Over the last years, many variations of the quadratic k -means clustering procedure have been proposed, all aiming to robustify the performance of the algorithm in the presence of outliers. In general terms, two main approaches have been developed: one based on penalized regularization methods, and one based on trimming functions. In this work, we present a theoretical analysis of the robustness and consistency properties of a variant of the classical quadratic k -means algorithm, the robust k -means, which borrows ideas from outlier detection in regression. We show that two outliers in a dataset are enough to breakdown this clustering procedure. However, if we focus on "well-structured" datasets, then robust k -means can recover the underlying cluster structure in spite of the outliers. Finally, we show that, with slight modifications, the most general non-asymptotic results for consistency of quadratic k -means remain valid for this robust variant.

Neural Information Processing Systems http://nips.cc/

The proximal map is the key step in gradient-type algorithms, which have become prevalent in large-scale high-dimensional problems. For simple functions this proximal map is available in closed-form while for more complicated functions it can become highly nontrivial. Motivated by the need of combining regularizers to simultaneously induce different types of structures, this paper initiates a systematic investigation of when the proximal map of a sum of functions decomposes into the composition of the proximal maps of the individual summands. We not only unify a few known results scattered in the literature but also discover several new decompositions obtained almost effortlessly from our theory.

The following instantiation of Corollary 4.2 is notable: setting

Modern deep learning models exhibit exceptional vision and language processing capabilities, but come with excessive sizes and demands on memory and computing. Quantization is an effective approach for model compression, which can significantly reduce their memory footprint, computing cost, as well as latency for inference (e.g., Han et al., 2016; Sze et al., 2017). There are two main classes of quantization methods: post-training quantization (PTQ) and quantization-aware training (QAT). Both are widely adopted and receive extensive research--see the recent survey papers (Gholami et al., 2022; Fournarakis et al., 2022) and references therein. PTQ converts the weights of a pre-trained model directly into lower precision without repeating the training pipeline; it thus has less overhead and is relatively easy to apply Nagel et al. (2020); Cai et al. (2020); Chee et al. (2024). However, it is mainly limited to 4 or more bit regimes and can suffer steep performance drops with fewer bits Yao et al. (2022); Dettmers & Zettlemoyer (2023). This is especially the case for transformer-based models, which prove harder to quantize Bai et al. (2021); Qin et al. (2022) compared to convolutional architectures Martinez et al. (2019); Qin et al. (2020). On the other hand, QAT integrates quantization into pre-training and/or fine-tuning processes and can produce low-bit (especially binary) models with mild performance degradation (e.g.

Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of regions and region-specific models. This paper proposes novel convex region-specific linear models, which we refer to as partition-wise linear models. Our key ideas are 1) assigning linear models not to regions but to partitions (region-specifiers) and representing region-specific linear models by linear combinations of partitionspecific models, and 2) optimizing regions via partition selection from a large number of given partition candidates by means of convex structured regularizations. In addition to providing initialization-free globally-optimal solutions, our convex formulation makes it possible to derive a generalization bound and to use such advanced optimization techniques as proximal methods and decomposition of the proximal maps for sparsity-inducing regularizations. Experimental results demonstrate that our partition-wise linear models perform better than or are at least competitive with state-of-the-art region-specific or locally linear models.

We consider convex optimization with non-smooth objective function and log-concave sampling with non-smooth potential (negative log density). In particular, we study two specific settings where the convex objective/potential function is either semi-smooth or in composite form as the finite sum of semi-smooth components. To overcome the challenges caused by non-smoothness, our algorithms employ two powerful proximal frameworks in optimization and sampling: the proximal point framework for optimization and the alternating sampling framework (ASF) that uses Gibbs sampling on an augmented distribution. A key component of both optimization and sampling algorithms is the efficient implementation of the proximal map by the regularized cutting-plane method. We establish the iteration-complexity of the proximal map in both semi-smooth and composite settings. We further propose an adaptive proximal bundle method for non-smooth optimization. The proposed method is universal since it does not need any problem parameters as input. Additionally, we develop a proximal sampling oracle that resembles the proximal map in optimization and establish its complexity using a novel technique (a modified Gaussian integral). Finally, we combine this proximal sampling oracle and ASF to obtain a Markov chain Monte Carlo method with non-asymptotic complexity bounds for sampling in semi-smooth and composite settings.