COLT 2019 Papers

In the past decade, sparse principal component analysis has emerged as an archetypal problem for illustrating statistical-computational tradeoffs. This trend has largely been driven by a line of research aiming to characterize the average-case complexity of sparse PCA through reductions from the planted clique ($\textsc{pc}$) conjecture. All previous reductions either fail to show tight computational lower bounds matching existing algorithms or show lower bounds for formulations of sparse PCA other than its canonical generative model, the spiked covariance model. Also, these lower bounds all quickly degrade given weak forms of the $\textsc{pc}$ conjecture. We give a reduction from $\textsc{pc}$ that yields the first full characterization of the computational barrier in the spiked covariance model, providing tight lower bounds at all sparsities $k$. We also show the surprising result that even a mild improvement in the signal strength needed by the best known polynomial-time sparse PCA algorithms would imply that the hardness threshold for $\textsc{pc}$ is $\text{polylog}(N)$, rather than on the order $N^{1/2}$ as is widely conjectured. This is the first instance of a suboptimal hardness assumption implying optimal lower bounds for another problem in unsupervised learning.

We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order queries required to optimize a convex Lipschitz function subject to a memory constraint.

We formalize a particular class of problems called \textit{survival multiarmed bandits} (S-MAB), which constitutes a modified version of \textit{budgeted multiarmed bandits} (B-MAB) where a true \textit{risk of ruin} must be considered, bringing it closer to \textit{risk-averse multiarmed bandits} (RA-MAB). In a S-MAB, pulling an arm can result in both positive and negative rewards. The agent has an initial budget that evolves in time with the received rewards. The goal is finding a good \textit{exploration-exploitation-safety} trade-off, maximizing rewards while minimizing the probability of getting ruined (i.e. hitting a negative budget). Such simple and until now neglected modification in the MAB statement changes the way to approach the problem, asking for adapted algorithms and specific analytical tools, and also make it more likely related to some important real-world applications. We are interested in the following open problems which stem from such new MAB definition: (a) how can the regret be meaningfully defined in formal terms for a S-MAB given its multiobjective optimization nature? (b) can a S-MAB be reduced to a RA-MAB or a B-MAB, transferring their theoretical guarantees? (c) what kind of method or strategy must an agent follow to optimally solve a S-MAB?

We pose the question to what extent a learning algorithm behaves monotonically in the following sense: does it perform better, in expectation, when adding one instance to the training set? We focus on empirical risk minimization and illustrate this property with several examples, two where it does hold and two where it does not. We also relate it to the notion of PAC-learnability.

We ask whether every class of Boolean functions that has polynomial margin complexity can be PAC learned efficiently by a non-interactive locally differentially private algorithm.

We ask how many measurements of the accuracy on a multiclass benchmark are needed to achieve a given amount of overfitting.

Folklore results in the theory of Stochastic Approximation indicates the (minimax) optimality of Stochastic Gradient Descent (SGD) (Robbins and Monro, 1951) with polynomially decaying stepsizes and iterate averaging (Ruppert, 1988; Polyak and Juditsky, 1992) for classes of stochastic convex optimization. Basing of these folkore results and some recent developments, this manuscript considers a more subtle question: does any algorithm necessarily (information theoretically) have to query iterates that are sub-optimal infinitely often?

In this paper we focus on the problem of Online Principal Component Analysis in the regret minimization framework. For this problem, all existing regret minimization algorithms for the fully-adversarial setting are based on a positive semidefinite convex relaxation, and hence require quadratic memory and SVD computation (either thin of full) on each iteration, which amounts to at least quadratic runtime per iteration. This is in stark contrast to a corresponding stochastic i.i.d. variant of the problem, which was studied extensively lately, and admits very efficient gradient ascent algorithms that work directly on the natural non-convex formulation of the problem, and hence require only linear memory and linear runtime per iteration. This raises the question: \textit{can non-convex online gradient ascent algorithms be shown to minimize regret in online adversarial settings?} In this paper we take a step forward towards answering this question. We introduce an \textit{adversarially-perturbed spiked-covariance model} in which, each data point is assumed to follow a fixed stochastic distribution with a non-zero spectral gap in the covariance matrix, but is then perturbed with some adversarial vector. This model is a natural extension of a well studied standard \textit{stochastic} setting that allows for non-stationary (adversarial) patterns to arise in the data and hence, might serve as a significantly better approximation for real-world data-streams. We show that in an interesting regime of parameters, when the non-convex online gradient ascent algorithm is initialized with a “warm-start" vector, it provably minimizes the regret with high probability. We further discuss the possibility of computing such a “warm-start" vector, and also the use of regularization to obtain fast regret rates. Our theoretical findings are supported by empirical experiments on both synthetic and real-world data.

