This paper proposes to develop a new variant of the two-time-scale stochastic approximation to find the roots of two coupled nonlinear operators, assuming only noisy samples of these operators can be observed. Our key idea is to leverage the classic Ruppert-Polyak averaging technique to dynamically estimate the operators through their samples. The estimated values of these averaging steps will then be used in the two-time-scale stochastic approximation updates to find the desired solution. Our main theoretical result is to show that under the strongly monotone condition of the underlying nonlinear operators the mean-squared errors of the iterates generated by the proposed method converge to zero at an optimal rate $O(1/k)$, where $k$ is the number of iterations. Our result significantly improves the existing result of two-time-scale stochastic approximation, where the best known finite-time convergence rate is $O(1/k^{2/3})$. We illustrate this result by applying the proposed method to develop new reinforcement learning algorithms with improved performance.
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The $2$-Wasserstein distance is sensitive to minor geometric differences between distributions, making it a very powerful dissimilarity metric. However, due to this sensitivity, a small outlier mass can also cause a significant increase in the $2$-Wasserstein distance between two similar distributions. Similarly, sampling discrepancy can cause the empirical $2$-Wasserstein distance on $n$ samples in $\mathbb{R}^2$ to converge to the true distance at a rate of $n^{-1/4}$, which is significantly slower than the rate of $n^{-1/2}$ for $1$-Wasserstein distance. We introduce a new family of distances parameterized by $k \ge 0$, called $k$-RPW, that is based on computing the partial $2$-Wasserstein distance. We show that (1) $k$-RPW satisfies the metric properties, (2) $k$-RPW is robust to small outlier mass while retaining the sensitivity of $2$-Wasserstein distance to minor geometric differences, and (3) when $k$ is a constant, $k$-RPW distance between empirical distributions on $n$ samples in $\mathbb{R}^2$ converges to the true distance at a rate of $n^{-1/3}$, which is faster than the convergence rate of $n^{-1/4}$ for the $2$-Wasserstein distance. Using the partial $p$-Wasserstein distance, we extend our distance to any $p \in [1,\infty]$. By setting parameters $k$ or $p$ appropriately, we can reduce our distance to the total variation, $p$-Wasserstein, and the L\'evy-Prokhorov distances. Experiments show that our distance function achieves higher accuracy in comparison to the $1$-Wasserstein, $2$-Wasserstein, and TV distances for image retrieval tasks on noisy real-world data sets.
The digitization of traffic sensing infrastructure has significantly accumulated an extensive traffic data warehouse, which presents unprecedented challenges for transportation analytics. The complexities associated with querying large-scale multi-table databases require specialized programming expertise and labor-intensive development. Additionally, traditional analysis methods have focused mainly on numerical data, often neglecting the semantic aspects that could enhance interpretability and understanding. Furthermore, real-time traffic data access is typically limited due to privacy concerns. To bridge this gap, the integration of Large Language Models (LLMs) into the domain of traffic management presents a transformative approach to addressing the complexities and challenges inherent in modern transportation systems. This paper proposes an intelligent online chatbot, TP-GPT, for efficient customized transportation surveillance and management empowered by a large real-time traffic database. The innovative framework leverages contextual and generative intelligence of language models to generate accurate SQL queries and natural language interpretations by employing transportation-specialized prompts, Chain-of-Thought prompting, few-shot learning, multi-agent collaboration strategy, and chat memory. Experimental study demonstrates that our approach outperforms state-of-the-art baselines such as GPT-4 and PaLM 2 on a challenging traffic-analysis benchmark TransQuery. TP-GPT would aid researchers and practitioners in real-time transportation surveillance and management in a privacy-preserving, equitable, and customizable manner.
Locally repairable codes (LRCs) are designed for distributed storage systems to reduce the repair bandwidth and disk I/O complexity during the storage node repair process. A code with $(r,\delta)$-locality (also called an $(r,\delta)$-LRC) can simultaneously repair up to $\delta-1$ symbols in a codeword by accessing at most $r$ other symbols in the codeword. In this paper, we propose a new method to calculate the $(r,\delta)$-locality of cyclic codes. Initially, we give a description of the algebraic structure of repeated-root cyclic codes of prime power lengths. Using this result, we derive a formula of $(r,\delta)$-locality of these cyclic codes for a wide range of $\delta$ values. Furthermore, we calculate the parameters of repeated-root cyclic codes of prime power lengths and obtain several infinite families of optimal cyclic $(r,\delta)$-LRCs, which exhibit new parameters compared with existing research on optimal $(r,\delta)$-LRCs with a cyclic structure. For the specific case of $\delta=2$, we have comprehensively identified all potential optimal cyclic $(r,2)$-LRCs of prime power lengths.
We study the problem of privacy-preserving $k$-means clustering in the horizontally federated setting. Existing federated approaches using secure computation, suffer from substantial overheads and do not offer output privacy. At the same time, differentially private (DP) $k$-means algorithms assume a trusted central curator and do not extend to federated settings. Naively combining the secure and DP solutions results in a protocol with impractical overhead. Instead, our work provides enhancements to both the DP and secure computation components, resulting in a design that is faster, more private, and more accurate than previous work. By utilizing the computational DP model, we design a lightweight, secure aggregation-based approach that achieves four orders of magnitude speed-up over state-of-the-art related work. Furthermore, we not only maintain the utility of the state-of-the-art in the central model of DP, but we improve the utility further by taking advantage of constrained clustering techniques.
