An influential 1990 paper of Hochbaum and Shanthikumar made it common wisdom that "convex separable optimization is not much harder than linear optimization" [JACM 1990]. We exhibit two fundamental classes of mixed integer (linear) programs that run counter this intuition. Namely those whose constraint matrices have small coefficients and small primal or dual treedepth: While linear optimization is easy [Brand, Kouteck\'y, Ordyniak, AAAI 2021], we prove that separable convex optimization IS much harder. Moreover, in the pure integer and mixed integer linear cases, these two classes have the same parameterized complexity. We show that they yet behave quite differently in the separable convex mixed integer case. Our approach employs the mixed Graver basis introduced by Hemmecke [Math. Prog. 2003]. We give the first non-trivial lower and upper bounds on the norm of mixed Graver basis elements. In previous works involving the integer Graver basis, such upper bounds have consistently resulted in efficient algorithms for integer programming. Curiously, this does not happen in our case. In fact, we even rule out such an algorithm.
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We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. Focusing on a stochastic query model that provides noisy evaluations of the operator, we analyze a variance-reduced stochastic approximation scheme, and establish non-asymptotic bounds for both the operator defect and the estimation error, measured in an arbitrary semi-norm. In contrast to worst-case guarantees, our bounds are instance-dependent, and achieve the local asymptotic minimax risk non-asymptotically. For linear operators, contractivity can be relaxed to multi-step contractivity, so that the theory can be applied to problems like average reward policy evaluation problem in reinforcement learning. We illustrate the theory via applications to stochastic shortest path problems, two-player zero-sum Markov games, as well as policy evaluation and $Q$-learning for tabular Markov decision processes.
We study sparse linear regression over a network of agents, modeled as an undirected graph and no server node. The estimation of the $s$-sparse parameter is formulated as a constrained LASSO problem wherein each agent owns a subset of the $N$ total observations. We analyze the convergence rate and statistical guarantees of a distributed projected gradient tracking-based algorithm under high-dimensional scaling, allowing the ambient dimension $d$ to grow with (and possibly exceed) the sample size $N$. Our theory shows that, under standard notions of restricted strong convexity and smoothness of the loss functions, suitable conditions on the network connectivity and algorithm tuning, the distributed algorithm converges globally at a {\it linear} rate to an estimate that is within the centralized {\it statistical precision} of the model, $O(s\log d/N)$. When $s\log d/N=o(1)$, a condition necessary for statistical consistency, an $\varepsilon$-optimal solution is attained after $\mathcal{O}(\kappa \log (1/\varepsilon))$ gradient computations and $O (\kappa/(1-\rho) \log (1/\varepsilon))$ communication rounds, where $\kappa$ is the restricted condition number of the loss function and $\rho$ measures the network connectivity. The computation cost matches that of the centralized projected gradient algorithm despite having data distributed; whereas the communication rounds reduce as the network connectivity improves. Overall, our study reveals interesting connections between statistical efficiency, network connectivity \& topology, and convergence rate in high dimensions.
In the standard Gaussian linear measurement model $Y=X\mu_0+\xi \in \mathbb{R}^m$ with a fixed noise level $\sigma>0$, we consider the problem of estimating the unknown signal $\mu_0$ under a convex constraint $\mu_0 \in K$, where $K$ is a closed convex set in $\mathbb{R}^n$. We show that the risk of the natural convex constrained least squares estimator (LSE) $\hat{\mu}(\sigma)$ can be characterized exactly in high dimensional limits, by that of the convex constrained LSE $\hat{\mu}_K^{\mathsf{seq}}$ in the corresponding Gaussian sequence model at a different noise level. The characterization holds (uniformly) for risks in the maximal regime that ranges from constant order all the way down to essentially the parametric rate, as long as certain necessary non-degeneracy condition is satisfied for $\hat{\mu}(\sigma)$. The precise risk characterization reveals a fundamental difference between noiseless (or low noise limit) and noisy linear inverse problems in terms of the sample complexity for signal recovery. A concrete example is given by the isotonic regression problem: While exact recovery of a general monotone signal requires $m\gg n^{1/3}$ samples in the noiseless setting, consistent signal recovery in the noisy setting requires as few as $m\gg \log n$ samples. Such a discrepancy occurs when the low and high noise risk behavior of $\hat{\mu}_K^{\mathsf{seq}}$ differ significantly. In statistical languages, this occurs when $\hat{\mu}_K^{\mathsf{seq}}$ estimates $0$ at a faster `adaptation rate' than the slower `worst-case rate' for general signals. Several other examples, including non-negative least squares and generalized Lasso (in constrained forms), are also worked out to demonstrate the concrete applicability of the theory in problems of different types.
A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of $H$. Many $\mathsf{NP}$-complete problems, including problems such as Dominating Set, Feedback Vertex Set, Induced Matching and List $k$-Colouring, become polynomial-time solvable for ${\mathcal H}$-convex graphs when ${\mathcal H}$ is the set of paths. In this case, the class of ${\mathcal H}$-convex graphs is known as the class of convex graphs. The underlying reason is that the class of convex graphs has bounded mim-width. We extend the latter result to families of ${\mathcal H}$-convex graphs where (i) ${\mathcal H}$ is the set of cycles, or (ii) ${\mathcal H}$ is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least $3$. As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of ${\mathcal H}$-convex graphs is unbounded if ${\mathcal H}$ is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least $3$. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of ${\cal H}$-convex graphs.
We prove complexity bounds for the primal-dual algorithm with random extrapolation and coordinate descent (PURE-CD), which has been shown to obtain good practical performance for solving convex-concave min-max problems with bilinear coupling. Our complexity bounds either match or improve the best-known results in the literature for both dense and sparse (strongly)-convex-(strongly)-concave problems.
We consider infinite-horizon discounted Markov decision problems with finite state and action spaces. We show that with direct parametrization in the policy space, the weighted value function, although non-convex in general, is both quasi-convex and quasi-concave. While quasi-convexity helps explain the convergence of policy gradient methods to global optima, quasi-concavity hints at their convergence guarantees using arbitrarily large step sizes that are not dictated by the Lipschitz constant charactering smoothness of the value function. In particular, we show that when using geometrically increasing step sizes, a general class of policy mirror descent methods, including the natural policy gradient method and a projected Q-descent method, all enjoy a linear rate of convergence without relying on entropy or other strongly convex regularization. In addition, we develop a theory of weak gradient-mapping dominance and use it to prove sharper sublinear convergence rate of the projected policy gradient method. Finally, we also analyze the convergence rate of an inexact policy mirror descent method and estimate its sample complexity under a simple generative model.
We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.
Deep reinforcement learning (RL) methods generally engage in exploratory behavior through noise injection in the action space. An alternative is to add noise directly to the agent's parameters, which can lead to more consistent exploration and a richer set of behaviors. Methods such as evolutionary strategies use parameter perturbations, but discard all temporal structure in the process and require significantly more samples. Combining parameter noise with traditional RL methods allows to combine the best of both worlds. We demonstrate that both off- and on-policy methods benefit from this approach through experimental comparison of DQN, DDPG, and TRPO on high-dimensional discrete action environments as well as continuous control tasks. Our results show that RL with parameter noise learns more efficiently than traditional RL with action space noise and evolutionary strategies individually.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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