Sampling from Gibbs distributions $p(x) \propto \exp(-V(x)/\varepsilon)$ and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. However, while efficient algorithms are known for convex potentials $V$, the situation is much more difficult in the non-convex case, where algorithms necessarily suffer from the curse of dimensionality in the worst case. For optimization, which can be seen as a low-temperature limit of sampling, it is known that smooth functions $V$ allow faster convergence rates. Specifically, for $m$-times differentiable functions in $d$ dimensions, the optimal rate for algorithms with $n$ function evaluations is known to be $O(n^{-m/d})$, where the constant can potentially depend on $m, d$ and the function to be optimized. Hence, the curse of dimensionality can be alleviated for smooth functions at least in terms of the convergence rate. Recently, it has been shown that similarly fast rates can also be achieved with polynomial runtime $O(n^{3.5})$, where the exponent $3.5$ is independent of $m$ or $d$. Hence, it is natural to ask whether similar rates for sampling and log-partition computation are possible, and whether they can be realized in polynomial time with an exponent independent of $m$ and $d$. We show that the optimal rates for sampling and log-partition computation are sometimes equal and sometimes faster than for optimization. We then analyze various polynomial-time sampling algorithms, including an extension of a recent promising optimization approach, and find that they sometimes exhibit interesting behavior but no near-optimal rates. Our results also give further insights on the relation between sampling, log-partition, and optimization problems.
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This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ($O(1 / k^2)$), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.
Performance of optimization on quadratic problems sensitively depends on the low-lying part of the spectrum. For large (effectively infinite-dimensional) problems, this part of the spectrum can often be naturally represented or approximated by power law distributions, resulting in power law convergence rates for iterative solutions of these problems by gradient-based algorithms. In this paper, we propose a new spectral condition providing tighter upper bounds for problems with power law optimization trajectories. We use this condition to build a complete picture of upper and lower bounds for a wide range of optimization algorithms -- Gradient Descent, Steepest Descent, Heavy Ball, and Conjugate Gradients -- with an emphasis on the underlying schedules of learning rate and momentum. In particular, we demonstrate how an optimally accelerated method, its schedule, and convergence upper bound can be obtained in a unified manner for a given shape of the spectrum. Also, we provide first proofs of tight lower bounds for convergence rates of Steepest Descent and Conjugate Gradients under spectral power laws with general exponents. Our experiments show that the obtained convergence bounds and acceleration strategies are not only relevant for exactly quadratic optimization problems, but also fairly accurate when applied to the training of neural networks.
We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient $A$ and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of $A$, fractional order $\alpha$ and the smoothness of the first initial condition, as well as to the properties of the equation's right-hand side $f(t)$. The resulting method possesses exponential convergence for positive sectorial $A$, any finite $t$, including $t = 0$, and the whole range $\alpha \in (0,2)$. It is suitable for a practically important case, when no knowledge of $f(t)$ is available outside the considered interval $t \in [0, T]$. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.
In this paper, we propose a gradient-based block coordinate descent (BCD-G) framework to solve the joint approximate diagonalization of matrices defined on the product of the complex Stiefel manifold and the special linear group. Instead of the cyclic fashion, we choose a block optimization based on the Riemannian gradient. To update the first block variable in the complex Stiefel manifold, we use the well-known line search descent method. To update the second block variable in the special linear group, based on four kinds of different elementary transformations, we construct three classes: GLU, GQU and GU, and then get three BCD-G algorithms: BCD-GLU, BCD-GQU and BCD-GU. We establish the global and weak convergence of these three algorithms using the \L{}ojasiewicz gradient inequality under the assumption that the iterates are bounded. We also propose a gradient-based Jacobi-type framework to solve the joint approximate diagonalization of matrices defined on the special linear group. As in the BCD-G case, using the GLU and GQU classes of elementary transformations, we focus on the Jacobi-GLU and Jacobi-GQU algorithms and establish their global and weak convergence. All the algorithms and convergence results described in this paper also apply to the real case.
We present a new general-purpose algorithm for learning classes of $[0,1]$-valued functions in a generalization of the prediction model, and prove a general upper bound on the expected absolute error of this algorithm in terms of a scale-sensitive generalization of the Vapnik dimension proposed by Alon, Ben-David, Cesa-Bianchi and Haussler. We give lower bounds implying that our upper bounds cannot be improved by more than a constant factor in general. We apply this result, together with techniques due to Haussler and to Benedek and Itai, to obtain new upper bounds on packing numbers in terms of this scale-sensitive notion of dimension. Using a different technique, we obtain new bounds on packing numbers in terms of Kearns and Schapire's fat-shattering function. We show how to apply both packing bounds to obtain improved general bounds on the sample complexity of agnostic learning. For each $\epsilon > 0$, we establish weaker sufficient and stronger necessary conditions for a class of $[0,1]$-valued functions to be agnostically learnable to within $\epsilon$, and to be an $\epsilon$-uniform Glivenko-Cantelli class. This is a manuscript that was accepted by JCSS, together with a correction.
