Peter Bartlett's Publications (1993-2022)

Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0-1 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 0-1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function: that it satisfy a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise. Finally, we present applications of our results to the estimation of convergence rates in the general setting of function classes that are scaled convex hulls of a finite-dimensional base class, with a variety of commonly used loss functions.

In this paper, we review two families of algorithms used to estimate large-scale statistical models for prediction problems, kernel methods and boosting algorithms. We focus on the computational and statistical properties of prediction algorithms of this kind. Convexity plays an important role for these algorithms, since they exploit the computational advantages of convex optimization procedures. However, in addition to its computational advantages, the use of convexity in these methods also confers some attractive statistical properties. We present some recent results that show the advantages of convexity for estimation rates, the rates at which the prediction accuracies approach their optimal values. In addition, we present results that quantify the cost of using a convex loss function in place of the real loss function of interest.

We present sharp bounds on the risk of the empirical minimization algorithm under mild assumptions on the class. We introduce the notion of isomorphic coordinate projections and show that this leads to a sharper error bound than the best previously known. The quantity which governs this bound on the empirical minimizer is the fixed point of the function (r) = {|f - _n f|: finF, f = r }. We prove that this is the best estimate one can obtain using `structural results', and that it is possible to estimate the error rate from data. We then prove that the bound on the empirical minimization algorithm can be improved further by a direct analysis, and that the correct error rate is the maximizer of (r) -r, where (r) = {f - _n f: finF, f = r }.

One of the nice properties of kernel classifiers such as SVMs is that they often produce sparse solutions. However, the decision functions of these classifiers cannot always be used to estimate the conditional probability of the class label. We investigate the relationship between these two properties and show that these are intimately related: sparseness does not occur when the conditional probabilities can be unambiguously estimated. We consider a family of convex loss functions and derive sharp asymptotic bounds for the number of support vectors. This enables us to characterize the exact trade-off between sparseness and the ability to estimate conditional probabilities for these loss functions.

Many classification algorithms, including the support vector machine, boosting and logistic regression, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0-1 loss function. We characterize the statistical consequences of using such a surrogate by providing a general quantitative relationship between the risk as assessed using the 0-1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial bounds under the weakest possible condition on the loss function---that it satisfy a pointwise form of Fisher consistency for classification. The relationship is based on a variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise. Finally, we present applications of our results to the estimation of convergence rates in the general setting of function classes that are scaled hulls of a finite-dimensional base class.

We propose new bounds on the error of learning algorithms in terms of a data-dependent notion of complexity. The estimates we establish give optimal rates and are based on a local and empirical version of Rademacher averages, in the sense that the Rademacher averages are computed from the data, on a subset of functions with small empirical error. We present some applications to prediction with bounded loss, and to regression with a convex loss function and a convex function class.

We consider the problem of structured classification, where the task is to predict a label y from an input x, and y has meaningful internal structure. Our framework includes supervised training of both Markov random fields and weighted context-free grammars as special cases. We describe an algorithm that solves the large-margin optimization problem defined by Taskar et al, using an exponential-family (Gibbs distribution) representation of structured objects. The algorithm is efficient -- even in cases where the number of labels y is exponential in size -- provided that certain expectations under Gibbs distributions can becalculated efficiently. The optimization method we use for structured labels relies on a more general result, specifically the application of exponentiated gradient (EG) updates to quadratic programs (QPs). We describe a new method for solving QPs based on these techniques, and give bounds on its rate of convergence. In addition to their application to the structured-labels task, the EG updates lead to simple algorithms for optimizing “conventional” binary or multiclass SVM problems. Finally, we give a new generalization bound for structured classification, using PAC-Bayesian methods for the analysis of large margin classifiers.

We consider the problem of binary classification where the classifier can, for a particular cost, choose not to classify an observation. Just as in the conventional classification problem, minimization of the sample average of the cost is a difficult optimization problem. As an alternative, we propose the optimization of a certain convex loss function f, analogous to the hinge loss used in support vector machines (SVMs). Its convexity ensures that the sample average of this surrogate loss can be efficiently minimized. We study its statistical properties. We show that minimizing the expected surrogate loss---the f-risk---also minimizes the risk. We also study the rate at which the f-risk approaches its minimum value. We show that fast rates are possible when the conditional probability Pr(Y=1|X) is unlikely to be close to certain critical values.

