A new algorithm for regret minimization in online convex optimization is described. The regret of the algorithm after $T$ time periods is $O(\sqrt{T \log T})$ - which is the minimum possible up to a logarithmic term. In addition, the new algorithm is adaptive, in the sense that the regret bounds hold not only for the time periods $1,\ldots,T$ but also for every sub-interval $s,s+1,\ldots,t$. The running time of the algorithm matches that of newly introduced interior point algorithms for regret minimization: in $n$-dimensional space, during each iteration the new algorithm essentially solves a system of linear equations of order $n$, rather than solving some constrained convex optimization problem in $n$ dimensions and possibly many constraints.
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We introduce a decentralized mechanism for pricing and exchanging alternatives constrained by transaction costs. We characterize the time-invariant solutions of a heat equation involving a (weighted) Tarski Laplacian operator, defined for max-plus matrix-weighted graphs, as approximate equilibria of the trading system. We study algebraic properties of the solution sets as well as convergence behavior of the dynamical system. We apply these tools to the "economic problem" of allocating scarce resources among competing uses. Our theory suggests differences in competitive equilibrium, bargaining, or cost-benefit analysis, depending on the context, are largely due to differences in the way that transaction costs are incorporated into the decision-making process. We present numerical simulations of the synchronization algorithm (RRAggU), demonstrating our theoretical findings.
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the extreme learning machine (ELM) approach from low to high dimensions. With the first method the unknown solution field in $d$ dimensions is represented by a randomized feed-forward neural network, in which the hidden-layer parameters are randomly assigned and fixed while the output-layer parameters are trained. The PDE and the boundary/initial conditions, as well as the continuity conditions (for the local variant of the method), are enforced on a set of random interior/boundary collocation points. The resultant linear or nonlinear algebraic system, through its least squares solution, provides the trained values for the network parameters. With the second method the high-dimensional PDE problem is reformulated through a constrained expression based on an Approximate variant of the Theory of Functional Connections (A-TFC), which avoids the exponential growth in the number of terms of TFC as the dimension increases. The free field function in the A-TFC constrained expression is represented by a randomized neural network and is trained by a procedure analogous to the first method. We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance. These methods can produce accurate solutions to high-dimensional PDEs, in particular with their errors reaching levels not far from the machine accuracy for relatively lower dimensions. Compared with the physics-informed neural network (PINN) method, the current method is both cost-effective and more accurate for high-dimensional PDEs.
Automatic related work generation must ground their outputs to the content of the cited papers to avoid non-factual hallucinations, but due to the length of scientific documents, existing abstractive approaches have conditioned only on the cited paper \textit{abstracts}. We demonstrate that the abstract is not always the most appropriate input for citation generation and that models trained in this way learn to hallucinate. We propose to condition instead on the \textit{cited text span} (CTS) as an alternative to the abstract. Because manual CTS annotation is extremely time- and labor-intensive, we experiment with automatic, ROUGE-based labeling of candidate CTS sentences, achieving sufficiently strong performance to substitute for expensive human annotations, and we propose a human-in-the-loop, keyword-based CTS retrieval approach that makes generating citation texts grounded in the full text of cited papers both promising and practical.
Distributed stochastic gradient descent (SGD) with gradient compression has become a popular communication-efficient solution for accelerating distributed learning. One commonly used method for gradient compression is Top-K sparsification, which sparsifies the gradients by a fixed degree during model training. However, there has been a lack of an adaptive approach to adjust the sparsification degree to maximize the potential of the model's performance or training speed. This paper proposes a novel adaptive Top-K in SGD framework that enables an adaptive degree of sparsification for each gradient descent step to optimize the convergence performance by balancing the trade-off between communication cost and convergence error. Firstly, an upper bound of convergence error is derived for the adaptive sparsification scheme and the loss function. Secondly, an algorithm is designed to minimize the convergence error under the communication cost constraints. Finally, numerical results on the MNIST and CIFAR-10 datasets demonstrate that the proposed adaptive Top-K algorithm in SGD achieves a significantly better convergence rate compared to state-of-the-art methods, even after considering error compensation.
We study the complexity of producing $(\delta,\epsilon)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $\Omega(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(d\delta^{-1}\epsilon^{-3})$, which is optimal (up to numerical constants) with respect to $d$ and also optimal with respect to the accuracy parameters $\delta,\epsilon$, thus solving an open question due to Lin et al. (NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also optimal for smooth objectives, proving that in the nonconvex stochastic zero-order setting, nonsmooth optimization is as easy as smooth optimization. We provide algorithms that achieve the aforementioned convergence rate in expectation as well as with high probability. Our analysis is based on a simple yet powerful geometric lemma regarding the Goldstein-subdifferential set, which allows utilizing recent advancements in first-order nonsmooth nonconvex optimization.
