Doubling Trick - A Clever Strategy to Handle Unknown Horizons

Online learning algorithms often rely on carefully chosen parameters, such as the learning rate (\(\eta\)), to achieve good performance. However, many theoretical analyses assume that the total time horizon (\(T\)) is known in advance. In practical scenarios, this is rarely the case.

The Doubling Trick is a clever method to overcome this limitation. By dividing time into intervals of exponentially increasing lengths and resetting parameters at the start of each interval, this method ensures that the algorithm performs almost as well as if \(T\) were known from the beginning.

Why Do We Need the Doubling Trick?

Imagine you’re tasked with predicting the weather for the next \(T\) days using an algorithm. If you knew \(T\), you could optimize your algorithm’s parameters to minimize prediction errors. But what if you don’t know how long you’ll be predicting for? Setting parameters without knowing \(T\) can lead to suboptimal performance.

The Doubling Trick resolves this problem by resetting the algorithm at specific intervals. These intervals grow exponentially, ensuring that the algorithm adapts dynamically without sacrificing much performance.

The Idea Behind the Doubling Trick

The Doubling Trick divides time into periods of exponentially increasing lengths:

\[I_k = [2^k, 2^{k+1} - 1],\]

where \(k = 0, 1, 2, \dots\) represents the interval index and \(T \geq 2^n-1\) [Why?]. Each interval is twice as long as the previous one. For example:

\(I_0 = [1, 1]\) (length = 1),
\(I_1 = [2, 3]\) (length = 2),
\(I_2 = [4, 7]\) (length = 4),
\(I_3 = [8, 15]\) (length = 8),
and so on.

Steps in the Doubling Trick

Divide Time into Intervals: Time is divided into intervals of lengths \(1, 2, 4, 8, \dots\). These intervals grow exponentially, covering the entire time horizon \(T\).
Choose Parameters for Each Interval: At the start of each interval, parameters (e.g., the learning rate \(\eta_k\)) are chosen based on the length of the interval. For instance, the learning rate could be set as:
\[\eta_k = \sqrt{\frac{8 \log N}{2^k}},\]
where \(N\) is the number of options (or experts) the algorithm is learning from.
Reset the Algorithm: The algorithm is reset at the beginning of each interval, treating it as a fresh start.
Run the Algorithm Independently in Each Interval: The algorithm operates independently in each interval, accumulating loss or regret for that specific period.
Sum Up Regret Across Intervals: After \(T\) rounds, the total regret is the sum of regrets from all intervals.

Regret Bound with the Doubling Trick

Theorem

Assume the same conditions as those in the Exponential Weighted Average Algorithm. For any total time horizon \(T\), the regret achieved by the Doubling Trick satisfies:

\[\text{Regret}(T) \leq \frac{\sqrt{2}}{\sqrt{2}-1} \cdot \sqrt{\frac{T}{2} \log N} + \sqrt{\log N / 2}\]

This result demonstrates that the Doubling Trick achieves regret bounds that are only slightly worse (by a constant factor) than if the total time horizon \(T\) were known in advance.

Proof Sketch

Setup

Time Intervals:
Divide the time horizon \(T\) into intervals \(I_k = [2^k, 2^{k+1} - 1]\), where \(k \in \{0, 1, \dots, n\}\) and \(n = \lfloor \log_2(T + 1) \rfloor\).
Each interval \(I_k\) has length \(2^k\), and the total time horizon \(T\) satisfies \(T \geq 2^n - 1\).
Learning Rate:
Within each interval \(I_k\), the learning rate is chosen as:
\[\eta_k = \sqrt{\frac{8 \log N}{2^k}}\]

For any interval \(I_k\), the regret is bounded using:

\[L_{I_k} - \min_{i \in [N]} L_{I_k, i} \leq \sqrt{\frac{2^k}{2} \log N}\]

Here:

\(L_{I_k}\) is the total loss incurred by the algorithm in interval \(I_k\).
\(L_{I_k, i}\) is the total loss incurred by expert \(i\) in interval \(I_k\).