We investigate the behavior of the empirical minimization algorithm using various methods. We first analyze it by comparing the empirical, random, structure and the original one on the class, either in an additive sense, via the uniform law of large numbers, or in a multiplicative sense, using isomorphic coordinate projections. We then show that a direct analysis of the empirical minimization algorithm yields a significantly better bound, and that the estimates we obtain are essentially sharp. The method of proof we use is based on Talagrand's concentration inequality for empirical processes.

Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0-1 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 0-1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function: that it satisfy a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise. Finally, we present applications of our results to the estimation of convergence rates in the general setting of function classes that are scaled convex hulls of a finite-dimensional base class, with a variety of commonly used loss functions.

Log-linear and maximum-margin models are two commonly used methods in supervised machine learning, and are frequently used in structured prediction problems. Efficient learning of parameters in these models is therefore an important problem, and becomes a key factor when learning from very large data sets. This paper describes exponentiated gradient (EG) algorithms for training such models, where EG updates are applied to the convex dual of either the log-linear or max-margin objective function; the dual in both the log-linear and max-margin cases corresponds to minimizing a convex function with simplex constraints. We study both batch and online variants of the algorithm, and provide rates of convergence for both cases. In the max-margin case, O(1ε) EG updates are required to reach a given accuracy ε in the dual; in contrast, for log-linear models only O(log( 1ε)) updates are required. For both the max-margin and log-linear cases, our bounds suggest that the online algorithm requires a factor of n less computation to reach a desired accuracy, where n is the number of training examples. Our experiments confirm that the online algorithms are much faster than the batch algorithms in practice. We describe how the EG updates factor in a convenient way for structured prediction problems, allowing the algorithms to be efficiently applied to problems such as sequence learning or natural language parsing. We perform extensive evaluation of the algorithms, comparing them to to L-BFGS and stochastic gradient descent for log-linear models, and to SVM-Struct for max-margin models. The algorithms are applied to multi-class problems as well as a more complex large-scale parsing task. In all these settings, the EG algorithms presented here outperform the other methods.

A number of learning problems can be cast as an Online Convex Game: on each round, a learner makes a prediction x from a convex set, the environment plays a loss function f, and the learner's long-term goal is to minimize regret. Algorithms have been proposed by Zinkevich, when f is assumed to be convex, and Hazan et al, when f is assumed to be strongly convex, that have provably low regret. We consider these two settings and analyze such games from a minimax perspective, proving lower bounds in each case. These results prove that the existing algorithms are essentially optimal.

We present worst-case log-loss regret bounds for the Bayesian model averaging algorithm in the regression setting. Our work generalizes earlier work that gives bounds of a similar form that only hold for Gaussian priors. Our bounds hold for arbitrary priors, though the regret term now includes a term involving the modulus of continuity of the prior, as well as the value of the prior at the point of comparison.

We present new expected risk bounds for binary and multiclass prediction, and resolve several recent conjectures on sample compressibility due to Kuzmin and Warmuth. By exploiting the combinatorial structure of concept class F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy. The key step in their proof is a d=VC(F) bound on the graph density of a subgraph of the hypercube - one-inclusion graph. The first main result of this report is a density bound of n choose(n-1,<=d-1)/choose(n,<=d) < d, which positively resolves a conjecture of Kuzmin and Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved one-inclusion mistake bound. The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is an algebraic topological property of maximum classes of VC-dimension d as being d-contractible simplicial complexes, extending the well-known characterization that d=1 maximum classes are trees. We negatively resolve a minimum degree conjecture of Kuzmin and Warmuth - the second part to a conjectured proof of correctness for Peeling - that every class has one-inclusion minimum degree at most its VC-dimension. Our final main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This result improves on known PAC-based expected risk bounds by a factor of O(log n) and is shown to be optimal up to a O(log k) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout.

The risk, or probability of error, of the classifier produced by the AdaBoost algorithm is investigated. In particular, we consider the stopping strategy to be used in AdaBoost to achieve universal consistency. We show that provided AdaBoost is stopped after n^(1-a) iterations---for sample size n and 0<a<1---the sequence of risks of the classifiers it produces approaches the Bayes risk.

We consider the problem of choosing, sequentially, a map which assigns elements of a set A to a few elements of a set B. On each round, the algorithm suffers some cost associated with the chosen assignment, and the goal is to minimize the cumulative loss of these choices relative to the best map on the entire sequence. Even though the offline problem of finding the best map is provably hard, we show that there is an equivalent online approximation algorithm, Randomized Map Prediction (RMP), that is efficient and performs nearly as well. While drawing upon results from the `Online Prediction with Expert Advice' setting, we show how RMP can be utilized as an online approach to several standard batch problems. We apply RMP to online clustering as well as online feature selection and, surprisingly, RMP often outperforms the standard batch algorithms on these problems.

