Risk-averse learning with delayed feedback

arXiv:2409.16866 · cs.LG, math.OC · Submitted 2024-09-25 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "Risk-averse learning with delayed feedback".

Jane: The gist In this paper, they investigate risk-averse learning using Conditional Value at Risk (CVaR) as a risk measure while incorporating feedback with random but bounded delays.

Tom: First, who's behind it and why it matters.

Title and authors: Tom: Alright, so let's look at who wrote this. We have Siyi Wang, Zifan Wang, Sandra Hirche, and Karl H. Johansson as the authors of this paper on "Risk-averse learning with delayed feedback".

Jane: It’s interesting to see that it brings together folks from different fields—the IEEE fellows—to tackle a very practical problem in online learning where things are messy because of those time delays.

Lu: The focus here is really on how to design two specific risk-averse learning algorithms using zeroth-order optimization to handle this delayed feedback structure.

Meng: Zeroth-order optimization tells us that we can estimate the gradient of a complex function without needing its exact derivative, which is helpful when we don't have perfect access to the cost function itself.

Tom: So they build two versions: one-point risk-averse learning and a two-point risk-averse learning approach, which is what we need to see the comparison between.

Jane: That comparison is key because it shows how much better you can get at managing that uncertainty when dealing with those random delays.

Lalam: The title itself points to the core issue: making learning choices when the information arrives late and randomly.

The paper's summary: Tom: Okay, so what’s actually in this paper? It summarizes how they use CVaR as the risk measure and then develop these two algorithms based on zeroth-order optimization to estimate that gradient.

Jane: They specifically address the delay model where feedback arrives at a certain time slot, and they sort those losses by their generating time to build what they call a virtual timeline for analysis.

Lu: The authors quantify the error introduced by this delayed feedback by computing what they call cumulative delay, which is just the total amount of time that has passed without receiving the necessary feedback.

Meng: They are trying to get a firm estimate of how much regret we're going to incur over time, not just looking at a snapshot in time.

Tom: The main result they show is that both algorithms achieve sublinear regrets, which means as time T goes to infinity, the dynamic regret divided by T goes to zero.

Jane: That’s the fundamental goal: ensuring that even with these delays and risks, the algorithm doesn't just keep making terrible mistakes forever.

Lalam: They provide specific regret bounds for both versions based on terms like cumulative delay D and total samplings, which gives us a concrete way to measure performance.

The paper's improvements: Tom: Now, the paper suggests two main improvements over the existing one-point approach: they introduce a two-point risk-averse learning algorithm.

Jane: This second version uses two perturbed actions—one slightly larger and one slightly smaller—to query the cost function under both to estimate that CVaR gradient.

Lu: That difference between those two queries is what they use to estimate the CVaR gradient, which they call gbar nt, and it helps them get a tighter bound on regret.

Meng: The paper states that this two-point algorithm outperforms the one-point algorithm by achieving a smaller regret bound in terms of the dynamic regret DR2(T).

Tom: So, when you look at their theoretical results, for the one-point version, they get a bound that depends on T to the power of one minus a over four plus some terms involving D and p bar d p PT <ref:2409.16866#pg1>.

Jane: But for the two-point algorithm, they manage to achieve DR2(T) which is bounded by T to the power of one minus a over four plus something involving square roots and D <ref:2409.16866#pg1>.

Lalam: The key improvement in their analysis is showing that this two-point version provides better performance under delay conditions compared to the simpler one-point version.

Conclusion: Tom: So, wrapping up this "Risk-averse learning with delayed feedback" paper, both algorithms successfully achieve sublinear regrets, meaning they don't grow linearly with time.

Jane: And crucially, the two-point risk-averse learning algorithm is superior because it achieves a smaller regret bound than the one-point version under these delayed feedback conditions.

Lu: It also suggests that this second algorithm produces more consistent and stable outcomes in practice when things are delayed, even though the convergence speed might be different.

Meng: From a production standpoint, that stability is important; you want predictable performance even if your data stream is occasionally noisy with delays.

Lalam: So, to summarize, they used zeroth-order optimization to tackle CVaR gradients under delayed feedback and found the two-point method gives better regret bounds.

Tom: That’s it for this one. It shows that when you have delayed feedback, just having a basic risk-averse learner isn't enough; you need a method like Algorithm two for better results <ref:2409.16866#pg2>.

Jane: A very solid piece of work showing how to adapt existing optimization tools to handle the messy reality of real-world decision making with time lags.

