A.1 Details of "stronger recipe" In Table 1 of our main paper, we evaluate the impact of limited model capacity [1] and small-scale dataset by comparing the results of using "previous training recipe" and our "stronger recipe". We summarize the details of "stronger recipe" and present them in Table 13. Table 13: Stronger training strategy used for distillation. "B" and "C" represent strategies for training students on ImageNet-1K and CIFAR100, respectively. A.2 Numerical results In Figure 1 of our main paper, we present a comparison of performance gaps among vanilla KD and two logits-based baselines, i.e., DKD [2] and DIST [3], on two datasets of varying scales, to demonstrate the underestimation of vanilla KD on small-scale datasets.

As action decoder their mentioned architectures of is multimodal adopted in the in to paper Figure information generate, the 1. visual-gr natural with languages cross-attention ounded alignment conditioned blocks, decoder on while the is visual applied the visual-grounded input. Based on these deeply fused representations, we finally generate the predicted answers with the visual-grounded generation decoder. In this section, we describe the settings used when fine-tuning the pretrained models on various downstream tasks. We use RandomAugment [1] for data augmentation. The default settings for finetuning on each dataset are shown in Table 1.

The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action $x_t \in \mathcal{X} \subset \mathbb{R}^d$, a convex loss function $f_t$ and a convex constraint function $g_t$ that drives the constraint $g_t(x)\le 0$ are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions $f_t$ and $g_t$ for all $t$ ahead of time, and chooses a static optimal action that is feasible with respect to all $g_t(x)\le 0$. In recent prior work Sinha and Vaze [2024], algorithms with simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ or (CCV of $O(1)$ in specific cases Vaze and Sinha [2025], e.g. when $d=1$) have been proposed. It is widely believed that CCV is $Ω(\sqrt{T})$ for all algorithms that ensure that regret is $O(\sqrt{T})$ with the worst case input for any $d\ge 2$. In this paper, we refute this and show that the algorithm of Vaze and Sinha [2025] simultaneously achieves regret of $O(\sqrt{T})$ regret and CCV of $O(T^{1/3})$ when $d=2$.

A well-studied generalization of the standard online convex optimization (OCO) framework is constrained online convex optimization (COCO). In COCO, on every round, a convex cost function and a convex constraint function are revealed to the learner after it chooses the action for that round. The objective is to design an online learning policy that simultaneously achieves a small regret while ensuring a small cumulative constraint violation (CCV) against an adaptive adversary interacting over a horizon of length $T$. A long-standing open question in COCO is whether an online policy can simultaneously achieve $O(\sqrt{T})$ regret and $\tilde{O}(\sqrt{T})$ CCV without any restrictive assumptions. For the first time, we answer this in the affirmative and show that a simple first-order policy can simultaneously achieve these bounds. Furthermore, in the case of strongly convex cost and convex constraint functions, the regret guarantee can be improved to $O(\log T)$ while keeping the CCV bound the same as above. We establish these results by effectively combining adaptive OCO policies as a blackbox with Lyapunov optimization - a classic tool from control theory. Surprisingly, the analysis is short and elegant.

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