We present and analyze an algorithm for optimizing smooth and convex or strongly convex objectives using minibatch stochastic gradient estimates. The algorithm is optimal with respect to its dependence on both the minibatch size and minimum expected loss simultaneously. This improves over the optimal method of Lan [17], which is insensitive to the minimum expected loss; over the optimistic acceleration of Cotter et al. [10], which has suboptimal dependence on the minibatch size; and over the algorithm of Liu and Belkin [19], which is limited to least squares problems and is also similarly suboptimal with respect to the minibatch size.

We organize our supplementary document as follows: 1. Results on additional dataset 2. Results for limited data settings on YFCC26k and GWS15k datasets 3. Additional Ablations (a) Gallery Size (b) Queue Length (c) ση for Batch GPS noise (d) ση for Queue GPS noise (e) σ for Random Fourier Features (f) Number of hierarchies (M) 4. Different selection choices for GPSGallery Construction (a) Evenly Spaced GPSCoordinates (b) Test Set GPSCoordinates 5. Analysis of Runtime and Memory Footprint 6. Motivations for using Pretrained CLIP as Image encoder Backbone 7. Qualitative Demonstration (a) Hierarchical learning in our location encoder L () (b) GeoCLIP with Image Query (c) Distribution of correct predictions of GeoCLIP on different datasets (d) GeoCLIP with Text Query 8. Discussion on Ethical Issues and Possible Mitigation In section 4.1 of the main paper, we demonstrated the performance of our GeoCLIP method on Im2GPS3k [2] and GWS15k [1] datasets and compared them with the state-of-the-art methods. Here, we perform experiments on another dataset YFCC26k [6]. The results are provided in Table 1. This result highlights that GeoCLIP performs well across datasets, being useful across different data distributions. GeoCLIP achieves decent performance across datasets even when the training data is significantly reduced. 2 We show the efficacy of GeoCLIP on limited training samples of Im2GPS3k in section 4.2 of the main paper. Now, we further investigate the performance of GeoCLIP for limited data settings on other datasets (YFCC26k and GWS15k).

We explore the minimax optimal error associated with a demographic parityconstrained regression problem within the context of a linear model. Our proposed model encompasses a broader range of discriminatory bias sources compared to the model presented by Chzhen and Schreuder [6]. Our analysis reveals that the minimax optimal error for the demographic parity-constrained regression problem under our model is characterized by Θ(dM/n), where ndenotes the sample size, d represents the dimensionality, and M signifies the number of demographic groups arising from sensitive attributes. Moreover, we demonstrate that the minimax error increases in conjunction with a larger bias present in the model.

We explore the minimax optimal error associated with a demographic parityconstrained regression problem within the context of a linear model. Our proposed model encompasses a broader range of discriminatory bias sources compared to the model presented by Chzhen and Schreuder [6]. Our analysis reveals that the minimax optimal error for the demographic parity-constrained regression problem under our model is characterized by Θ(dM/n), where ndenotes the sample size, d represents the dimensionality, and M signifies the number of demographic groups arising from sensitive attributes. Moreover, we demonstrate that the minimax error increases in conjunction with a larger bias present in the model.

Double Q-learning (Hasselt, 2010) has gained significant success in practice due to its effectiveness in overcoming the overestimation issue of Q-learning. However, the theoretical understanding of double Q-learning is rather limited. The only existing finite-time analysis was recently established in (Xiong et al., 2020), where the polynomial learning rate adopted in the analysis typically yields a slower convergence rate. This paper tackles the more challenging case of a constant learning rate, and develops new analytical tools that improve the existing convergence rate by orders of magnitude.

The recent success of transformer-based image generative models in object-centric learning highlights the importance of powerful image generators for handling complex scenes. However, despite the high expressiveness of diffusion models in image generation, their integration into object-centric learning remains largely unexplored in this domain. In this paper, we explore the feasibility and potential of integrating diffusion models into object-centric learning and investigate the pros and cons of this approach. We introduce Latent Slot Diffusion (LSD), a novel model that serves dual purposes: it is the first object-centric learning model to replace conventional slot decoders with a latent diffusion model conditioned on object slots, and it is also the first unsupervised compositional conditional diffusion model that operates without the need for supervised annotations like text. Through experiments on various object-centric tasks, including the first application of the FFHQ dataset in this field, we demonstrate that LSD significantly outperforms state-of-the-art transformer-based decoders, particularly in more complex scenes, and exhibits superior unsupervised compositional generation quality. In addition, we conduct a preliminary investigation into the integration of pre-trained diffusion models in LSD and demonstrate its effectiveness in real-world image segmentation and generation.