Overcoming Brittleness in Pareto-Optimal Learning-Augmented Algorithms

Summary

In the burgeoning field of learning-augmented online algorithms, the ideal case is to design an algorithm that achieves a competitive ratio (CR) as a function of prediction errors, without knowing the prediction error in advance. To tackle this problem, many existing works focus on two extreme metrics: consistency (i.e., the CR when the prediction error is zero) and robustness (i.e., the worst CR over all prediction errors). These works aim to derive algorithms that achieve the Pareto-optimal trade-off between consistency and robustness. This paper introduces and studies the concept of brittleness in Pareto-optimal learning-augmented algorithms, highlighting that the CR of Pareto-optimal algorithms may sharply degrade to the robustness guarantee with only a small prediction error. Specifically, the work formally defines and demonstrates the brittleness of the max-rate prediction in the one-way trading problem. To overcome this brittleness, it extends the consistency to a concept of consistency by profile F, which specifies the target CR over different prediction errors. A profile-based algorithm is proposed to check the feasibility of the consistency by profile F and find the online algorithm if feasible algorithms exist to attain it. Finally, the paper presents an adaptive algorithm that can improve the performance of Pareto-optimal algorithms for particular instances. However, this algorithm also suffers from brittleness.

Strengths

- This paper formalizes the brittleness issues in Pareto-optimal algorithms for the one-way trading problem. This helps to better understand the limitations of existing works in learning-augmented algorithms, which is particularly important for practical applications where perfect predictions are nearly impossible. - The concept of consistency by profile is a natural yet useful extension of the classic consistency. The proposed algorithm, which can quickly identify feasibility and find a feasible online strategy, is an interesting extension of classic threshold algorithms.

Weaknesses

- The paper focuses solely on the brittleness issues for the one-way trading problem. It is unclear whether many other problems exhibit similar brittleness and whether profile-based algorithms can also be designed to address such brittleness. The paper mentions the contract scheduling problem in the appendix, which deserves more formal treatment. In contrast, the paper uses an entire section (Section 5) to propose an adaptive Pareto-optimal algorithm (which is still brittle) for one-way trading, which deviates from the central topic of overcoming brittleness in learning-augmented algorithms. - The concept of consistency by profile is relatively easier to define for single-value predictions, raising the question of whether this concept would generalize when considering multiple predictions. Specifying the profile may also be challenging for users. - The algorithmic techniques used to design profile-based algorithms seem similar to existing approaches, which involve designing thresholds to maintain the target consistency profile while being prepared for the worst-case scenario if exchange rates drop to 1. The complexity arises from maintaining consistency over the user-specified profile instead of a single perfect prediction point.

Questions

- Can you explain how Section 5 (adaptive brittle algorithm for one-way trading) is connected to the theme of the paper (overcoming brittleness in learning-augmented algorithms)? How significant are the results for contract scheduling in the appendix, and can these results be used to validate the generalizability of the proposed concepts and approaches? - For a given user profile, there may exist multiple algorithms that can ensure such a profile; however, these algorithms may exhibit different instance-dependent performances. Can you make some formal statements on how to select the profile-based algorithm in practice? It seems the algorithm making $\omega_{l+1} = 1$ can be a good candidate algorithm. - The definition of brittleness (in Definition 3.1) is specific to the maximum rate prediction of one-way trading. Can this be a more general definition for the brittleness of learning-augmented algorithms? - in line 216 page 5, should the integral in $\tilde{s}_i = \int_1^{w_i}\Phi(u)du$ start from $0$?

