In this paper, we consider the counting function $E_P(y) = |P_{y} \cap Z^{n_x}|$ for a parametric polyhedron $P_{y} = \{x \in R^{n_x} \colon A x \leq b + B y\}$, where $y \in R^{n_y}$. We give a new representation of $E_P(y)$, called a \emph{piece-wise step-polynomial with periodic coefficients}, which is a generalization of piece-wise step-polynomials and integer/rational Ehrhart's quasi-polynomials. In terms of the computational complexity, our result gives the fastest way to calculate $E_P(y)$ in certain scenarios. The most remarkable cases are the following: 1) Consider a parametric polyhedron $P_y$ defined by a standard-form system $A x = y,\, x \geq 0$ with a fixed number of equalities. We show that there exists an $poly\bigl(n, \|A\|_{\infty}\bigr)$ preprocessing-algorithm that returns a polynomial-time computable representation of $E_P(y)$. That is, $E_(y)$ can be computed by a polynomial-time algorithm for any given $y \in Q^k$; 2) Again, assuming that the co-dimension is fixed, we show that integer/rational Ehrhart's quasi-polynomials of a polytope can be computed by FPT-algorithms, parameterized by sub-determinants of $A$ or its elements; 3) Our representation of $E_P(y)$ is more efficient than other known approaches, if the matrix $A$ has bounded elements, especially if the matrix $A$ is sparse in addition; Additionally, we provide a discussion about possible applications in the area of compiler optimization. In some "natural" assumptions on a program code, our approach has the fastest complexity bounds.
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We study dynamic algorithms in the model of algorithms with predictions. We assume the algorithm is given imperfect predictions regarding future updates, and we ask how such predictions can be used to improve the running time. This can be seen as a model interpolating between classic online and offline dynamic algorithms. Our results give smooth tradeoffs between these two extreme settings. First, we give algorithms for incremental and decremental transitive closure and approximate APSP that take as an additional input a predicted sequence of updates (edge insertions, or edge deletions, respectively). They preprocess it in $\tilde{O}(n^{(3+\omega)/2})$ time, and then handle updates in $\tilde{O}(1)$ worst-case time and queries in $\tilde{O}(\eta^2)$ worst-case time. Here $\eta$ is an error measure that can be bounded by the maximum difference between the predicted and actual insertion (deletion) time of an edge, i.e., by the $\ell_\infty$-error of the predictions. The second group of results concerns fully dynamic problems with vertex updates, where the algorithm has access to a predicted sequence of the next $n$ updates. We show how to solve fully dynamic triangle detection, maximum matching, single-source reachability, and more, in $O(n^{\omega-1}+n\eta_i)$ worst-case update time. Here $\eta_i$ denotes how much earlier the $i$-th update occurs than predicted. Our last result is a reduction that transforms a worst-case incremental algorithm without predictions into a fully dynamic algorithm which is given a predicted deletion time for each element at the time of its insertion. As a consequence we can, e.g., maintain fully dynamic exact APSP with such predictions in $\tilde{O}(n^2)$ worst-case vertex insertion time and $\tilde{O}(n^2 (1+\eta_i))$ worst-case vertex deletion time (for the prediction error $\eta_i$ defined as above).
We consider two-player zero-sum stochastic games and propose a two-timescale $Q$-learning algorithm with function approximation that is payoff-based, convergent, rational, and symmetric between the two players. In two-timescale $Q$-learning, the fast-timescale iterates are updated in spirit to the stochastic gradient descent and the slow-timescale iterates (which we use to compute the policies) are updated by taking a convex combination between its previous iterate and the latest fast-timescale iterate. Introducing the slow timescale as well as its update equation marks as our main algorithmic novelty. In the special case of linear function approximation, we establish, to the best of our knowledge, the first last-iterate finite-sample bound for payoff-based independent learning dynamics of these types. The result implies a polynomial sample complexity to find a Nash equilibrium in such stochastic games. To establish the results, we model our proposed algorithm as a two-timescale stochastic approximation and derive the finite-sample bound through a Lyapunov-based approach. The key novelty lies in constructing a valid Lyapunov function to capture the evolution of the slow-timescale iterates. Specifically, through a change of variable, we show that the update equation of the slow-timescale iterates resembles the classical smoothed best-response dynamics, where the regularized Nash gap serves as a valid Lyapunov function. This insight enables us to construct a valid Lyapunov function via a generalized variant of the Moreau envelope of the regularized Nash gap. The construction of our Lyapunov function might be of broad independent interest in studying the behavior of stochastic approximation algorithms.
The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.
