Determine the probability function for selecting experts in a Follow-the-perturbed-leader prediction algorithm | Step-by-Step Solution

Hello! Let's dive into this fascinating problem about the Follow-the-Perturbed-Leader (FPL) algorithm. This is a beautiful example of how randomness can actually help us make better decisions in machine learning!

1. What We're Solving:

We need to understand how the FPL algorithm determines the probability of selecting each expert when making predictions, specifically when we add exponentially distributed random noise to perturb our decision-making process.

2. The Approach:

The FPL algorithm is clever because it doesn't just pick the expert with the lowest cumulative loss (which could be too rigid). Instead, it adds random "perturbations" to each expert's loss and then picks the one with the lowest perturbed loss. This randomness helps us:

• Avoid getting stuck with one expert too early
• Maintain some exploration while still favoring better-performing experts
• Achieve better theoretical guarantees for regret bounds

3. Step-by-Step Solution:

Step 1: Understanding the Setup

• We have K experts, each with cumulative losses L₁, L₂, ..., Lₖ
• We add independent exponentially distributed random variables Z₁, Z₂, ..., Zₖ to each loss
• Each Zᵢ has rate parameter η (so density function f(z) = ηe^(-ηz) for z ≥ 0)

Step 2: The Selection Rule The FPL algorithm selects expert i if: Lᵢ + Zᵢ ≤ Lⱼ + Zⱼ for all j ≠ i

Step 3: Finding the Probability To find P(expert i is selected), we need: P(Lᵢ + Zᵢ ≤ Lⱼ + Zⱼ for all j ≠ i)

This is equivalent to: P(Zⱼ - Zᵢ ≥ Lᵢ - Lⱼ for all j ≠ i)

Step 4: Using Properties of Exponential Distributions The probability that Zᵢ is the minimum among {Z₁ + L₁, Z₂ + L₂, ..., Zₖ + Lₖ} is:

P(select expert i) = e^(-ηLᵢ) / Σⱼ₌₁ᵏ e^(-ηLⱼ)

Step 5: Understanding Why This Works This formula emerges from the memoryless property of exponential distributions and some elegant probability theory involving order statistics.

4. The Answer:

The probability function for selecting expert i in the FPL algorithm is:

P(i) = e^(-ηLᵢ) / Σⱼ₌₁ᵏ e^(-ηLⱼ)

Where:

• Lᵢ is the cumulative loss of expert i
• η is the rate parameter of the exponential distribution
• The denominator normalizes to ensure probabilities sum to 1

Notice how this is essentially a softmax function applied to the negative losses! Higher losses get lower probabilities, but the randomness ensures every expert has some chance of being selected.

5. Memory Tip:

Think of it as "exponentially favoring better experts" - the probability literally uses the exponential function with negative losses. The parameter η controls how "sharp" our preferences are: larger η means we're more deterministic (strongly favor the best expert), while smaller η means we're more exploratory (more uniform selection). It's like adjusting the "temperature" in a softmax!

The beauty of FPL is that it bridges the gap between pure exploration (random selection) and pure exploitation (always picking the current best) in an mathematically principled way. Pretty cool, right?