Estimating parameters of functional ARMA, GARCH and invertible processes requires estimating lagged covariance and cross-covariance operators of Cartesian product Hilbert space-valued processes. Asymptotic results have been derived in recent years, either less generally or under a strict condition. This article derives upper bounds of the estimation errors for such operators based on the mild condition Lp-m-approximability for each lag, Cartesian power(s) and sample size, where the two processes can take values in different spaces in the context of lagged cross-covariance operators. Implications of our results on eigenelements, parameters in functional AR(MA) models and other general situations are also discussed.
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Robot learning holds tremendous promise to unlock the full potential of flexible, general, and dexterous robot systems, as well as to address some of the deepest questions in artificial intelligence. However, bringing robot learning to the level of generality required for effective real-world systems faces major obstacles in terms of data, generalization, and robustness. In this paper, we discuss how generalist robot policies (i.e., robot foundation models) can address these challenges, and how we can design effective generalist robot policies for complex and highly dexterous tasks. We propose a novel flow matching architecture built on top of a pre-trained vision-language model (VLM) to inherit Internet-scale semantic knowledge. We then discuss how this model can be trained on a large and diverse dataset from multiple dexterous robot platforms, including single-arm robots, dual-arm robots, and mobile manipulators. We evaluate our model in terms of its ability to perform tasks in zero shot after pre-training, follow language instructions from people and from a high-level VLM policy, and its ability to acquire new skills via fine-tuning. Our results cover a wide variety of tasks, such as laundry folding, table cleaning, and assembling boxes.
Federated Learning (FL) is a distributed machine learning framework that trains accurate global models while preserving clients' privacy-sensitive data. However, most FL approaches assume that clients possess labeled data, which is often not the case in practice. Federated Semi-Supervised Learning (FSSL) addresses this label deficiency problem, targeting situations where only the server has a small amount of labeled data while clients do not. However, a significant performance gap exists between Centralized Semi-Supervised Learning (SSL) and FSSL. This gap arises from confirmation bias, which is more pronounced in FSSL due to multiple local training epochs and the separation of labeled and unlabeled data. We propose $(FL)^2$, a robust training method for unlabeled clients using sharpness-aware consistency regularization. We show that regularizing the original pseudo-labeling loss is suboptimal, and hence we carefully select unlabeled samples for regularization. We further introduce client-specific adaptive thresholding and learning status-aware aggregation to adjust the training process based on the learning progress of each client. Our experiments on three benchmark datasets demonstrate that our approach significantly improves performance and bridges the gap with SSL, particularly in scenarios with scarce labeled data.
Retrieval-augmented generation (RAG) is a promising approach to address the limitations of fixed knowledge in large language models (LLMs). However, current benchmarks for evaluating RAG systems suffer from two key deficiencies: (1) they fail to adequately measure LLMs' capability in handling \emph{long-context retrieval} due to a lack of datasets that reflect the characteristics of retrieved documents, and (2) they lack a comprehensive evaluation method for assessing LLMs' ability to generate \emph{long-form responses} that effectively exploits retrieved information. To address these shortcomings, we introduce the \textsc{Long$^2$RAG} benchmark and the Key Point Recall (\textit{KPR}) metric. \textsc{Long$^2$RAG} comprises 280 questions spanning 10 domains and across 8 question categories, each associated with 5 retrieved documents with an average length of 2,444 words. \textit{KPR} evaluates the extent to which LLMs incorporate key points extracted from the retrieved documents into their generated responses, providing a more nuanced assessment of their ability to exploit retrieved information. Our dataset and scripts are available at //github.com/QZH-777/longrag.
In this research, we introduce an algorithm that produces what appears to be a new mathematical object as a consequence of projecting the \( n \)-dimensional \( Z \)-curve onto an \( n \)-dimensional sphere. The first part presents the algorithm that enables this transformation, and the second part focuses on studying its properties.
Researchers in empirical machine learning recently spotlighted their fears of so-called Model Collapse. They imagined a discard workflow, where an initial generative model is trained with real data, after which the real data are discarded, and subsequently, the model generates synthetic data on which a new model is trained. They came to the conclusion that models degenerate as model-fitting generations proceed. However, other researchers considered an augment workflow, where the original real data continue to be used in each generation of training, augmented by synthetic data from models fit in all earlier generations. Empirical results on canonical datasets and learning procedures confirmed the occurrence of model collapse under the discard workflow and avoidance of model collapse under the augment workflow. Under the augment workflow, theoretical evidence also confirmed avoidance in particular instances; specifically, Gerstgrasser et al. (2024) found that for classical Linear Regression, test risk at any later generation is bounded by a moderate multiple, viz. pi-squared-over-6 of the test risk of training with the original real data alone. Some commentators questioned the generality of theoretical conclusions based on the generative model assumed in Gerstgrasser et al. (2024): could similar conclusions be reached for other task/model pairings? In this work, we demonstrate the universality of the pi-squared-over-6 augment risk bound across a large family of canonical statistical models, offering key insights into exactly why collapse happens under the discard workflow and is avoided under the augment workflow. In the process, we provide a framework that is able to accommodate a large variety of workflows (beyond discard and augment), thereby enabling an experimenter to judge the comparative merits of multiple different workflows by simulating a simple Gaussian process.