Stable diffusion, a generative model used in text-to-image synthesis, frequently encounters resolution-induced composition problems when generating images of varying sizes. This issue primarily stems from the model being trained on pairs of single-scale images and their corresponding text descriptions. Moreover, direct training on images of unlimited sizes is unfeasible, as it would require an immense number of text-image pairs and entail substantial computational expenses. To overcome these challenges, we propose a two-stage pipeline named Any-Size-Diffusion (ASD), designed to efficiently generate well-composed images of any size, while minimizing the need for high-memory GPU resources. Specifically, the initial stage, dubbed Any Ratio Adaptability Diffusion (ARAD), leverages a selected set of images with a restricted range of ratios to optimize the text-conditional diffusion model, thereby improving its ability to adjust composition to accommodate diverse image sizes. To support the creation of images at any desired size, we further introduce a technique called Fast Seamless Tiled Diffusion (FSTD) at the subsequent stage. This method allows for the rapid enlargement of the ASD output to any high-resolution size, avoiding seaming artifacts or memory overloads. Experimental results on the LAION-COCO and MM-CelebA-HQ benchmarks demonstrate that ASD can produce well-structured images of arbitrary sizes, cutting down the inference time by 2x compared to the traditional tiled algorithm.
A well-known approach in the design of efficient algorithms, called matrix sparsification, approximates a matrix $A$ with a sparse matrix $A'$. Achlioptas and McSherry [2007] initiated a long line of work on spectral-norm sparsification, which aims to guarantee that $\|A'-A\|\leq \epsilon \|A\|$ for error parameter $\epsilon>0$. Various forms of matrix approximation motivate considering this problem with a guarantee according to the Schatten $p$-norm for general $p$, which includes the spectral norm as the special case $p=\infty$. We investigate the relation between fixed but different $p\neq q$, that is, whether sparsification in the Schatten $p$-norm implies (existentially and/or algorithmically) sparsification in the Schatten $q\text{-norm}$ with similar sparsity. An affirmative answer could be tremendously useful, as it will identify which value of $p$ to focus on. Our main finding is a surprising contrast between this question and the analogous case of $\ell_p$-norm sparsification for vectors: For vectors, the answer is affirmative for $p<q$ and negative for $p>q$, but for matrices we answer negatively for almost all sufficiently distinct $p\neq q$. In addition, our explicit constructions may be of independent interest.
A dictionary data structure maintains a set of at most $n$ keys from the universe $[U]$ under key insertions and deletions, such that given a query $x \in [U]$, it returns if $x$ is in the set. Some variants also store values associated to the keys such that given a query $x$, the value associated to $x$ is returned when $x$ is in the set. This fundamental data structure problem has been studied for six decades since the introduction of hash tables in 1953. A hash table occupies $O(n\log U)$ bits of space with constant time per operation in expectation. There has been a vast literature on improving its time and space usage. The state-of-the-art dictionary by Bender, Farach-Colton, Kuszmaul, Kuszmaul and Liu [BFCK+22] has space consumption close to the information-theoretic optimum, using a total of \[ \log\binom{U}{n}+O(n\log^{(k)} n) \] bits, while supporting all operations in $O(k)$ time, for any parameter $k \leq \log^* n$. The term $O(\log^{(k)} n) = O(\underbrace{\log\cdots\log}_k n)$ is referred to as the wasted bits per key. In this paper, we prove a matching cell-probe lower bound: For $U=n^{1+\Theta(1)}$, any dictionary with $O(\log^{(k)} n)$ wasted bits per key must have expected operational time $\Omega(k)$, in the cell-probe model with word-size $w=\Theta(\log U)$. Furthermore, if a dictionary stores values of $\Theta(\log U)$ bits, we show that regardless of the query time, it must have $\Omega(k)$ expected update time. It is worth noting that this is the first cell-probe lower bound on the trade-off between space and update time for general data structures.
Cold-start problems are long-standing challenges for practical recommendations. Most existing recommendation algorithms rely on extensive observed data and are brittle to recommendation scenarios with few interactions. This paper addresses such problems using few-shot learning and meta learning. Our approach is based on the insight that having a good generalization from a few examples relies on both a generic model initialization and an effective strategy for adapting this model to newly arising tasks. To accomplish this, we combine the scenario-specific learning with a model-agnostic sequential meta-learning and unify them into an integrated end-to-end framework, namely Scenario-specific Sequential Meta learner (or s^2 meta). By doing so, our meta-learner produces a generic initial model through aggregating contextual information from a variety of prediction tasks while effectively adapting to specific tasks by leveraging learning-to-learn knowledge. Extensive experiments on various real-world datasets demonstrate that our proposed model can achieve significant gains over the state-of-the-arts for cold-start problems in online recommendation. Deployment is at the Guess You Like session, the front page of the Mobile Taobao.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis, thereby allowing manual manipulation in predicting the final answer.
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