We utilize the sparsity operation proposed in Section 3.1 for ResNet-50. For ViT, we also apply the spatial Top-K operation as described in the general response. We can observe an increase in both ResNet-50 and ViT-B architectures, furthering closing the gap between human and existing models. We generalize section 4.2 in the main text to ResNet-50 and ViT-B architectures (Figure 1). The ResNet-50's sparsity definition is the same as AlexNet and VGG. For ViT-B, we reshape the intermediate activation response from [n, h * w, d] to [n, d, h * w] and apply the Top-K selection over dimension 2 before the activation is passed through the multiple head attention (Note that h and w is the height and weight of the latent tensor after reshape it to 2d, for ViT-B with patch size 16 on the 224x224 images, h=w=14, n denotes the batch size).

A.1 Creation of the Multimodal Web Document Dataset A.1.1 Collecting of a Large Number of HTML Files Our data collection process begins by considering the 25 most recent Common Crawl It contains webpages spanning from February 2020 to January/February 2023. This process yields a total of 41.2 billion documents. Selection of English content To identify non-English content, we apply the FastText classifier (Joulin et al., 2017) to the extracted text, e ectively filtering out 63.6% of the documents. Early text deduplication Often, a set of URLs is crawled repeatedly across di erent Common Crawl snapshots. However, the content of these websites may vary as web administrators make changes over time. Hence, at this stage, we refrain from deduplicating documents based on their URLs. Instead, we perform MinHash (Broder, 1997) deduplication with 16 hashes calculated over 5-grams. To further refine the data, we eliminate documents containing substantial proportions of repeated paragraphs and n-grams, employing the methodology described in MassiveText (Rae et al., 2022).

Recent works have demonstrated that the sample complexity of gradient-based learning of single index models, i.e. functions that depend on a 1-dimensional projection of the input data, is governed by their information exponent. However, these results are only concerned with isotropic data, while in practice the input often contains additional structure which can implicitly guide the algorithm. In this work, we investigate the effect of a spiked covariance structure and reveal several interesting phenomena. First, we show that in the anisotropic setting, the commonly used spherical gradient dynamics may fail to recover the true direction, even when the spike is perfectly aligned with the target direction. Next, we show that appropriate weight normalization that is reminiscent of batch normalization can alleviate this issue. Further, by exploiting the alignment between the (spiked) input covariance and the target, we obtain improved sample complexity compared to the isotropic case. In particular, under the spiked model with a suitably large spike, the sample complexity of gradient-based training can be made independent of the information exponent while also outperforming lower bounds for rotationally invariant kernel methods.

Given a collection of feature maps indexed by a set T, we study the performance of empirical risk minimization (ERM) on regression problems with square loss over the union of the linear classes induced by these feature maps. This setup aims at capturing the simplest instance of feature learning, where the model is expected to jointly learn from the data an appropriate feature map and a linear predictor. We start by studying the asymptotic quantiles of the excess risk of sequences of empirical risk minimizers. Remarkably, we show that when the set T is not too large and when there is a unique optimal feature map, these quantiles coincide, up to a factor of two, with those of the excess risk of the oracle procedure, which knows a priori this optimal feature map and deterministically outputs an empirical risk minimizer from the associated optimal linear class. We complement this asymptotic result with a non-asymptotic analysis that quantifies the decaying effect of the global complexity of the set T on the excess risk of ERM, and relates it to the size of the sublevel sets of the suboptimality of the feature maps. As an application of our results, we obtain new guarantees on the performance of the best subset selection procedure in sparse linear regression under general assumptions.

Interpretability methods are valuable only if their explanations faithfully describe the explained model. In this work, we consider neural networks whose predictions are invariant under a specific symmetry group. This includes popular architectures, ranging from convolutional to graph neural networks. Any explanation that faithfully explains this type of model needs to be in agreement with this invariance property.

To strengthen the design rationale for incorporating prompts instead of following recent methods [3] Table 1: Comparisons under all-in-one setting: between the usage of degradation embedding extracted from the Contrastive-learning Based Degradation Encoder (CBDE) of the Airnet [3] Model and the usage of prompt tokens in the PromptIR framework. We show that it is important the prompt block is only used on the decoder side. Table 2: Comparisons under the all-in-one setting: between the usage of the Prompt-block on both the encoder branch and encoder branch with using the prompt block only on the decoder branch. Figure 1: Overview of the Transformer block used in the PromptIR framework. The Transformer block is composed of two sub modules,the Multi Dconv head transposed attention module(MDTA) and the Gated Dconv feed-forward network(GDFN).

Recently [1] gave the first and only known result that achieves sublinear bounds in both the sampling complexity and the query time while preserving polynomial data structure space. However, their improvement over linear samples and time is only by subpolynomial factors. Our main result is a lower bound showing that, for a broad class of data structures, their bounds cannot be significantly improved.