Rating

6

Confidence

4

Soundness

3

Presentation

3

Contribution

3

Limitations

Yes

Summary

The authors consider learning-augmented algorithms for the one-way trading problem. In this problem, we are given a budget of 1 and a sequence of exchanges rates between 1 and M that are revealed in an online manner. Whenever an exchange rate is revealed, we have to decide whether to exchange a fraction of our remaining budget at this rate or not. The goal is to maximize the overall profit. In the learning-augmented setting, we are additionally given a prediction on the maximum exchange rate, which is equivalent to predicting the optimum, as an optimal solution will exchange the complete budget at the maximum rate. Usually, learning-augmented algorithms are analyzed using consistency, the competitive ratio for a perfectly accurate prediction, and robustness, the worst-case competitive ratio for any input. For the one-way trading problem, a pareto optimal algorithm w.r.t. consistency and robustness is already known from previous works. The main point of the paper is to address two weaknesses of such pareto-optimal algorithms and analyses via consistency and robustness in general. The first weakness is called *brittleness* and describes prediction models were even a wrong prediction that is arbitrarily close to the correct value leads to any pareto-optimal algorithm having a competitive ratio matching the robustness. This means that only completely perfect predictions allow for an improved performance. The authors prove that maximum rate predictions for the one-way trading problem are brittle and address the brittleness by proposing an analysis via *profiles*, a generalization of consistency and robustness. A $profile$ partitions the range of possible maximum exchange rates into intervals and, for each interval, defines a target competitive ratio. The interval that contains the predicted maximum exchange rate has the best target competitive ratio and the ratio degrades for intervals that are farther away. An algorithm respects the profile if it always achieves the target competitive ratio of the interval that contains the actual maximum exchange rate. As a main contribution, the paper gives a constructive algorithm that decides whether there exists an algorithm that respects a given profile. The second weakness of pareto-optimal algorithms is that such algorithms are often tailored to worst-case instances w.r.t. the consistency and robustness tradeoff. To address this, the authors give a pareto-optimal algorithm that dominates all other pareto-optimal algorithms in the sense that it does not perform worse than any other pareto-optimal algorithm on instances where the actual maximum exchange rate is larger than the predicted optimal exchange rate.

Strengths

* The paper identifies and addresses two reasonable and realistic potential drawbacks of pareto-optimal learning-augmented algorithms. Since the framework of learning-augmented algorithms is ultimately a tool to analyze algorithms beyond the worst-case, the proposed generalizations that extend the ability to do just that are certainly of interest to the community. * The authors give a proof of concept that profile-based algorithms are indeed possible by introducing such algorithms for the one-way trading problem (and contract scheduling in the appendix). It is a nice idea to consider the offline problem of deciding whether a given profile is feasible or not. I am not completely convinced that this idea also works for problems of a different flavor (see weaknesses), but at the very least it inspires future work to settle this question, which already is an important contribution. * The paper is well-written, the problem statement is properly motivated, and the results are presented clearly and concisely.

Weaknesses

* The definition of profiles seems to be tailored to predictions models where only a single value is predicted. For more complex predictions, one could define profiles w.r.t. the prediction error. However, specifying a profile upfront would not always be possible as the range for the predictions error often depends on the unknown online input. Even for the maximum exchange rate predictions, the specification of a profile requires knowledge of the maximum exchange rate M. * As one of the main technical contributions, the authors give an algorithm that decides whether a given profile is feasible. This algorithm heavily relies on the simple characterization of worst-case instances as given in Remark 2.1. For many other online problems, no such simple characterization of worst-case instances are known. This could limit the impact of the proposed analysis framework via profiles as deciding whether a profile is feasible is likely more difficult (or even not possible) for problems without such a simple worst-case characterization. * I am not an expert regarding the literature on the one-way exchange problem. However, the paper does not seem to introduce too many new algorithmic ideas. Instead, the algorithmic contributions of the paper seem like a very natural extension of known ideas. In particular, Algorithm 1 seems to be the canonical way of extending threshold-based algorithms to profiles. * Regarding the experiments, it would be interesting to also see results for different profiles. The used profile is quite similar to the consistency and robustness cases. While this makes the comparison to the pareto-optimal algorithm more fair, it would be interesting to see results for profiles that emulate a smooth error dependency. * Minor comment: Line 86: I do not think that not having access to the prediction ahead of time is a novelty. In the context of online algorithms there are several examples where the predictions also are revealed over time. One example would be reference [9].

Questions

-

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

In my opinion, all limitations have been properly addressed.

Summary

The paper considers the learning-augmented one-way trading problem. In the problem, we are given a starting budget equal to 1 and a sequence of exchange rates $p_1,...,p_n \in [1,M]$ arriving online. When each $p_i$ arrives, we need to. decide the amount to be exchanged to the secondary currency. Our goal is to maximize the total profit under the given budget. In the learning-augmented setting, the algorithm can access an imperfect prediction $\hat{p}$ of the largest rate. The authors first show that Pareto-optimality is very fragile for comparing online trading algorithms and then motivates a new metric called performance profile. This new metric incorporates the structural information of instances, rather than a simple comparison based on consistency-robustness values. They further develop an online algorithm that can satisfy the given performance profile (if it is feasible). The authors also discuss another generalization of Pareto-optimality and provide empirical evaluations of the proposed algorithms in the end.

