arXiv subjects

Hamed Hassani

Publications and source records attributed to Hamed Hassani.

140 records · Page 8Linked to original sources

arXiv2018

Stochastic Conditional Gradient Methods: From Convex Minimization to Submodular Maximization

This paper considers stochastic optimization problems for a large class of objective functions, including convex and continuous submodular. Stochastic proximal gradient methods have been widely used to solve such problems; however, their applicability remains limited when the problem dimension is large and the projection onto a convex set is costly. Instead, stochastic conditional gradient methods are proposed as an alternative solution relying on (i) Approximating gradients via a simple averaging technique requiring a single stochastic gradient evaluation per iteration; (ii) Solving a linear program to compute the descent/ascent direction. The averaging technique reduces the noise of gradient approximations as time progresses, and replacing projection step in proximal methods by a linear program lowers the computational complexity of each iteration. We show that under convexity and smoothness assumptions, our proposed method converges to the optimal objective function value at a sublinear rate of $O(1/t^{1/3})$. Further, for a monotone and continuous DR-submodular function and subject to a general convex body constraint, we prove that our proposed method achieves a $((1-1/e)OPT-\eps)$ guarantee with $O(1/\eps^3)$ stochastic gradient computations. This guarantee matches the known hardness results and closes the gap between deterministic and stochastic continuous submodular maximization. Additionally, we obtain $((1/e)OPT -\eps)$ guarantee after using $O(1/\eps^3)$ stochastic gradients for the case that the objective function is continuous DR-submodular but non-monotone and the constraint set is down-closed. By using stochastic continuous optimization as an interface, we provide the first $(1-1/e)$ tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a matroid constraint and $(1/e)$ approximation guarantee for the non-monotone case.

math.OC↗

arXiv2018

Latency-Reliability Tradeoffs for State Estimation

The emerging interest in low-latency high-reliability applications, such as connected vehicles, necessitates a new abstraction between communication and control. Thanks to advances in cyber-physical systems over the past decades, we understand this interface for classical bit-rate models of channels as well as packet-loss-type channels. This work proposes a new abstraction characterized as a tradeoff curve between latency, reliability and rate. Our aim is to understand: Do we (control engineers) prefer faster but less reliable communications (with shorter codes), or slower but more reliable communications (with longer codes)? In this paper we examine the tradeoffs between latency and reliability for the problem of estimating dynamical systems over communication channels. Employing different latency-reliability curves derived from practical coding schemes, we develop a co-design methodology, i.e., select the code length depending on the system dynamics to optimize system performance.

math.OC↗

arXiv2018

Discrete Sampling using Semigradient-based Product Mixtures

We consider the problem of inference in discrete probabilistic models, that is, distributions over subsets of a finite ground set. These encompass a range of well-known models in machine learning, such as determinantal point processes and Ising models. Locally-moving Markov chain Monte Carlo algorithms, such as the Gibbs sampler, are commonly used for inference in such models, but their convergence is, at times, prohibitively slow. This is often caused by state-space bottlenecks that greatly hinder the movement of such samplers. We propose a novel sampling strategy that uses a specific mixture of product distributions to propose global moves and, thus, accelerate convergence. Furthermore, we show how to construct such a mixture using semigradient information. We illustrate the effectiveness of combining our sampler with existing ones, both theoretically on an example model, as well as practically on three models learned from real-world data sets.

cs.LG↗

arXiv2018

Projection-Free Online Optimization with Stochastic Gradient: From Convexity to Submodularity

Online optimization has been a successful framework for solving large-scale problems under computational constraints and partial information. Current methods for online convex optimization require either a projection or exact gradient computation at each step, both of which can be prohibitively expensive for large-scale applications. At the same time, there is a growing trend of non-convex optimization in machine learning community and a need for online methods. Continuous DR-submodular functions, which exhibit a natural diminishing returns condition, have recently been proposed as a broad class of non-convex functions which may be efficiently optimized. Although online methods have been introduced, they suffer from similar problems. In this work, we propose Meta-Frank-Wolfe, the first online projection-free algorithm that uses stochastic gradient estimates. The algorithm relies on a careful sampling of gradients in each round and achieves the optimal $O( \sqrt{T})$ adversarial regret bounds for convex and continuous submodular optimization. We also propose One-Shot Frank-Wolfe, a simpler algorithm which requires only a single stochastic gradient estimate in each round and achieves an $O(T^{2/3})$ stochastic regret bound for convex and continuous submodular optimization. We apply our methods to develop a novel "lifting" framework for the online discrete submodular maximization and also see that they outperform current state-of-the-art techniques on various experiments.