Real-world recommender systems often need to balance multiple objectives when deciding which recommendations to present to users. These include behavioural signals (e.g. clicks, shares, dwell time), as well as broader objectives (e.g. diversity, fairness). Scalarisation methods are commonly used to handle this balancing task, where a weighted average of per-objective reward signals determines the final score used for ranking. Naturally, how these weights are computed exactly, is key to success for any online platform. We frame this as a decision-making task, where the scalarisation weights are actions taken to maximise an overall North Star reward (e.g. long-term user retention or growth). We extend existing policy learning methods to the continuous multivariate action domain, proposing to maximise a pessimistic lower bound on the North Star reward that the learnt policy will yield. Typical lower bounds based on normal approximations suffer from insufficient coverage, and we propose an efficient and effective policy-dependent correction for this. We provide guidance to design stochastic data collection policies, as well as highly sensitive reward signals. Empirical observations from simulations, offline and online experiments highlight the efficacy of our deployed approach.
We consider a novel algorithm, for the completion of partially observed low-rank matrices in a structured setting where each entry can be chosen from a finite discrete alphabet set, such as in common recommender systems. The proposed low-rank matrix completion (MC) method is an improved variation of state-of-the-art (SotA) discrete aware matrix completion method which we previously proposed, in which discreteness is enforced by an $\ell_0$-norm regularizer, not by replaced with the $\ell_1$-norm, but instead approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework. Simulation results demonstrate the superior performance of the new method compared to the SotA techniques as well as the earlier $\ell_1$-norm-based discrete-aware matrix completion approach.
Coresets are arguably the most popular compression paradigm for center-based clustering objectives such as $k$-means. Given a point set $P$, a coreset $\Omega$ is a small, weighted summary that preserves the cost of all candidate solutions $S$ up to a $(1\pm \varepsilon)$ factor. For $k$-means in $d$-dimensional Euclidean space the cost for solution $S$ is $\sum_{p\in P}\min_{s\in S}\|p-s\|^2$. A very popular method for coreset construction, both in theory and practice, is Sensitivity Sampling, where points are sampled in proportion to their importance. We show that Sensitivity Sampling yields optimal coresets of size $\tilde{O}(k/\varepsilon^2\min(\sqrt{k},\varepsilon^{-2}))$ for worst-case instances. Uniquely among all known coreset algorithms, for well-clusterable data sets with $\Omega(1)$ cost stability, Sensitivity Sampling gives coresets of size $\tilde{O}(k/\varepsilon^2)$, improving over the worst-case lower bound. Notably, Sensitivity Sampling does not have to know the cost stability in order to exploit it: It is appropriately sensitive to the clusterability of the data set while being oblivious to it. We also show that any coreset for stable instances consisting of only input points must have size $\Omega(k/\varepsilon^2)$. Our results for Sensitivity Sampling also extend to the $k$-median problem, and more general metric spaces.
Logically constrained term rewriting is a relatively new formalism where rules are equipped with constraints over some arbitrary theory. Although there are many recent advances with respect to rewriting induction, completion, complexity analysis and confluence analysis for logically constrained term rewriting, these works solely focus on the syntactic side of the formalism lacking detailed investigations on semantics. In this paper, we investigate a semantic side of logically constrained term rewriting. To this end, we first define constrained equations, constrained equational theories and validity of the former based on the latter. After presenting the relationship of validity and conversion of rewriting, we then construct a sound inference system to prove validity of constrained equations in constrained equational theories. Finally, we give an algebraic semantics, which enables one to establish invalidity of constrained equations in constrained equational theories. This algebraic semantics derive a new notion of consistency for constrained equational theories.
Generative commonsense reasoning which aims to empower machines to generate sentences with the capacity of reasoning over a set of concepts is a critical bottleneck for text generation. Even the state-of-the-art pre-trained language generation models struggle at this task and often produce implausible and anomalous sentences. One reason is that they rarely consider incorporating the knowledge graph which can provide rich relational information among the commonsense concepts. To promote the ability of commonsense reasoning for text generation, we propose a novel knowledge graph augmented pre-trained language generation model KG-BART, which encompasses the complex relations of concepts through the knowledge graph and produces more logical and natural sentences as output. Moreover, KG-BART can leverage the graph attention to aggregate the rich concept semantics that enhances the model generalization on unseen concept sets. Experiments on benchmark CommonGen dataset verify the effectiveness of our proposed approach by comparing with several strong pre-trained language generation models, particularly KG-BART outperforms BART by 5.80, 4.60, in terms of BLEU-3, 4. Moreover, we also show that the generated context by our model can work as background scenarios to benefit downstream commonsense QA tasks.
Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling. We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence. It consists of a segment-level recurrence mechanism and a novel positional encoding scheme. Our method not only enables capturing longer-term dependency, but also resolves the context fragmentation problem. As a result, Transformer-XL learns dependency that is 80% longer than RNNs and 450% longer than vanilla Transformers, achieves better performance on both short and long sequences, and is up to 1,800+ times faster than vanilla Transformers during evaluation. Notably, we improve the state-of-the-art results of bpc/perplexity to 0.99 on enwiki8, 1.08 on text8, 18.3 on WikiText-103, 21.8 on One Billion Word, and 54.5 on Penn Treebank (without finetuning). When trained only on WikiText-103, Transformer-XL manages to generate reasonably coherent, novel text articles with thousands of tokens. Our code, pretrained models, and hyperparameters are available in both Tensorflow and PyTorch.
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