We consider the problem of prediction with expert advice in the setting where a forecaster is presented with several online prediction tasks. Instead of competing against the best expert separately on each task, we assume the tasks are related, and thus we expect that a few experts will perform well on the entire set of tasks. That is, our forecaster would like, on each task, to compete against the best expert chosen from a small set of experts. While we describe the `ideal' algorithm and its performance bound, we show that the computation required for this algorithm is as hard as computation of a matrix permanent. We present an efficient algorithm based on mixing priors, and prove a bound that is nearly as good for the sequential task presentation case. We also consider a harder case where the task may change arbitrarily from round to round, and we develop an efficient randomized algorithm based on Markov chain Monte Carlo techniques.

The risk, or probability of error, of the classifier produced by the AdaBoost algorithm is investigated. In particular, we consider the stopping strategy to be used in AdaBoost to achieve universal consistency. We show that provided AdaBoost is stopped after n^(1-a) iterations---for sample size n and a>0---the sequence of risks of the classifiers it produces approaches the Bayes risk.

In the multi-view approach to semi-supervised learning, we choose one predictor from each of multiple hypothesis classes, and we `co-regularize' our choices by penalizing disagreement among the predictors on the unlabeled data. In this paper we examine the co-regularization method used in the recently proposed co-regularized least squares (CoRLS) algorithm. In this method we have two hypothesis classes, each a reproducing kernel Hilbert space (RKHS), and we co-regularize by penalizing the average squared difference in predictions on the unlabeled data. We get our final predictor by taking the pointwise average of the predictors from each view. We call the set of predictors that can result from this procedure the co-regularized hypothesis class. The main result of this paper is a tight bound on the Rademacher complexity of the co-regularized hypothesis class in terms of the kernel matrices of each RKHS. We find that the co-regularization reduces the Rademacher complexity of the hypothesis class by an amount depending on how different the two views are, measured by a data dependent metric. We then use standard techniques to bound the gap between training error and test error for the CoRLS algorithm. Experimentally, we find that the amount of reduction in complexity introduced by co-regularization correlates with the amount of improvement that co-regularization gives in the CoRLS algorithm

We consider complexity penalization methods for model selection. These methods aim to choose a model to optimally trade off estimation and approximation errors by minimizing the sum of an empirical risk term and a complexity penalty. It is well known that if we use a bound on the maximal deviation between empirical and true risks as a complexity penalty, then the risk of our choice is no more than the approximation error plus twice the complexity penalty. There are many cases, however, where complexity penalties like this give loose upper bounds on the estimation error. In particular, if we choose a function from a suitably simple convex function class with a strictly convex loss function, then the estimation error (the difference between the risk of the empirical risk minimizer and the minimal risk in the class) approaches zero at a faster rate than the maximal deviation between empirical and true risks. In this note, we address the question of whether it is possible to design a complexity penalized model selection method for these situations. We show that, provided the sequence of models is ordered by inclusion, in these cases we can use tight upper bounds on estimation error as a complexity penalty. Surprisingly, this is the case even in situations when the difference between the empirical risk and true risk (and indeed the error of any estimate of the approximation error) decreases much more slowly than the complexity penalty. We give an oracle inequality showing that the resulting model selection method chooses a function with risk no more than the approximation error plus a constant times the complexity penalty.

We study sample-based estimates of the expectation of the function produced by the empirical minimization algorithm. We investigate the extent to which one can estimate the rate of convergence of the empirical minimizer in a data dependent manner. We establish three main results. First, we provide an algorithm that upper bounds the expectation of the empirical minimizer in a completely data-dependent manner. This bound is based on a structural result in http://www.stat.berkeley.edu/~bartlett/papers/bm-em-03.pdf, which relates expectations to sample averages. Second, we show that these structural upper bounds can be loose. In particular, we demonstrate a class for which the expectation of the empirical minimizer decreases as O(1/n) for sample size n, although the upper bound based on structural properties is Ω(1). Third, we show that this looseness of the bound is inevitable: we present an example that shows that a sharp bound cannot be universally recovered from empirical data.

