With the growth of large data as well as large-scale learning tasks, the need for efficient and robust linear system solvers is greater than ever. The randomized Kaczmarz method (RK) and similar stochastic iterative methods have received considerable recent attention due to their efficient implementation and memory footprint. These methods can tolerate streaming data, accessing only part of the data at a time, and can also approximate the least squares solution even if the system is affected by noise. However, when data is instead affected by large (possibly adversarial) corruptions, these methods fail to converge, as corrupted data points draw iterates far from the true solution. A recently proposed solution to this is the QuantileRK method, which avoids harmful corrupted data by exploring the space carefully as the method iterates. The exploration component requires the computation of quantiles of large samples from the system and is computationally much heavier than the subsequent iteration update. In this paper, we propose an approach that better uses the information obtained during exploration by incorporating an averaged version of the block Kaczmarz method. This significantly speeds up convergence, while still allowing for a constant fraction of the equations to be arbitrarily corrupted. We provide theoretical convergence guarantees as well as experimental supporting evidence. We also demonstrate that the classical projection-based block Kaczmarz method cannot be robust to sparse adversarial corruptions, but rather the blocking has to be carried out by averaging one-dimensional projections.
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The aim of this paper is to design computationally-efficient and optimal algorithms for the online and stochastic exp-concave optimization settings. Typical algorithms for these settings, such as the Online Newton Step (ONS), can guarantee a $O(d\ln T)$ bound on their regret after $T$ rounds, where $d$ is the dimension of the feasible set. However, such algorithms perform so-called generalized projections whenever their iterates step outside the feasible set. Such generalized projections require $\Omega(d^3)$ arithmetic operations even for simple sets such a Euclidean ball, making the total runtime of ONS of order $d^3 T$ after $T$ rounds, in the worst-case. In this paper, we side-step generalized projections by using a self-concordant barrier as a regularizer to compute the Newton steps. This ensures that the iterates are always within the feasible set without requiring projections. This approach still requires the computation of the inverse of the Hessian of the barrier at every step. However, using the stability properties of the Newton steps, we show that the inverse of the Hessians can be efficiently approximated via Taylor expansions for most rounds, resulting in a $O(d^2 T +d^\omega \sqrt{T})$ total computational complexity, where $\omega$ is the exponent of matrix multiplication. In the stochastic setting, we show that this translates into a $O(d^3/\epsilon)$ computational complexity for finding an $\epsilon$-suboptimal point, answering an open question by Koren 2013. We first show these new results for the simple case where the feasible set is a Euclidean ball. Then, to move to general convex set, we use a reduction to Online Convex Optimization over the Euclidean ball. Our final algorithm can be viewed as a more efficient version of ONS.
Incomplete covariate vectors are known to be problematic for estimation and inferences on model parameters, but their impact on prediction performance is less understood. We develop an imputation-free method that builds on a random partition model admitting variable-dimension covariates. Cluster-specific response models further incorporate covariates via linear predictors, facilitating estimation of smooth prediction surfaces with relatively few clusters. We exploit marginalization techniques of Gaussian kernels to analytically project response distributions according to any pattern of missing covariates, yielding a local regression with internally consistent uncertainty propagation that utilizes only one set of coefficients per cluster. Aggressive shrinkage of these coefficients regulates uncertainty due to missing covariates. The method allows in- and out-of-sample prediction for any missingness pattern, even if the pattern in a new subject's incomplete covariate vector was not seen in the training data. We develop an MCMC algorithm for posterior sampling that improves a computationally expensive update for latent cluster allocation. Finally, we demonstrate the model's effectiveness for nonlinear point and density prediction under various circumstances by comparing with other recent methods for regression of variable dimensions on synthetic and real data.
We consider the stochastic optimization problem with smooth but not necessarily convex objectives in the heavy-tailed noise regime, where the stochastic gradient's noise is assumed to have bounded $p$th moment ($p\in(1,2]$). Zhang et al. (2020) is the first to prove the $\Omega(T^{\frac{1-p}{3p-2}})$ lower bound for convergence (in expectation) and provides a simple clipping algorithm that matches this optimal rate. Cutkosky and Mehta (2021) proposes another algorithm, which is shown to achieve the nearly optimal high-probability convergence guarantee $O(\log(T/\delta)T^{\frac{1-p}{3p-2}})$, where $\delta$ is the probability of failure. However, this desirable guarantee is only established under the additional assumption that the stochastic gradient itself is bounded in $p$th moment, which fails to hold even for quadratic objectives and centered Gaussian noise. In this work, we first improve the analysis of the algorithm in Cutkosky and Mehta (2021) to obtain the same nearly optimal high-probability convergence rate $O(\log(T/\delta)T^{\frac{1-p}{3p-2}})$, without the above-mentioned restrictive assumption. Next, and curiously, we show that one can achieve a faster rate than that dictated by the lower bound $\Omega(T^{\frac{1-p}{3p-2}})$ with only a tiny bit of structure, i.e., when the objective function $F(x)$ is assumed to be in the form of $\mathbb{E}_{\Xi\sim\mathcal{D}}[f(x,\Xi)]$, arguably the most widely applicable class of stochastic optimization problems. For this class of problems, we propose the first variance-reduced accelerated algorithm and establish that it guarantees a high-probability convergence rate of $O(\log(T/\delta)T^{\frac{1-p}{2p-1}})$ under a mild condition, which is faster than $\Omega(T^{\frac{1-p}{3p-2}})$. Notably, even when specialized to the finite-variance case, our result yields the (near-)optimal high-probability rate $O(\log(T/\delta)T^{-1/3})$.