Siyi Wang, Zifan Wang, Sandra Hirche, Karl H. Johansson

KTH Royal Institute of Technology · Technical University of Munich

cs.LG, math.OC

Submitted: 2024-09-25

Updated: 2026-10-19

License: http://creativecommons.org/publicdomain/zero/1.0/

The gist: The gist In this paper, they investigate risk-averse learning using Conditional Value at Risk (CVaR) as a risk measure while incorporating feedback with random but bounded delays.

Key concepts

Conditional Value at Risk (CVaR): CVaR is a risk measure used to assess the potential tail risk of learning outcomes. It focuses on quantifying the expected loss beyond a certain high threshold. This approach helps in making decisions that are robust against worst-case scenarios, which is crucial in online learning where adverse events can occur.
Zeroth-Order Optimization: Since exact gradients for CVaR are often unavailable, this method estimates them by querying the objective function at slightly perturbed actions. This involves using a smoothed approximation of the CVaR function to derive an estimate of its gradient. It allows the algorithm to learn effectively even when analytical derivatives are too complex.
Dynamic Regret: Instead of static regret, dynamic regret is used to measure performance in environments where feedback arrives with delays. This metric accounts for the time-dependent nature of learning and incorporates the cumulative delay experienced by the system. It provides a more accurate measure of how well the algorithm adapts over time despite delayed observations.

Terminology

Regret — 3× in this paper · explained in 4 episodes

Summary

The gist In this paper, they investigate risk-averse learning using Conditional Value at Risk (CVaR) as a risk measure while incorporating feedback with random but bounded delays.

Risk Measure and Optimization Approach

The work uses Conditional Value at Risk (CVaR) as the risk measure to mitigate potential adverse outcomes in online learning In this context, CVaR is widely adopted because it offers a coherent risk assessment by considering the tail end of the loss distribution Since exact CVaR gradients are generally unavailable, they construct an estimate using zeroth-order optimization approaches to query the objective function at perturbed actions Specifically, they use a smoothed approximation of the CVaR function and derive its gradient.

Handling Delayed Feedback

The study addresses real-world scenarios where observations may arrive with delays, which necessitates the use of dynamic regret as a more relevant metric than static regret The delay model assumes that the loss will be observed after a certain time slot, and they sort received losses according to their generating time to construct a virtual timeline. They quantify the error introduced by delayed feedback by computing the cumulative delay, which is defined as the aggregate of delays experienced by the feedback.

Algorithm Development and Performance Analysis

Two risk-averse learning algorithms are developed based on zeroth-order optimization:

One-point risk-averse learning relies on querying the stochastic cost for multiple times under a perturbed action to estimate the CVaR gradient The regret bound is analyzed in terms of the cumulative delay and the number of total samplings.
Two-point risk-averse learning employs two perturbed actions, querying costs under both to estimate the CVaR gradient, which they denote as gbar nt. They demonstrate that the two-point algorithm outperforms the one-point algorithm by achieving a smaller regret bound.

Theoretical Results and Regret Bounds

The theoretical analysis shows that both algorithms achieve sublinear regrets, meaning limT→∞ DR(T)/T = 0 The regret bound for the one-point algorithm is shown to be O˜ T(1− a/4 + (T + D)(3a/4 p¯dPT) + (T + D)(−3a/4 T a 4 D) with high probability The two-point algorithm achieves DR2(T) = O˜ T(1− a/4 + (T+D)(1/2 p¯dPT)) with high probability.

Numerical Experiments

Numerical experiments on a dynamic pricing problem were conducted to demonstrate the performance of the algorithms The results show that Algorithm 2 outperforms Algorithm 1 by achieving a lower regret and also yields results with lower variance compared to Algorithm 1. Furthermore, the two-point risk-averse learning algorithm has a higher tolerance for delay compared to the one-point algorithm.

Conclusion

The paper concludes that both algorithms achieve sublinear regrets, and the twopoint risk-averse learning algorithm outperforms the one-point risk-averse learning algorithm by achieving a smaller regret bound. The study also suggests that Algorithm 2 produces more consistent and stable outcomes than Algorithm 1. The dependence on the sampling parameter 'a' is from using queried cost values to construct the empirical distribution functions for the stochastic costs.

How it works

The one-point risk-averse learning algorithm involves selecting a sampling number mt that satisfies a specific condition related to T At each time slot, the learner queries the stochastic cost for mt times under the perturbed action and obtains J(ˆxt, ξi t). The CVaR estimate is then constructed using an empirical distribution function Fˆ snt(y).