In this paper we examine the problem of prediction with expert advice in a setup where the learner is presented with a sequence of examples coming from different tasks. In order for the learner to be able to benefit from performing multiple tasks simultaneously, we make assumptions of task relatedness by constraining the comparator to use a lesser number of best experts than the number of tasks. We show how this corresponds naturally to learning under spectral or structural matrix constraints, and propose regularization techniques to enforce the constraints. The regularization techniques proposed here are interesting in their own right and multitask learning is just one application for the ideas. A theoretical analysis of one such regularizer is performed, and a regret bound that shows benefits of this setup is reported.

We consider complexity penalization methods for model selection. These methods aim to choose a model to optimally trade off estimation and approximation errors by minimizing the sum of an empirical risk term and a complexity penalty. It is well known that if we use a bound on the maximal deviation between empirical and true risks as a complexity penalty, then the risk of our choice is no more than the approximation error plus twice the complexity penalty. There are many cases, however, where complexity penalties like this give loose upper bounds on the estimation error. In particular, if we choose a function from a suitably simple convex function class with a strictly convex loss function, then the estimation error (the difference between the risk of the empirical risk minimizer and the minimal risk in the class) approaches zero at a faster rate than the maximal deviation between empirical and true risks. In this note, we address the question of whether it is possible to design a complexity penalized model selection method for these situations. We show that, provided the sequence of models is ordered by inclusion, in these cases we can use tight upper bounds on estimation error as a complexity penalty. Surprisingly, this is the case even in situations when the difference between the empirical risk and true risk (and indeed the error of any estimate of the approximation error) decreases much more slowly than the complexity penalty. We give an oracle inequality showing that the resulting model selection method chooses a function with risk no more than the approximation error plus a constant times the complexity penalty.

Log-linear and maximum-margin models are two commonly used methods in supervised machine learning, and are frequently used in structured prediction problems. Efficient learning of parameters in these models is therefore an important problem, and becomes a key factor when learning from very large data sets. This paper describes exponentiated gradient (EG) algorithms for training such models, where EG updates are applied to the convex dual of either the log-linear or max-margin objective function; the dual in both the log-linear and max-margin cases corresponds to minimizing a convex function with simplex constraints. We study both batch and online variants of the algorithm, and provide rates of convergence for both cases. In the max-margin case, O(1ε) EG updates are required to reach a given accuracy ε in the dual; in contrast, for log-linear models only O(log( 1ε)) updates are required. For both the max-margin and log-linear cases, our bounds suggest that the online algorithm requires a factor of n less computation to reach a desired accuracy, where n is the number of training examples. Our experiments confirm that the online algorithms are much faster than the batch algorithms in practice. We describe how the EG updates factor in a convenient way for structured prediction problems, allowing the algorithms to be efficiently applied to problems such as sequence learning or natural language parsing. We perform extensive evaluation of the algorithms, comparing them to to L-BFGS and stochastic gradient descent for log-linear models, and to SVM-Struct for max-margin models. The algorithms are applied to multi-class problems as well as a more complex large-scale parsing task. In all these settings, the EG algorithms presented here outperform the other methods.

We consider the problem of binary classification where the classifier can, for a particular cost, choose not to classify an observation. Just as in the conventional classification problem, minimization of the sample average of the cost is a difficult optimization problem. As an alternative, we propose the optimization of a certain convex loss function f, analogous to the hinge loss used in support vector machines (SVMs). Its convexity ensures that the sample average of this surrogate loss can be efficiently minimized. We study its statistical properties. We show that minimizing the expected surrogate loss---the f-risk---also minimizes the risk. We also study the rate at which the f-risk approaches its minimum value. We show that fast rates are possible when the conditional probability Pr(Y=1|X) is unlikely to be close to certain critical values.

We present an algorithm called Optimistic Linear Programming (OLP) for learning to optimize average reward in an irreducible but otherwise unknown Markov decision process (MDP). OLP uses its experience so far to estimate the MDP. It chooses actions by optimistically maximizing estimated future rewards over a set of next-state transition probabilities that are close to the estimates: a computation that corresponds to solving linear programs. We show that the total expected reward obtained by OLP up to time T is within C(P)logT of the reward obtained by the optimal policy, where C(P) is an explicit, MDP-dependent constant. OLP is closely related to an algorithm proposed by Burnetas and Katehakis with four key differences: OLP is simpler, it does not require knowledge of the supports of transition probabilities and the proof of the regret bound is simpler, but our regret bound is a constant factor larger than the regret of their algorithm. OLP is also similar in flavor to an algorithm recently proposed by Auer and Ortner. But OLP is simpler and its regret bound has a better dependence on the size of the MDP.

