Multi-Objective $\textit{min-max}$ Online Convex Optimization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914317334478848 |
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| author | Vaze, Rahul Mishra, Sumiran |
| author_facet | Vaze, Rahul Mishra, Sumiran |
| contents | In this paper, we broaden the horizon of online convex optimization (OCO), and consider multi-objective OCO, where there are $K$ distinct loss function sequences, and an algorithm has to choose its action at time $t$, before the $K$ loss functions at time $t$ are revealed. To capture the tradeoff between tracking the $K$ different sequences, we consider the {\it min-max} regret, where the benchmark (optimal offline algorithm) takes a static action across all time slots that minimizes the maximum of the total loss (summed across time slots) incurred by each of the $K$ sequences. An online algorithm is allowed to change its action across time slots, and its {\it min-max} regret is defined as the difference between its {\it min-max} cost and that of the benchmark. The {\it min-max} regret is a stringent performance measure and an algorithm with small regret needs to `track' all loss functions simultaneously.
We first show that with adversarial input, {\it min-max} regret scales linearly with the time horizon $T$ for any online algorithm. Consequently, we consider a stochastic i.i.d. input model where all loss functions are i.i.d. generated from an unknown joint distribution and propose a simple algorithm that combines the well-known {\it Hedge} and online gradient descent (OGD) and show via a remarkably simple proof that its expected {\it min-max} regret is $O(\sqrt{T \log (T K)})$. Analogous results are also derived for Martingale difference and Markov input models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-Objective $\textit{min-max}$ Online Convex Optimization Vaze, Rahul Mishra, Sumiran Machine Learning In this paper, we broaden the horizon of online convex optimization (OCO), and consider multi-objective OCO, where there are $K$ distinct loss function sequences, and an algorithm has to choose its action at time $t$, before the $K$ loss functions at time $t$ are revealed. To capture the tradeoff between tracking the $K$ different sequences, we consider the {\it min-max} regret, where the benchmark (optimal offline algorithm) takes a static action across all time slots that minimizes the maximum of the total loss (summed across time slots) incurred by each of the $K$ sequences. An online algorithm is allowed to change its action across time slots, and its {\it min-max} regret is defined as the difference between its {\it min-max} cost and that of the benchmark. The {\it min-max} regret is a stringent performance measure and an algorithm with small regret needs to `track' all loss functions simultaneously. We first show that with adversarial input, {\it min-max} regret scales linearly with the time horizon $T$ for any online algorithm. Consequently, we consider a stochastic i.i.d. input model where all loss functions are i.i.d. generated from an unknown joint distribution and propose a simple algorithm that combines the well-known {\it Hedge} and online gradient descent (OGD) and show via a remarkably simple proof that its expected {\it min-max} regret is $O(\sqrt{T \log (T K)})$. Analogous results are also derived for Martingale difference and Markov input models. |
| title | Multi-Objective $\textit{min-max}$ Online Convex Optimization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.13560 |