stat.ML↗

arXiv2018

Online Continuous Submodular Maximization

In this paper, we consider an online optimization process, where the objective functions are not convex (nor concave) but instead belong to a broad class of continuous submodular functions. We first propose a variant of the Frank-Wolfe algorithm that has access to the full gradient of the objective functions. We show that it achieves a regret bound of $O(\sqrt{T})$ (where $T$ is the horizon of the online optimization problem) against a $(1-1/e)$-approximation to the best feasible solution in hindsight. However, in many scenarios, only an unbiased estimate of the gradients are available. For such settings, we then propose an online stochastic gradient ascent algorithm that also achieves a regret bound of $O(\sqrt{T})$ regret, albeit against a weaker $1/2$-approximation to the best feasible solution in hindsight. We also generalize our results to $γ$-weakly submodular functions and prove the same sublinear regret bounds. Finally, we demonstrate the efficiency of our algorithms on a few problem instances, including non-convex/non-concave quadratic programs, multilinear extensions of submodular set functions, and D-optimal design.

stat.ML↗

arXiv2018

Decentralized Submodular Maximization: Bridging Discrete and Continuous Settings

In this paper, we showcase the interplay between discrete and continuous optimization in network-structured settings. We propose the first fully decentralized optimization method for a wide class of non-convex objective functions that possess a diminishing returns property. More specifically, given an arbitrary connected network and a global continuous submodular function, formed by a sum of local functions, we develop Decentralized Continuous Greedy (DCG), a message passing algorithm that converges to the tight (1-1/e) approximation factor of the optimum global solution using only local computation and communication. We also provide strong convergence bounds as a function of network size and spectral characteristics of the underlying topology. Interestingly, DCG readily provides a simple recipe for decentralized discrete submodular maximization through the means of continuous relaxations. Formally, we demonstrate that by lifting the local discrete functions to continuous domains and using DCG as an interface we can develop a consensus algorithm that also achieves the tight (1-1/e) approximation guarantee of the global discrete solution once a proper rounding scheme is applied.

math.OC↗

arXiv2018

Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels

Consider a binary linear code of length $N$, minimum distance $d_{\text{min}}$, transmission over the binary erasure channel with parameter $0 < ε< 1$ or the binary symmetric channel with parameter $0 < ε< \frac12$, and block-MAP decoding. It was shown by Tillich and Zemor that in this case the error probability of the block-MAP decoder transitions "quickly" from $δ$ to $1-δ$ for any $δ>0$ if the minimum distance is large. In particular the width of the transition is of order $O(1/\sqrt{d_{\text{min}}})$. We strengthen this result by showing that under suitable conditions on the weight distribution of the code, the transition width can be as small as $Θ(1/N^{\frac12-κ})$, for any $κ>0$, even if the minimum distance of the code is not linear. This condition applies e.g., to Reed-Mueller codes. Since $Θ(1/N^{\frac12})$ is the smallest transition possible for any code, we speak of "almost" optimal scaling. We emphasize that the width of the transition says nothing about the location of the transition. Therefore this result has no bearing on whether a code is capacity-achieving or not. As a second contribution, we present a new estimate on the derivative of the EXIT function, the proof of which is based on the Blowing-Up Lemma.

cs.IT↗

arXiv2017

Stochastic Submodular Maximization: The Case of Coverage Functions

Stochastic optimization of continuous objectives is at the heart of modern machine learning. However, many important problems are of discrete nature and often involve submodular objectives. We seek to unleash the power of stochastic continuous optimization, namely stochastic gradient descent and its variants, to such discrete problems. We first introduce the problem of stochastic submodular optimization, where one needs to optimize a submodular objective which is given as an expectation. Our model captures situations where the discrete objective arises as an empirical risk (e.g., in the case of exemplar-based clustering), or is given as an explicit stochastic model (e.g., in the case of influence maximization in social networks). By exploiting that common extensions act linearly on the class of submodular functions, we employ projected stochastic gradient ascent and its variants in the continuous domain, and perform rounding to obtain discrete solutions. We focus on the rich and widely used family of weighted coverage functions. We show that our approach yields solutions that are guaranteed to match the optimal approximation guarantees, while reducing the computational cost by several orders of magnitude, as we demonstrate empirically.

cs.LG↗

arXiv2017

Conditional Gradient Method for Stochastic Submodular Maximization: Closing the Gap

In this paper, we study the problem of \textit{constrained} and \textit{stochastic} continuous submodular maximization. Even though the objective function is not concave (nor convex) and is defined in terms of an expectation, we develop a variant of the conditional gradient method, called \alg, which achieves a \textit{tight} approximation guarantee. More precisely, for a monotone and continuous DR-submodular function and subject to a \textit{general} convex body constraint, we prove that \alg achieves a $[(1-1/e)\text{OPT} -\eps]$ guarantee (in expectation) with $\mathcal{O}{(1/\eps^3)}$ stochastic gradient computations. This guarantee matches the known hardness results and closes the gap between deterministic and stochastic continuous submodular maximization. By using stochastic continuous optimization as an interface, we also provide the first $(1-1/e)$ tight approximation guarantee for maximizing a \textit{monotone but stochastic} submodular \textit{set} function subject to a general matroid constraint.