The total loss after \(T\) rounds is the sum of losses over all intervals:

\[L_T = \sum_{k=0}^n L_{I_k}\]

The total regret can then be expressed as:

\[R_T = \sum_{k=0}^n \left( L_{I_k} - \min_{i \in [N]} L_{I_k, i} \right)\]

Substituting the bound for \(L_{I_k} - \min_{i \in [N]} L_{I_k, i}\):

\[R_T \leq \sum_{k=0}^n \sqrt{\frac{2^k}{2} \log N}\]

Simplifying the Geometric Sum

The term \(\sum_{k=0}^n \sqrt{\frac{2^k}{2} \log N}\) can be simplified as:

\[\sum_{k=0}^n \sqrt{\frac{2^k}{2} \log N} = \sqrt{\frac{\log N}{2}} \cdot \sum_{k=0}^n \sqrt{2^k}\]

The sum \(\sum_{k=0}^n \sqrt{2^k}\) forms a geometric series:

\[\sum_{k=0}^n 2^{k/2} = \frac{2^{(n+1)/2} - 1}{\sqrt{2} - 1} \tag{How?}\] \[\leq \frac{\sqrt{2} \sqrt{T + 1} - 1}{\sqrt{2} - 1}\] \[\leq \frac{\sqrt{2} (\sqrt{T} + 1) - 1}{\sqrt{2} - 1}\] \[= \frac{\sqrt{2} \sqrt{T}}{\sqrt{2} - 1} + 1\]

Plugging this result back into the regret expression, we get:

\[\text{Regret}(T) \leq \frac{\sqrt{2}}{\sqrt{2}-1} \cdot \sqrt{\frac{T}{2} \log N} + \sqrt{\log N / 2}\]

This establishes the regret bound for the EWA algorithm with the doubling trick.

There is one question to address before we can confidently say we fully understand this proof, specifically the equation tagged with ‘How?’. However, the answer is actually quite simple.

The sum you are dealing with is of the form:

\[S_n = \sum_{k=0}^n 2^{k/2}\]

To solve this, we notice that the series is geometric in nature, where each term is of the form \(2^{k/2}\). This suggests a geometric series with the first term \(a = 2^0 = 1\) and the common ratio \(r = 2^{1/2} = \sqrt{2}\).

A standard formula for the sum of the first \(n\) terms of a geometric series is:

\[S_n = \frac{a(r^{n+1} - 1)}{r - 1}\]

Substituting \(a = 1\) and \(r = \sqrt{2}\), we get:

\[S_n = \frac{(\sqrt{2})^{n+1} - 1}{\sqrt{2} - 1}\]

Now, simplify \((\sqrt{2})^{n+1}\) to \(2^{(n+1)/2}\):

\[S_n = \frac{2^{(n+1)/2} - 1}{\sqrt{2} - 1}\]

Thus, we arrive at the desired result:

\[\sum_{k=0}^n 2^{k/2} = \frac{2^{(n+1)/2} - 1}{\sqrt{2} - 1}\]

Now we’re all set. Yes. Next, we’ll tackle a few more questions and wrap up this blog.

Note: The \(O(\sqrt{T})\) dependency on \(T\) presented in this bound cannot be improved for general loss functions. This is because the case we discussed here is the Exponentiated Weighted Average (EWA), where we assumed a convex loss with respect to its first parameter. The Doubling Trick is simply an additional technique applied on top of this, and that’s all.

Why Does the Doubling Trick Work?

Exponential Growth Covers Any Horizon: By doubling the length of intervals, the Doubling Trick ensures that any unknown time horizon \(T\) will fall within the union of the intervals.
Dynamic Parameter Adjustment: Since parameters are reset at the start of each interval, the algorithm adapts to the growing time horizon. This avoids overcommitting to a parameter setting based on an incorrect estimate of \(T\).
Minimal Regret Accumulation: The regret within each interval is bounded, and the total regret is the sum of these bounds. Using properties of geometric sums, the total regret remains manageable.

Advantages of the Doubling Trick

No Prior Knowledge of \(T\): The Doubling Trick eliminates the need to know \(T\) in advance, making it practical for real-world applications.
Near-Optimal Performance: The regret bound is close to the best possible bound achievable with full knowledge of \(T\).
Simplicity: The method is straightforward to implement and applies to a wide range of online learning algorithms.

Applications of the Doubling Trick

The Doubling Trick, while commonly used in regret minimization, also finds applications in scenarios where the time horizon is uncertain or unknown. Here are a few key areas where this trick proves useful;

[I have this question but haven’t come up with a convincing answer or a way to phrase it clearly yet. I’ll update once I find it.]

Conclusion

The Doubling Trick is a powerful and versatile technique that addresses the challenge of unknown time horizons in online learning. By dividing time into exponentially growing intervals and resetting parameters dynamically, it achieves near-optimal regret bounds with minimal computational overhead. This method underscores the elegance of combining mathematical rigor with practical adaptability.

Next, we’ll define a general setting for online learning and briefly touch on the different setups within it. Stay tuned for more!