The k-means algorithm can simplify large-scale spatial vectors, such as 2D geo-locations and 3D point clouds, to support fast analytics and learning. However, when processing large-scale datasets, existing k-means algorithms have been developed to achieve high performance with significant computational resources, such as memory and CPU usage time. These algorithms, though effective, are not well-suited for resource-constrained devices. In this paper, we propose a fast, memory-efficient, and cost-predictable k-means called Dask-means. We first accelerate k-means by designing a memory-efficient accelerator, which utilizes an optimized nearest neighbor search over a memory-tunable index to assign spatial vectors to clusters in batches. We then design a lightweight cost estimator to predict the memory cost and runtime of the k-means task, allowing it to request appropriate memory from devices or adjust the accelerator's required space to meet memory constraints, and ensure sufficient CPU time for running k-means. Experiments show that when simplifying datasets with scale such as $10^6$, Dask-means uses less than $30$MB of memory, achieves over $168$ times speedup compared to the widely-used Lloyd's algorithm. We also validate Dask-means on mobile devices, where it demonstrates significant speedup and low memory cost compared to other state-of-the-art (SOTA) k-means algorithms. Our cost estimator estimates the memory cost with a difference of less than $3\%$ from the actual ones and predicts runtime with an MSE up to $33.3\%$ lower than SOTA methods.

In location-based resource allocation scenarios, the distances between each individual and the facility are desired to be approximately equal, thereby ensuring fairness. Individually fair clustering is often employed to achieve the principle of treating all points equally, which can be applied in these scenarios. This paper proposes a novel algorithm, tilted k-means (TKM), aiming to achieve individual fairness in clustering. We integrate the exponential tilting into the sum of squared errors (SSE) to formulate a novel objective function called tilted SSE. We demonstrate that the tilted SSE can generalize to SSE and employ the coordinate descent and first-order gradient method for optimization. We propose a novel fairness metric, the variance of the distances within each cluster, which can alleviate the Matthew Effect typically caused by existing fairness metrics. Our theoretical analysis demonstrates that the well-known k-means++ incurs a multiplicative error of O(k log k), and we establish the convergence of TKM under mild conditions. In terms of fairness, we prove that the variance generated by TKM decreases with a scaled hyperparameter. In terms of efficiency, we demonstrate the time complexity is linear with the dataset size. Our experiments demonstrate that TKM outperforms state-of-the-art methods in effectiveness, fairness, and efficiency.

Personalized federated learning (PFL) is an approach proposed to address the issue of poor convergence on heterogeneous data. However, most existing PFL frameworks require strong assumptions for convergence. In this paper, we propose an alternating direction method of multipliers (ADMM) for training PFL models with Moreau envelope (FLAME), which achieves a sublinear convergence rate, relying on the relatively weak assumption of gradient Lipschitz continuity. Moreover, due to the gradient-free nature of ADMM, FLAME alleviates the need for hyperparameter tuning, particularly in avoiding the adjustment of the learning rate when training the global model. In addition, we propose a biased client selection strategy to expedite the convergence of training of PFL models. Our theoretical analysis establishes the global convergence under both unbiased and biased client selection strategies. Our experiments validate that FLAME, when trained on heterogeneous data, outperforms state-of-the-art methods in terms of model performance. Regarding communication efficiency, it exhibits an average speedup of 3.75x compared to the baselines. Furthermore, experimental results validate that the biased client selection strategy speeds up the convergence of both personalized and global models.