The deployment of large language models (LLMs) is often hindered by the extensive memory requirements of the Key-Value (KV) cache, especially as context lengths increase. Existing approaches to reduce the KV cache size involve either fine-tuning the model to learn a compression strategy or leveraging attention scores to reduce the sequence length. We analyse the attention distributions in decoder-only Transformers-based models and observe that attention allocation patterns stay consistent across most layers. Surprisingly, we find a clear correlation between the $L_2$ and the attention scores over cached KV pairs, where a low $L_2$ of a key embedding usually leads to a high attention score during decoding. This finding indicates that the influence of a KV pair is potentially determined by the key embedding itself before being queried. Based on this observation, we compress the KV cache based on the $L_2$ of key embeddings. Our experimental results show that this simple strategy can reduce the KV cache size by 50% on language modelling and needle-in-a-haystack tasks and 90% on passkey retrieval tasks without losing accuracy. Moreover, without relying on the attention scores, this approach remains compatible with FlashAttention, enabling broader applicability.
The Johnson-Lindenstrauss (JL) Lemma introduced the concept of dimension reduction via a random linear map, which has become a fundamental technique in many computational settings. For a set of $n$ points in $\mathbb{R}^d$ and any fixed $\epsilon>0$, it reduces the dimension $d$ to $O(\log n)$ while preserving, with high probability, all the pairwise Euclidean distances within factor $1+\epsilon$. Perhaps surprisingly, the target dimension can be lower if one only wishes to preserve the optimal value of a certain problem on the pointset, e.g., Euclidean max-cut or $k$-means. However, for some notorious problems, like diameter (aka furthest pair), dimension reduction via the JL map to below $O(\log n)$ does not preserve the optimal value within factor $1+\epsilon$. We propose to focus on another regime, of \emph{moderate dimension reduction}, where a problem's value is preserved within factor $\alpha>1$ using target dimension $\tfrac{\log n}{poly(\alpha)}$. We establish the viability of this approach and show that the famous $k$-center problem is $\alpha$-approximated when reducing to dimension $O(\tfrac{\log n}{\alpha^2}+\log k)$. Along the way, we address the diameter problem via the special case $k=1$. Our result extends to several important variants of $k$-center (with outliers, capacities, or fairness constraints), and the bound improves further with the input's doubling dimension. While our $poly(\alpha)$-factor improvement in the dimension may seem small, it actually has significant implications for streaming algorithms, and easily yields an algorithm for $k$-center in dynamic geometric streams, that achieves $O(\alpha)$-approximation using space $poly(kdn^{1/\alpha^2})$. This is the first algorithm to beat $O(n)$ space in high dimension $d$, as all previous algorithms require space at least $\exp(d)$. Furthermore, it extends to the $k$-center variants mentioned above.
Quantum error correction is essential for the development of any scalable quantum computer. In this work we introduce a generalization of a quantum interleaving method for combating clusters of errors in toric quantum error-correcting codes. We present new $n$-dimensional toric quantum codes, where $n\geq 5$, which are featured by lattice codes and apply the proposed quantum interleaving method to such new $n$-dimensional toric quantum codes. Through the application of this method to these novel $n$-dimensional toric quantum codes we derive new $n$-dimensional quantum burst-error-correcting codes. Consequently, $n$-dimensional toric quantum codes and burst-error-correcting quantum codes are provided offering both a good code rate and a significant coding gain when it comes to toric quantum codes. Another important consequence from the presented $n$-dimensional toric quantum codes is that if the Golomb and Welch conjecture in \cite{perfcodes} regarding the Lee sphere in $n$ dimensions for the respective close packings holds true, then it follows that these $n$-dimensional toric quantum codes are the only possible ones to be obtained from lattice codes. Moreover, such a methodology can be applied for burst error correction in cases involving localized errors, quantum data storage and quantum channels with memory.
We provide the first proof of convergence for normalized error feedback algorithms across a wide range of machine learning problems. Despite their popularity and efficiency in training deep neural networks, traditional analyses of error feedback algorithms rely on the smoothness assumption that does not capture the properties of objective functions in these problems. Rather, these problems have recently been shown to satisfy generalized smoothness assumptions, and the theoretical understanding of error feedback algorithms under these assumptions remains largely unexplored. Moreover, to the best of our knowledge, all existing analyses under generalized smoothness either i) focus on single-node settings or ii) make unrealistically strong assumptions for distributed settings, such as requiring data heterogeneity, and almost surely bounded stochastic gradient noise variance. In this paper, we propose distributed error feedback algorithms that utilize normalization to achieve the $O(1/\sqrt{K})$ convergence rate for nonconvex problems under generalized smoothness. Our analyses apply for distributed settings without data heterogeneity conditions, and enable stepsize tuning that is independent of problem parameters. Additionally, we provide strong convergence guarantees of normalized error feedback algorithms for stochastic settings. Finally, we show that due to their larger allowable stepsizes, our new normalized error feedback algorithms outperform their non-normalized counterparts on various tasks, including the minimization of polynomial functions, logistic regression, and ResNet-20 training.
Non-negative Matrix Factorization (NMF) is an effective algorithm for multivariate data analysis, including applications to feature selection, pattern recognition, and computer vision. Its variant, Semi-Nonnegative Matrix Factorization (SNF), extends the ability of NMF to render parts-based data representations to include mixed-sign data. Graph Regularized SNF builds upon this paradigm by adding a graph regularization term to preserve the local geometrical structure of the data space. Despite their successes, SNF-related algorithms to date still suffer from instability caused by the Frobenius norm due to the effects of outliers and noise. In this paper, we present a new $L_{2,1}$ SNF algorithm that utilizes the noise-insensitive $L_{2,1}$ norm. We provide monotonic convergence analysis of the $L_{2,1}$ SNF algorithm. In addition, we conduct numerical experiments on three benchmark mixed-sign datasets as well as several randomized mixed-sign matrices to demonstrate the performance superiority of $L_{2,1}$ SNF over conventional SNF algorithms under the influence of Gaussian noise at different levels.
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