We study the logistic bandit, in which rewards are binary with success probability $\exp(\beta a^\top \theta) / (1 + \exp(\beta a^\top \theta))$ and actions $a$ and coefficients $\theta$ are within the $d$-dimensional unit ball. While prior regret bounds for algorithms that address the logistic bandit exhibit exponential dependence on the slope parameter $\beta$, we establish a regret bound for Thompson sampling that is independent of $\beta$. Specifically, we establish that, when the set of feasible actions is identical to the set of possible coefficient vectors, the Bayesian regret of Thompson sampling is $\tilde{O}(d\sqrt{T})$. We also establish a $\tilde{O}(\sqrt{d\eta T}/\Delta)$ bound that applies more broadly, where $\Delta$ is the worst-case optimal log-odds and $\eta$ is the “fragility dimension,” a new statistic we define to capture the degree to which an optimal action for one model fails to satisfice for others. We demonstrate that the fragility dimension plays an essential role by showing that, for any $\epsilon > 0$, no algorithm can achieve $\mathrm{poly}(d, 1/\Delta)\cdot T^{1-\epsilon}$ regret.

We prove that the evolution of weight vectors in online gradient descent can encode arbitrary polynomial-space computations, even in very simple learning settings. Our results imply that, under weak complexity-theoretic assumptions, it is impossible to reason efficiently about the fine-grained behavior of online gradient descent.

We study the problem of mean estimation for high-dimensional distributions given access to a statistical query oracle. For a normed space $X = (\mathbb{R}^d, \|\cdot\|_X)$ and a distribution supported on vectors $x \in \mathbb{R}^d$ with $\|x\|_{X} \leq 1$, the task is to output an estimate $\hat{\mu} \in \mathbb{R}^d$ which is $\varepsilon$-close in the distance induced by $\|\cdot\|_X$ to the true mean of the distribution. We obtain sharp upper and lower bounds for the statistical query complexity of this problem when the the underlying norm is \emph{symmetric} as well as for Schatten-$p$ norms, answering two questions raised by Feldman, Guzmán, and Vempala (SODA 2017).

This work studies distributed learning in the spirit of Yao’s model of communication complexity: consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform some learning task. To naturally fit into the framework of learning theory, the players can send each other examples (as well as bits) where each example/bit costs one unit of communication. This enables a uniform treatment of infinite classes such as half-spaces in $\R^d$, which are ubiquitous in machine learning. We study several fundamental questions in this model. For example, we provide combinatorial characterizations of the classes that can be learned with efficient communication in the proper-case as well as in the improper-case. These findings imply unconditional separations in this context between various learning tasks, e.g. realizable versus agnostic learning, proper versus improper learning, etcetera. %They also imply lower bounds that match the performance %of algorithm from previous works. The derivation of these results hinges on a type of decision problems we term “{\it realizability problems}” where the goal is deciding whether a distributed input sample is consistent with an hypothesis from a pre-specified class. From a technical perspective, the protocols we devise (i.e. the upper bounds) are based on ideas from machine learning and the impossibility results (i.e. the lower bounds) are based on ideas from communication complexity.