This paper presents $\textbf{R}$epresentation-$\textbf{C}$onditioned image $\textbf{G}$eneration (RCG), a simple yet effective image generation framework which sets a new benchmark in class-unconditional image generation. RCG does not condition on any human annotations. Instead, it conditions on a self-supervised representation distribution which is mapped from the image distribution using a pre-trained encoder. During generation, RCG samples from such representation distribution using a representation diffusion model (RDM), and employs a pixel generator to craft image pixels conditioned on the sampled representation. Such a design provides substantial guidance during the generative process, resulting in high-quality image generation. Tested on ImageNet 256$\times$256, RCG achieves a Frechet Inception Distance (FID) of 3.31 and an Inception Score (IS) of 253.4. These results not only significantly improve the state-of-the-art of class-unconditional image generation but also rival the current leading methods in class-conditional image generation, bridging the long-standing performance gap between these two tasks. Code is available at //github.com/LTH14/rcg.
We present a best-response based algorithm for computing verifiable $\varepsilon$-perfect Bayesian equilibria for sequential auctions with combinatorial bidding spaces and incomplete information. Previous work has focused only on computing Bayes-Nash equilibria for static single-round auctions, which our work captures as a special case. Additionally, we prove an upper bound $\varepsilon$ on the utility loss of our approximate equilibria and present an algorithm to efficiently compute $\varepsilon$ based on the immediate loss at each subgame. We evaluate the performance of our algorithm by reproducing known results from several auctions previously introduced in the literature, including a model of combinatorial split-award auctions used in procurement.
Estimation of quantum relative entropy and its R\'{e}nyi generalizations is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of these divergences have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fr\'{e}chet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor's theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.
We propose a new framework to design and analyze accelerated methods that solve general monotone equation (ME) problems $F(x)=0$. Traditional approaches include generalized steepest descent methods and inexact Newton-type methods. If $F$ is uniformly monotone and twice differentiable, these methods achieve local convergence rates while the latter methods are globally convergent thanks to line search and hyperplane projection. However, a global rate is unknown for these methods. The variational inequality methods can be applied to yield a global rate that is expressed in terms of $\|F(x)\|$ but these results are restricted to first-order methods and a Lipschitz continuous operator. It has not been clear how to obtain global acceleration using high-order Lipschitz continuity. This paper takes a continuous-time perspective where accelerated methods are viewed as the discretization of dynamical systems. Our contribution is to propose accelerated rescaled gradient systems and prove that they are equivalent to closed-loop control systems. Based on this connection, we establish the properties of solution trajectories. Moreover, we provide a unified algorithmic framework obtained from discretization of our system, which together with two approximation subroutines yields both existing high-order methods and new first-order methods. We prove that the $p^{th}$-order method achieves a global rate of $O(k^{-p/2})$ in terms of $\|F(x)\|$ if $F$ is $p^{th}$-order Lipschitz continuous and the first-order method achieves the same rate if $F$ is $p^{th}$-order strongly Lipschitz continuous. If $F$ is strongly monotone, the restarted versions achieve local convergence with order $p$ when $p \geq 2$. Our discrete-time analysis is largely motivated by the continuous-time analysis and demonstrates the fundamental role that rescaled gradients play in global acceleration for solving ME problems.
We investigate algorithms for testing whether an image is connected. Given a proximity parameter $\epsilon\in(0,1)$ and query access to a black-and-white image represented by an $n\times n$ matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is $\epsilon$-far from connected. We show that connectedness can be tested nonadaptively with $O(\frac 1{\epsilon^2})$ queries and adaptively with $O(\frac{1}{\epsilon^{3/2}} \sqrt{\log\frac{1}{\epsilon}})$ queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity $O(\frac 1{\epsilon^2}\log \frac 1{\epsilon})$ and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make $\Omega(\frac 1\epsilon\log \frac 1\epsilon)$ queries.
We present TIGERScore, a \textbf{T}rained metric that follows \textbf{I}nstruction \textbf{G}uidance to perform \textbf{E}xplainable, and \textbf{R}eference-free evaluation over a wide spectrum of text generation tasks. Different from other automatic evaluation methods that only provide arcane scores, TIGERScore is guided by natural language instruction to provide error analysis to pinpoint the mistakes in the generated text. Our metric is based on LLaMA-2, trained on our meticulously curated instruction-tuning dataset MetricInstruct which covers 6 text generation tasks and 23 text generation datasets. The dataset consists of 42K quadruple in the form of (instruction, input, system output $\rightarrow$ error analysis). We collected the `system outputs' through from a large variety of models to cover different types of errors. To quantitatively assess our metric, we evaluate its correlation with human ratings on 5 held-in datasets, 2 held-out datasets and show that TIGERScore can achieve the open-source SoTA correlation with human ratings across these datasets and almost approaches GPT-4 evaluator. As a reference-free metric, its correlation can even surpass the best existing reference-based metrics. To further qualitatively assess the rationale generated by our metric, we conduct human evaluation on the generated explanations and found that the explanations are 70.8\% accurate. Through these experimental results, we believe TIGERScore demonstrates the possibility of building universal explainable metrics to evaluate any text generation task.
It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.
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