math.OC↗

arXiv2017

Gradient Methods for Submodular Maximization

In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor $1/2$ approximation to the global maxima. We also study stochastic gradient and mirror methods and show that after $\mathcal{O}(1/ε^2)$ iterations these methods reach solutions which achieve in expectation objective values exceeding $(\frac{\text{OPT}}{2}-ε)$. An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor $1/2$ approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient descent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort.

cs.LG↗

arXiv2017

Accelerated Dual Learning by Homotopic Initialization

Gradient descent and coordinate descent are well understood in terms of their asymptotic behavior, but less so in a transient regime often used for approximations in machine learning. We investigate how proper initialization can have a profound effect on finding near-optimal solutions quickly. We show that a certain property of a data set, namely the boundedness of the correlations between eigenfeatures and the response variable, can lead to faster initial progress than expected by commonplace analysis. Convex optimization problems can tacitly benefit from that, but this automatism does not apply to their dual formulation. We analyze this phenomenon and devise provably good initialization strategies for dual optimization as well as heuristics for the non-convex case, relevant for deep learning. We find our predictions and methods to be experimentally well-supported.

cs.LG↗

arXiv2017

Learning to Use Learners' Advice

In this paper, we study a variant of the framework of online learning using expert advice with limited/bandit feedback. We consider each expert as a learning entity, seeking to more accurately reflecting certain real-world applications. In our setting, the feedback at any time $t$ is limited in a sense that it is only available to the expert $i^t$ that has been selected by the central algorithm (forecaster), \emph{i.e.}, only the expert $i^t$ receives feedback from the environment and gets to learn at time $t$. We consider a generic black-box approach whereby the forecaster does not control or know the learning dynamics of the experts apart from knowing the following no-regret learning property: the average regret of any expert $j$ vanishes at a rate of at least $O(t_j^{\regretRate-1})$ with $t_j$ learning steps where $\regretRate \in [0, 1]$ is a parameter. In the spirit of competing against the best action in hindsight in multi-armed bandits problem, our goal here is to be competitive w.r.t. the cumulative losses the algorithm could receive by following the policy of always selecting one expert. We prove the following hardness result: without any coordination between the forecaster and the experts, it is impossible to design a forecaster achieving no-regret guarantees. In order to circumvent this hardness result, we consider a practical assumption allowing the forecaster to "guide" the learning process of the experts by filtering/blocking some of the feedbacks observed by them from the environment, \emph{i.e.}, not allowing the selected expert $i^t$ to learn at time $t$ for some time steps. Then, we design a novel no-regret learning algorithm \algo for this problem setting by carefully guiding the feedbacks observed by experts. We prove that \algo achieves the worst-case expected cumulative regret of $O(\Time^\frac{1}{2 - \regretRate})$ after $\Time$ time steps.

cs.LG↗

arXiv2017

Time-Invariant LDPC Convolutional Codes

Spatially coupled codes have been shown to universally achieve the capacity for a large class of channels. Many variants of such codes have been introduced to date. We discuss a further such variant that is particularly simple and is determined by a very small number of parameters. More precisely, we consider time-invariant low-density convolutional codes with very large constraint lengths. We show via simulations that, despite their extreme simplicity, such codes still show the threshold saturation behavior known from the spatially coupled codes discussed in the literature. Further, we show how the size of the typical minimum stopping set is related to basic parameters of the code. Due to their simplicity and good performance, these codes might be attractive from an implementation perspective.

cs.IT↗

arXiv2016

Near-Optimal Active Learning of Halfspaces via Query Synthesis in the Noisy Setting

In this paper, we consider the problem of actively learning a linear classifier through query synthesis where the learner can construct artificial queries in order to estimate the true decision boundaries. This problem has recently gained a lot of interest in automated science and adversarial reverse engineering for which only heuristic algorithms are known. In such applications, queries can be constructed de novo to elicit information (e.g., automated science) or to evade detection with minimal cost (e.g., adversarial reverse engineering). We develop a general framework, called dimension coupling (DC), that 1) reduces a d-dimensional learning problem to d-1 low dimensional sub-problems, 2) solves each sub-problem efficiently, 3) appropriately aggregates the results and outputs a linear classifier, and 4) provides a theoretical guarantee for all possible schemes of aggregation. The proposed method is proved resilient to noise. We show that the DC framework avoids the curse of dimensionality: its computational complexity scales linearly with the dimension. Moreover, we show that the query complexity of DC is near optimal (within a constant factor of the optimum algorithm). To further support our theoretical analysis, we compare the performance of DC with the existing work. We observe that DC consistently outperforms the prior arts in terms of query complexity while often running orders of magnitude faster.

cs.AI↗