The two-point risk-averse learning algorithm queries the stochastic costs under two perturbed actions, xˆ 1 t = xt + δu and xˆ 2 t = xt - δu The CVaR gradient estimate gbar nt is given by d 2δ / (CVaRα[Fˆ1 snt] - CVaRα[Fˆ2 snt]).

The proof of Lemma 5 shows that the optimal decisions are related to the delayed feedback by X LT τ=2 x˜∗ τ−1 − x˜∗ τ ≤ 2¯dPT. This inequality is crucial for bounding the regret in both algorithms.

The simulation results on the dynamic pricing problem demonstrate that Algorithm 2 tracks desired values faster compared to Algorithm 1 Fig. 3 depicts parking prices and occupancies generated by Algorithms 1 and 2. Fig. 4 shows that the two-point risk-averse learning algorithm outperforms the one-point algorithm by achieving lower regret. Fig. 7 depicts the resulting occupancies and cumulated losses generated by Algorithm 2 under maximal individual delays dbar = 50. The occupancy under Algorithm 2 converges to the desired value slower as the delay increases, and the variance of the resulting occupancy and the cumulated loss both increase.

Conclusion

The paper investigates risk-averse learning with delayed feedback where delays are random yet bounded They developed one-point and two-point risk-averse learning algorithms employing zeroth-order optimization approaches to estimate the CVaR gradient The theoretical results demonstrate that both algorithms achieve sublinear regrets Furthermore, the twopoint risk-averse learning algorithm outperforms the one-point algorithm by achieving a smaller regret bound. The study also suggests that Algorithm 2 produces more consistent and stable outcomes than Algorithm 1.

How it works

The dynamic regret bounds of the algorithms are analyzed in terms of the delay, the total sampling number, and the path length of comparators The cumulative delay D is defined to account for practical scenarios where some feedback may not arrive before time T. The analysis shows that R1(T) ≤ (3 + Dx/r)δL0T + 2U ¯d + ηL0dUδD + ηd 2U squared / 2δ squared T + 3D squared x + 4Dx¯dPT 2η+ 2dU¯dPT δ+4cdUDxαδ r ln(1/γ) squared T(1− a/4) with probability 1−γ. The regret order of Algorithm 1 decreases as the sampling number parameter a increases and increases as the delay or path length of comparator increases.

The two-point risk-averse learning algorithm achieves DR2(T) = O˜ T(1− a/4 + (T+D)(1/2 p¯dPT)) with high probability. This result is achieved by selecting δ = T - a 4 p¯dPT and η = (T + D)(− 1/2 p¯dPT). The proof of Theorem 2 follows the same approach as in Theorem 1.

The simulation results show that Algorithm 2 converges to the desired value slower as the delay increases, and the variance of the resulting occupancy and the cumulated loss both increase <ref:2409.

Improvements for AI systems

Bold header: One-point Risk-Averse Learning with Delayed Feedback (Algorithm 1)

This algorithm can be implemented in real-world online decision-making scenarios where feedback arrives with random but bounded delays, such as in dynamic pricing problems, to mitigate catastrophic outcomes using Conditional Value at Risk (CVaR) as the risk measure.

Bold header: Two-point Risk-Averse Learning with Delayed Feedback (Algorithm 2)

This algorithm provides a performance improvement over the one-point version by achieving a smaller regret bound, making it more effective for risk mitigation in complex, delayed environments.

Bold header: Improved Regret Bounds under Delay and Path Length

The theoretical results demonstrate that the regret bounds decrease as the total sampling number increases or the delays decrease, and specifically, the two-point risk-averse learning algorithm outperforms the one-point algorithm by achieving a smaller regret bound.

Abstract

In real-world scenarios, risk-averse learning is valuable for mitigating potential adverse outcomes. However, the delayed feedback makes it challenging to assess and manage risk effectively. In this paper, we investigate risk-averse learning using Conditional Value at Risk (CVaR) as risk measure, while incorporating feedback with random but bounded delays. We develop two risk-averse learning algorithms that rely on one-point and two-point zeroth-order optimization approaches, respectively. The dynamic regrets of the algorithms are analyzed in terms of the cumulative delay and the number of total samplings. In the absence of delay, the regret bounds match the established bounds of zeroth-order stochastic gradient methods for risk-averse learning. Furthermore, the two-point risk-averse learning outperforms the one-point algorithm by achieving a smaller regret bound. We provide numerical experiments on a dynamic pricing problem to demonstrate the performance of the algorithms.