Strengths

- The brittleness issue considered in the paper is well-motivated. Actually, this is a highly significant concern in the field of learning-augmented algorithms. Previous efforts mainly focused on achieving smoothness in ratios by defining new error metrics, while this paper takes a novel approach by introducing the concept of profiles to tackle this issue. - Both theoretical analysis and experimental evaluation are provided in the paper.

Weaknesses

- The main weakness of this work is that the concept of performance profile may be hard to extend to other online problems, which makes this work less interesting. It would be better if the authors could demonstrate the applicability of this technique to a wider range of problems. - In Line 131, I didn't see why $w_{A,i}=\sum_{j=1}^{i-1}w_{A,j}$. Is this a typo?

Questions

See the weakness above.

Rating

6

Confidence

4

Soundness

3

Presentation

2

Contribution

3

Limitations

I didn't see any potential negative societal impact.

Summary

In the context of learning augmented algorithms, two widely used metrics are robustness (i.e: the performance when the prediction is adversarially chosen) and consistency (i.e: the performance when the prediction is perfect). This work analyzes the interplay between these two metrics in the one way trading problem with imperfect predictions. In particular, it starts by showing that Pareto-optimal algorithms, that is, algorithms that for a given robustness upper bound r achieve the smallest possible consistency c(r), are dramatically sensitive to small prediction errors. Specifically, the authors show that any prediction error guarantees the existence of an input sequence for which the performance ratio reaches its robustness, a behavior denominated ‘britleness’. This implies that the competitive ratio of any Pareto-Optimal algorithm is either its consistency (if the prediction is indeed perfect) or its worse possible performance, suggesting that Pareto-Optimality might not be a sensible algorithm design criteria. In light of this result, this work puts forward the notion of profiles, which maps rate prediction intervals to desired competitive ratios, allowing for the performance of algorithms to degrade smoothly with respect to prediction errors. *V*iewing this, the a*u*thors present an a*l*gorithm that establishes the *f*easibility of a profile in an offline fashion and yields an online procedure that, in the feasible case, satisfies the enforced performance constraints. Lastly, the authors propose an adaptive algorithm that is not designed to handle worst-case predictions but leverages the deviations from the predicted exchange rate to navigate the robustness and consistency tradeoff.

Strengths

- The paper is clearly written. - The theoretical results (brittleness of Pareto-Optimality, feasibility determination of profile-following and correctness of online algorithm for profile following) are relevant and sound. - The analysis technique used to determine the feasibility and solve the profile-based one-way trading problem seems novel. In particular, the constraints associated to a profile can be written as a set of linear differential differential equation whose solution yields a profile-following exchange strategy. - The authors provide intuition regarding the PROFILE algorithm, in particular, on the behavior of the threshold function $\phi$ with respect to transitions in the desired performance ratio.

Weaknesses

- The experimental setting might be limited. Specifically, the profile used for evaluation if fairly simple, with only three intervals, two of which map to the worse possible ratio. - Is there any practical use of determining feasibility if it's in an offline fashion ?

Questions

- Is there a way to characterize the likelihood of the sequences that severely degrade the performance of Pareto-Optimal algorithms ? If those sequences are very unlikely then Pareto-Optimality might not be that fragile a criteria. - Why is the function \phi increasing ? That is, why do larger utilizations necessarily map to larger reservation rates ? (Is it only under the assumption that rates are increasing and then drop to 1?) Minor comment: Fix figure ratios.

Rating

8

Confidence

3

Soundness

4

Presentation

3

Contribution

3

Limitations

- N/A

Response to rebuttal

Thank you for your rebuttal on my review and the other reviews. My opinion on the paper and my score remain unchanged.

Response to rebuttal

Thank you for your reply. I still find that Section 5 is disconnected from the core theme related to overcoming the brittleness of learning-augmented algorithms, unless this section can formally address the brittleness of the adaptive algorithm, as the authors believe it can. After further consideration of the paper's potential impact, I believe it makes a valuable contribution by bringing the issue of brittleness in Pareto-optimal algorithms to the attention of the field. Therefore, I have increased my score from 5 to 6.