We study the rates of growth of the regret in online convex optimization. First, we show that a simple extension of the algorithm of Hazan et al eliminates the need for a priori knowledge of the lower bound on the second derivatives of the observed functions. We then provide an algorithm, Adaptive Online Gradient Descent, which interpolates between the results of Zinkevich for linear functions and of Hazan et al for strongly convex functions, achieving intermediate rates between sqrt(T) and logT. Furthermore, we show strong optimality of the algorithm. Finally, we provide an extension of our results to general norms.

Several recent studies in privacy-preserving learning have considered the trade-off between utility or risk and the level of differential privacy guaranteed by mechanisms for statistical query processing. In this paper we study this trade-off in private Support Vector Machine (SVM) learning. We present two efficient mechanisms, one for the case of finite-dimensional feature mappings and one for potentially infinite-dimensional feature mappings with translation-invariant kernels. For the case of translation-invariant kernels, the proposed mechanism minimizes regularized empirical risk in a random Reproducing Kernel Hilbert Space whose kernel uniformly approximates the desired kernel with high probability. This technique, borrowed from large-scale learning, allows the mechanism to respond with a finite encoding of the classifier, even when the function class is of infinite VC dimension. Differential privacy is established using a proof technique from algorithmic stability. Utility--the mechanism's response function is pointwise epsilon-close to non-private SVM with probability 1-delta--is proven by appealing to the smoothness of regularized empirical risk minimization with respect to small perturbations to the feature mapping. We conclude with a lower bound on the optimal differential privacy of the SVM. This negative result states that for any delta, no mechanism can be simultaneously (epsilon,delta)-useful and beta-differentially private for small epsilon and small beta.

Despite the conventional wisdom that proactive security is superior to reactive security, we show that reactive security can be competitive with proactive security as long as the reactive defender learns from past attacks instead of myopically overreacting to the last attack. Our game-theoretic model follows common practice in the security literature by making worst-case assumptions about the attacker: we grant the attacker complete knowledge of the defender's strategy and do not require the attacker to act rationally. In this model, we bound the competitive ratio between a reactive defense algorithm (which is inspired by online learning theory) and the best fixed proactive defense. Additionally, we show that, unlike proactive defenses, this reactive strategy is robust to a lack of information about the attacker's incentives and knowledge.

We study the regret of optimal strategies for online convex optimization games. Using von Neumann's minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary's action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen's inequality for a concave functional--the minimizer over the player's actions of expected loss--defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary.

Despite the large amount of literature on upper bounds on complexity of convex analysis, surprisingly little is known about the fundamental hardness of these problems. The extensive use of convex optimization in machine learning and statistics makes such an understanding very critical to understand fundamental computational limits of learning and estimation. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for some function classes. We also discuss implications of these results to learning and estimation.

We study the regret of optimal strategies for online convex optimization games. Using von Neumann's minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary's action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen's inequality for a concave functional---the minimizer over the player's actions of expected loss---defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary.

We study sample-based estimates of the expectation of the function produced by the empirical minimization algorithm. We investigate the extent to which one can estimate the rate of convergence of the empirical minimizer in a data dependent manner. We establish three main results. First, we provide an algorithm that upper bounds the expectation of the empirical minimizer in a completely data-dependent manner. This bound is based on a structural result in http://www.stat.berkeley.edu/~bartlett/papers/bm-em-03.pdf, which relates expectations to sample averages. Second, we show that these structural upper bounds can be loose. In particular, we demonstrate a class for which the expectation of the empirical minimizer decreases as O(1/n) for sample size n, although the upper bound based on structural properties is Ω(1). Third, we show that this looseness of the bound is inevitable: we present an example that shows that a sharp bound cannot be universally recovered from empirical data.

We consider the celebrated Blackwell Approachability Theorem for two-player games with vector payoffs. We show that Blackwell's result is equivalent, via efficient reductions, to the existence of 'no-regret' algorithms for Online Linear Optimization. Indeed, we show that any algorithm for one such problem can be efficiently converted into an algorithm for the other. We provide a useful application of this reduction: the first efficient algorithm for calibrated forecasting.

Recent research on multiple kernel learning has lead to a number of approaches for combining kernels in regularized risk minimization. The proposed approaches include different formulations of objectives and varying regularization strategies. In this paper we present a unifying general optimization criterion for multiple kernel learning and show how existing formulations are subsumed as special cases. We also derive the criterion's dual representation, which is suitable for general smooth optimization algorithms. Finally, we evaluate multiple kernel learning in this framework analytically using a Rademacher complexity bound on the generalization error and empirically in a set of experiments.