In building practical applications of evolutionary computation (EC), two optimizations are essential. First, the parameters of the search method need to be tuned to the domain in order to balance exploration and exploitation effectively. Second, the search method needs to be distributed to take advantage of parallel computing resources. This paper presents BLADE (BLAnket Distributed Evolution) as an approach to achieving both goals simultaneously. BLADE uses blankets (i.e., masks on the genetic representation) to tune the evolutionary operators during the search, and implements the search through hub-and-spoke distribution. In the paper, (1) the blanket method is formalized for the (1 + 1)EA case as a Markov chain process. Its effectiveness is then demonstrated by analyzing dominant and subdominant eigenvalues of stochastic matrices, suggesting a generalizable theory; (2) the fitness-level theory is used to analyze the distribution method; and (3) these insights are verified experimentally on three benchmark problems, showing that both blankets and distribution lead to accelerated evolution. Moreover, a surprising synergy emerges between them: When combined with distribution, the blanket approach achieves more than $n$-fold speedup with $n$ clients in some cases. The work thus highlights the importance and potential of optimizing evolutionary computation in practical applications.
Conscious states (states that there is something it is like to be in) seem both rich or full of detail, and ineffable or hard to fully describe or recall. The problem of ineffability, in particular, is a longstanding issue in philosophy that partly motivates the explanatory gap: the belief that consciousness cannot be reduced to underlying physical processes. Here, we provide an information theoretic dynamical systems perspective on the richness and ineffability of consciousness. In our framework, the richness of conscious experience corresponds to the amount of information in a conscious state and ineffability corresponds to the amount of information lost at different stages of processing. We describe how attractor dynamics in working memory would induce impoverished recollections of our original experiences, how the discrete symbolic nature of language is insufficient for describing the rich and high-dimensional structure of experiences, and how similarity in the cognitive function of two individuals relates to improved communicability of their experiences to each other. While our model may not settle all questions relating to the explanatory gap, it makes progress toward a fully physicalist explanation of the richness and ineffability of conscious experience: two important aspects that seem to be part of what makes qualitative character so puzzling.
The field of Automatic Machine Learning (AutoML) has recently attained impressive results, including the discovery of state-of-the-art machine learning solutions, such as neural image classifiers. This is often done by applying an evolutionary search method, which samples multiple candidate solutions from a large space and evaluates the quality of each candidate through a long training process. As a result, the search tends to be slow. In this paper, we show that large efficiency gains can be obtained by employing a fast unified functional hash, especially through the functional equivalence caching technique, which we also present. The central idea is to detect by hashing when the search method produces equivalent candidates, which occurs very frequently, and this way avoid their costly re-evaluation. Our hash is "functional" in that it identifies equivalent candidates even if they were represented or coded differently, and it is "unified" in that the same algorithm can hash arbitrary representations; e.g. compute graphs, imperative code, or lambda functions. As evidence, we show dramatic improvements on multiple AutoML domains, including neural architecture search and algorithm discovery. Finally, we consider the effect of hash collisions, evaluation noise, and search distribution through empirical analysis. Altogether, we hope this paper may serve as a guide to hashing techniques in AutoML.
In this paper, we study fast first-order algorithms that approximately solve linear programs (LPs). More specifically, we apply algorithms from online linear programming to offline LPs and derive algorithms that are free of any matrix multiplication. To further improve the applicability of the proposed methods, we propose a variable-duplication technique that achieves $\mathcal{O}(\sqrt{mn/K})$ optimality gap by copying each variable $K$ times. Moreover, we identify that online algorithms can be efficiently incorporated into a column generation framework for large-scale LPs. Finally, numerical experiments show that our proposed methods can be applied either as an approximate direct solver or as an initialization subroutine in frameworks of exact LP solving.
Efficient and accurate separable models for discrete material optimization: A continuous perspective
Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two -- at a first sight -- very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline phase. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.
We give query complexity lower bounds for convex optimization and the related feasibility problem. We show that quadratic memory is necessary to achieve the optimal oracle complexity for first-order convex optimization. In particular, this shows that center-of-mass cutting-planes algorithms in dimension $d$ which use $\tilde O(d^2)$ memory and $\tilde O(d)$ queries are Pareto-optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, we prove that to minimize $1$-Lipschitz convex functions over the unit ball to $1/d^4$ accuracy, any deterministic first-order algorithms using at most $d^{2-\delta}$ bits of memory must make $\tilde\Omega(d^{1+\delta/3})$ queries, for any $\delta\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-\delta}$ memory, the number of queries required is $\tilde\Omega(d^{1+\delta})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.
In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code length n, we can compute the weight distribution of any polar cosets in time O(n^2). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length n = 128.
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