We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein?s method and Lindeberg?s argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense

The Langevin Markov chain algorithms are widely deployed methods to sample from distributions in challenging high-dimensional and non-convex statistics and machine learning applications. Despite this, current bounds for the Langevin algorithms are worse than those of competing algorithms in many important situations, for instance when sampling from weakly log-concave distributions, or when sampling or optimizing non-convex log-densities. We obtain improved bounds in many of these situations, showing that the Metropolis-adjusted Langevin algorithm (MALA) is faster than the best bounds for its competitor algorithms when the target distribution satisfies weak third- and fourth- order regularity properties associated with the input data. In many settings, our regularity conditions are weaker than the usual Euclidean operator norm regularity properties, allowing us to show faster bounds for a much larger class of distributions than would be possible with the usual Euclidean operator norm approach, including in statistics and machine learning applications where the data satisfy a certain incoherence condition. In particular, we show that using our regularity conditions one can obtain faster bounds for applications which include sampling problems in Bayesian logistic regression with weakly convex priors, and the nonconvex optimization problem of learning linear classifiers with zero-one loss functions. Our main technical contribution is an analysis of the Metropolis acceptance probability of MALA in terms of its “energy-conservation error," and a bound for this error in terms of third- and fourth- order regularity conditions. The combination of this higher-order analysis of the energy conservation error with the conductance method is key to obtaining bounds which have a sub-linear dependence on the dimension $d$ in the non-strongly logconcave setting.

Stochastic approximation (SA) is a key method used in statistical learning. Recently, its non-asymptotic convergence analysis has been considered in many papers. However, most of the prior analyses are made under restrictive assumptions such as unbiased gradient estimates and convex objective function, which significantly limit their applications to sophisticated tasks such as online and reinforcement learning. These restrictions are all essentially relaxed in this work. In particular, we analyze a general SA scheme to minimize a non-convex, smooth objective function. We consider update procedure whose drift term depends on a state-dependent Markov chain and the mean field is not necessarily of gradient type, covering approximate second-order method and allowing asymptotic bias for the one-step updates. We illustrate these settings with the online EM algorithm and the policy-gradient method for average reward maximization in reinforcement learning.

We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $\Omega(\sqrt{dT \log T \log n})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results by one $\sqrt{\log T}$ factor, revealing a regret scaling quite different from classical multi-armed bandits in which no logarithmic $T$ term is present in minimax regret. Our proof techniques include variable confidence levels and a careful analysis of layer sizes of SupLinUCB on the upper bound side, and delicately constructed adversarial sequences showing the tightness of elliptical potential lemmas on the lower bound side.

We propose a near-optimal method for highly smooth convex optimization. More precisely, in the oracle model where one obtains the $p^{th}$ order Taylor expansion of a function at the query point, we propose a method with rate of convergence $\tilde{O}(1/k^{\frac{ 3p +1}{2}})$ after $k$ queries to the oracle for any convex function whose $p^{th}$ order derivative is Lipschitz.

In this merged paper, we consider the problem of minimizing a convex function with Lipschitz-continuous $p$-th order derivatives. Given an oracle which when queried at a point returns the first $p$-derivatives of the function at that point we provide some methods which compute an $\e$ approximate minimizer in $O\left(\e^{-\frac{2}{3p+1}} \right)$ iterations. These methods match known lower bounds up to polylogarithmic factors for constant $p$.

We study a strategic version of the multi-armed bandit problem, where each arm is an individual strategic agent and we, the principal, pull one arm each round. When pulled, the arm receives some private reward $v_a$ and can choose an amount $x_a$ to pass on to the principal (keeping $v_a-x_a$ for itself). All non-pulled arms get reward $0$. Each strategic arm tries to maximize its own utility over the course of $T$ rounds. Our goal is to design an algorithm for the principal incentivizing these arms to pass on as much of their private rewards as possible. When private rewards are stochastically drawn each round ($v_a^t \leftarrow D_a$), we show that: \begin{itemize} \item Algorithms that perform well in the classic adversarial multi-armed bandit setting necessarily perform poorly: For all algorithms that guarantee low regret in an adversarial setting, there exist distributions $D_1,\ldots,D_k$ and an $o(T)$-approximate Nash equilibrium for the arms where the principal receives reward $o(T)$. \item There exists an algorithm for the principal that induces a game among the arms where each arm has a dominant strategy. Moreover, for every $o(T)$-approximate Nash equilibrium, the principal receives expected reward $\mu’T - o(T)$, where $\mu’$ is the second-largest of the means $\mathbb{E}[D_{a}]$. This algorithm maintains its guarantee if the arms are non-strategic ($x_a = v_a$), and also if there is a mix of strategic and non-strategic arms. \end{itemize}