Evolution strategy (ES) is one of promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally $L$-strongly convex functions with $U$-Lipschitz continuous gradient are derived as $\exp\left(-\Omega_{d\to\infty}\left(\frac{L}{d\cdot U}\right)\right)$ and $\exp\left(-\frac1d\right)$, respectively. Notably, any prior knowledge on the mathematical properties of the objective function such as Lipschitz constant is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require them.
The idea of embedding optimization problems into deep neural networks as optimization layers to encode constraints and inductive priors has taken hold in recent years. Most existing methods focus on implicitly differentiating Karush-Kuhn-Tucker (KKT) conditions in a way that requires expensive computations on the Jacobian matrix, which can be slow and memory-intensive. In this paper, we developed a new framework, named Alternating Differentiation (Alt-Diff), that differentiates optimization problems (here, specifically in the form of convex optimization problems with polyhedral constraints) in a fast and recursive way. Alt-Diff decouples the differentiation procedure into a primal update and a dual update in an alternating way. Accordingly, Alt-Diff substantially decreases the dimensions of the Jacobian matrix especially for optimization with large-scale constraints and thus increases the computational speed of implicit differentiation. We show that the gradients obtained by Alt-Diff are consistent with those obtained by differentiating KKT conditions. In addition, we propose to truncate Alt-Diff to further accelerate the computational speed. Under some standard assumptions, we show that the truncation error of gradients is upper bounded by the same order of variables' estimation error. Therefore, Alt-Diff can be truncated to further increase computational speed without sacrificing much accuracy. A series of comprehensive experiments validate the superiority of Alt-Diff.
Decentralized minimax optimization has been actively studied in the past few years due to its application in a wide range of machine learning models. However, the current theoretical understanding of its convergence rate is far from satisfactory since existing works only focus on the nonconvex-strongly-concave problem. This motivates us to study decentralized minimax optimization algorithms for the nonconvex-nonconcave problem. To this end, we develop two novel decentralized stochastic variance-reduced gradient descent ascent algorithms for the finite-sum nonconvex-nonconcave problem that satisfies the Polyak-{\L}ojasiewicz (PL) condition. In particular, our theoretical analyses demonstrate how to conduct local updates and perform communication to achieve the linear convergence rate. To the best of our knowledge, this is the first work achieving linear convergence rates for decentralized nonconvex-nonconcave problems. Finally, we verify the performance of our algorithms on both synthetic and real-world datasets. The experimental results confirm the efficacy of our algorithms.
On Bakhvalov-type mesh, uniform convergence analysis of finite element method for a 2-D singularly perturbed convection-diffusion problem with exponential layers is still an open problem. Previous attempts have been unsuccessful. The primary challenges are the width of the mesh subdomain in the layer adjacent to the transition point, the restriction of the Dirichlet boundary condition, and the structure of exponential layers. To address these challenges, a novel analysis technique is introduced for the first time, which takes full advantage of the characteristics of interpolation and the connection between the smooth function and the layer function on the boundary. Utilizing this technique in conjunction with a new interpolation featuring a simple structure, uniform convergence of optimal order k+1 under an energy norm can be proven for finite element method of any order k. Numerical experiments confirm our theoretical results.
Sorting is a fundamental algorithmic pre-processing technique which often allows to represent data more compactly and, at the same time, speeds up search queries on it. In this paper, we focus on the well-studied problem of sorting and indexing string sets. Since the introduction of suffix trees in 1973, dozens of suffix sorting algorithms have been described in the literature. In 2017, these techniques were extended to sets of strings described by means of finite automata: the theory of Wheeler graphs [Gagie et al., TCS'17] introduced automata whose states can be totally-sorted according to the co-lexicographic (co-lex in the following) order of the prefixes of words accepted by the automaton. More recently, in [Cotumaccio, Prezza, SODA'21] it was shown how to extend these ideas to arbitrary automata by means of partial co-lex orders. This work showed that a co-lex order of minimum width (thus optimizing search query times) on deterministic finite automata (DFAs) can be computed in $O(m^2 + n^{5/2})$ time, $m$ being the number of transitions and $n$ the number of states of the input DFA. In this paper, we exhibit new combinatorial properties of the minimum-width co-lex order of DFAs and exploit them to design faster prefix sorting algorithms. In particular, we describe two algorithms sorting arbitrary DFAs in $O(mn)$ and $O(n^2\log n)$ time, respectively, and an algorithm sorting acyclic DFAs in $O(m\log n)$ time. Within these running times, all algorithms compute also a smallest chain partition of the partial order (required to index the DFA). We present an experiment result to show that an optimized implementation of the $O(n^2\log n)$-time algorithm exhibits a nearly-linear behaviour on large deterministic pan-genomic graphs and is thus also of practical interest.
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