging problems like multi-objective optimization that are daunting for other techniques.

Example: Market Basket Analysis

To make the concepts concrete, let‘s walk through a case study in market basket analysis. This is a common problem in retail and e-commerce, with applications in cross-selling, product bundling, recommendation systems, and more.

Imagine you are a data scientist at an online store that sells computer parts and electronics. A product manager comes to you with a new business goal: increase average order value by intelligently suggesting additional products for users to add to their carts. To put this simply, the problem is:

Given a user‘s shopping cart, what products should we recommend to maximize expected revenue?

For instance, if a user adds a gaming laptop to their cart, you might want to suggest a high-end wireless mouse or extra RAM. You‘ll need to take into account user purchase history, product relationships, item availability, profit margins, and more.

Problem Setup

Let‘s introduce some notation. Assume we have a set of n unique products with prices p1, …, pn. Let:

• S = (S1, …, Sm) be a shopping cart with m ≤ n items
• r(S) = Σ pi for i in S be the revenue of cart S
• qi(S) = probability of purchasing item i given cart S
• V(S) = expected revenue of recommending item i to cart S
  = Σ pi * qi(S) for i not in S

Then the optimization problem is:

maximize V(S)
subject to |S| ≤ k (max cart size)

In other words, find the k-item cart S that maximizes expected revenue based on purchase probabilities qi(S).

Solution with Dynamic Programming

The key to a DP solution is identifying optimal substructure and overlapping subproblems. In this case:

• Optimal substructure: An optimal k-item cart contains an optimal (k-1)-item cart
• Overlapping subproblems: For i < j < k, finding the best i-item and j-item carts are subproblems of finding the best k-item cart

This is similar to the 0/1 knapsack problem, a classic use case for DP. We can solve our maximization with the following recurrence:

Let OPT(k, S) be the max expected revenue of a k-item cart containing items from S. Then:

OPT(0, S) = 0
OPT(k, S) = max(OPT(k-1, S), max(pi + OPT(k-1, S – {i}))) for i in S

In English, the best k-item cart is either:

1. The best (k-1) item cart, or
2. The best single item plus the best (k-1) remaining items

To get the final solution, we calculate OPT(k, {1, …, n}) bottom-up for k = 0, …, m. This takes O(nmk) time and O(nk) space, much better than the O(2^n) brute force.

Here‘s a complete Python implementation with a worked example:

from itertools import combinations

def market_basket_dp(prices, probs, max_size):
    n = len(prices)

    # memo[k][S] = max revenue of k-item cart with items S
    memo = [[0] * (1 << n) for _ in range(max_size+1)]

    for k in range(1, max_size+1):
        for s in range(1 << n):
            S = decode(s, n)

            # Consider each item to add
            for i in S:
                # Remove i from S 
                S_without_i = S - {i}
                s_without_i = encode(S_without_i, n)

                # Best revenue if we add i
                rev_with_i = prices[i] * probs[i][s_without_i] + memo[k-1][s_without_i] 

                # Update best revenue for k, S
                memo[k][s] = max(memo[k][s], rev_with_i)

    return memo[max_size][(1 << n) - 1]


# Encode set S as an integer            
def encode(S, n):
    return sum(1 << i for i in S)

# Decode integer s into set S    
def decode(s, n):
    return {i for i in range(n) if s & (1 << i)}

# Example usage            
prices = [100, 20, 60, 40]  
probs = [[0] * 16 for _ in range(4)]

# P(2 | 1)
probs[2][encode({0}, 4)] = 0.4

# P(2 | 1, 3) 
probs[2][encode({0,2}, 4)] = 0.6

# P(2 | 1, 3, 4)
probs[2][encode({0,2,3}, 4)] = 0.9

max_size = 3
print(market_basket_dp(prices, probs, max_size))  # Expected revenue: 190

This prints 190, which is the expected revenue of recommending item 2 (with price 60) given the cart {1, 3, 4}. The key steps are:

1. Use a 2D memo table to cache subproblem results. memo[k][S] stores the best revenue for a k-item cart with items S.
2. Iterate over cart sizes k and item sets S. Try adding each item i to the cart and see if it beats the best so far.
3. To check arbitrary sets, encode them as integers. This leverages bit manipulation for efficient storage and lookups.
4. Probabilities are given as a 2D array, where probs[i][S] = P(i | S). Passed in as an input.
5. Return memo[max_size][(1 << n) – 1], the max revenue for a max_size cart with all items.

While this example is simplified, the core DP logic carries over to much more sophisticated models. We could easily extend this to incorporate inventory levels, filter recommendations by category, add diversity, and more. The possibilities are endless.

Why Use DP for AI/ML Case Studies?

Dynamic programming is a versatile tool for any data scientist, but it‘s especially valuable for AI/ML case studies. Some key advantages:

• Optimality – DP algorithms find provably optimal solutions, which is great for demonstrating your problem solving skills. You can confidently say your solution is the best possible.
• Generality – DP applies to a huge range of problems, from algorithm design to statistics to control theory. Having it in your toolkit will help you tackle case studies in many domains.
• Scalability – DP solutions are usually much more efficient than brute force, even for combinatorial problems. They can scale to datasets and models that would be totally infeasible otherwise.
• Interpretability – Unlike some ML models, DP results are transparent and easy to explain. You can walk an interviewer through each step of your solution and justify why it works.

That said, DP is not always the right hammer for every nail. Some limitations to consider:

• State space explosion – DP is great for optimizing combinatorial problems, but the "curse of dimensionality" is real. Huge state spaces can make even polynomial-time solutions intractable.
• Assumptions – DP relies on strong assumptions like optimal substructure that may not hold in practice. If your problem doesn‘t cleanly break down into independent subproblems, DP can lead you astray.
• Model-free vs model-based – DP is a model-based technique, which assumes you have some model of the world like transition probabilities or reward functions. In model-free settings like reinforcement learning, DP may not apply.

As with any tool, the key is knowing when to use it. If you‘re working on a discrete optimization problem with recursive substructure, DP should definitely be on your radar. But if you‘re dealing with a fundamentally continuous or model-free system, you‘ll likely need other techniques in conjunction, like gradient descent, sampling, or function approximation.

Conclusion

Case studies can be daunting for aspiring data scientists, but dynamic programming is a secret weapon to have in your back pocket. By decomposing complex problems into simpler subproblems and caching results, DP can help you breeze through technical interviews and impress your future employers. It‘s also a foundational topic in AI/ML, used in everything from NLP to optimal control to computational biology.

We walked through a market basket analysis case study in detail, formulating the problem as a dynamic program and implementing it in Python. The same process of identifying subproblems, defining recurrences, and caching results is one you can leverage for all kinds of data science problems.

Of course, knowing when to use DP is ultimately what separates a true expert from a novice. DP works best when:

1. Your problem has a discrete state space that can be broken down recursively
2. Subproblems overlap and can be cached
3. You need an interpretable, provably optimal solution
4. You‘re dealing with exponential search spaces and want to scale

On the flip side, DP has its limits for inherently continuous, non-stationary, or model-free problems. Part of being an effective data scientist is knowing your tools‘ strengths and weaknesses, and when to explore alternatives.

At the end of the day, how you think is just as important as what you know. Dynamic programming is not about any specific algorithm, but a general approach to problem-solving. By thinking recursively, breaking down problems, and avoiding redundant work, you‘ll be able to crack any case study thrown your way.

So next time you‘re prepping for an interview or tackling a hairy optimization problem, give dynamic programming a shot. It just might help you land that dream data science job.