We study the problem of allocating stocks to dark pools. We propose and analyze an optimal approach for allocations, if continuous-valued allocations are allowed. We also propose a modification for the case when only integer-valued allocations are possible. We extend the previous work on this problem to adversarial scenarios, while also improving on those results in the iid setup. The resulting algorithms are efficient, and perform well in simulations under stochastic and adversarial inputs.

We analyze general model selection procedures using penalized empirical loss minimization under computational constraints. While classical model selection approaches do not consider computational aspects of performing model selection, we argue that any practical model selection procedure must not only trade off estimation and approximation error, but also the effects of computational effort required to compute empirical minimizers for different function classes. We provide a framework for analyzing such problems, and we give algorithms for model selection under a computational budget.

By combining randomized smoothing techniques with accelerated gradient methods, we obtain convergence rates for stochastic optimization procedures, both in expectation and with high probability, that have optimal dependence on the variance of the gradient estimates. To the best of our knowledge, these are the first variance-based rates for non-smooth optimization. A combination of our techniques with recent work on decentralized optimization yields order-optimal parallel stochastic optimization algorithms. We give applications of our results to several statistical machine learning problems, providing experimental results (in the full version of the paper) demonstrating the effectiveness of our algorithms.

We analyze general model selection procedures using penalized empirical loss minimization under computational constraints. While classical model selection approaches do not consider computational aspects of performing model selection, we argue that any practical model selection procedure must not only trade off estimation and approximation error, but also the computational effort required to compute empirical minimizers for different function classes. We provide a framework for analyzing such problems, and we give algorithms for model selection under a computational budget. These algorithms satisfy oracle inequalities that show that the risk of the selected model is not much worse than if we had devoted all of our computational budget to the optimal function class.

We study online prediction of individual sequences under logarithmic loss with parametric experts. The optimal strategy, normalized maximum likelihood (NML), is computationally demanding and requires the length of the game to be known. We consider two simpler strategies: sequential normalized maximum likelihood (SNML), which computes the NML forecasts at each round as if it were the last round, and Bayesian prediction. Under appropriate conditions, both are known to achieve near-optimal regret. In this paper, we investigate when these strategies are optimal. We show that SNML is optimal iff the joint distribution on sequences defined by SNML is exchangeable. In the case of exponential families, this is equivalent to the optimality of any Bayesian prediction strategy, and the optimal prior is Jeffreys prior.

Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for various function classes.

We study online learning under logarithmic loss with regular parametric models. We show that a Bayesian strategy predicts optimally only if it uses Jeffreys prior. This result was known for canonical exponential families; we extend it to parametric models for which the maximum likelihood estimator is asymptotically normal. The optimal prediction strategy, normalized maximum likelihood, depends on the number n of rounds of the game, in general. However, when a Bayesian strategy is optimal, normalized maximum likelihood becomes independent of n. Our proof uses this to exploit the asymptotics of normalized maximum likelihood. The asymptotic normality of the maximum likelihood estimator is responsible for the necessity of Jeffreys prior.

We analyze convergence rates of stochastic optimization algorithms for nonsmooth convex optimization problems. By combining randomized smoothing techniques with accelerated gradient methods, we obtain convergence rates of stochastic optimization procedures, both in expectation and with high probability, that have optimal dependence on the variance of the gradient estimates. To the best of our knowledge, these are the first variance-based rates for nonsmooth optimization. We give several applications of our results to statistical estimation problems and provide experimental results that demonstrate the effectiveness of the proposed algorithms. We also describe how a combination of our algorithm with recent work on decentralized optimization yields a distributed stochastic optimization algorithm that is order-optimal.

Despite the conventional wisdom that proactive security is superior to reactive security, we show that reactive security can be competitive with proactive security as long as the reactive defender learns from past attacks instead of myopically overreacting to the last attack. Our game-theoretic model follows common practice in the security literature by making worst-case assumptions about the attacker: we grant the attacker complete knowledge of the defender’s strategy and do not require the attacker to act rationally. In this model, we bound the competitive ratio between a reactive defense algorithm (which is inspired by online learning theory) and the best fixed proactive defense. Additionally, we show that, unlike proactive defenses, this reactive strategy is robust to a lack of information about the attacker’s incentives and knowledge.

We study the predictive performance of _1-regularized linear regression in a model-free setting, including the case where the number of covariates is substantially larger than the sample size. We introduce a new analysis method that avoids the boundedness problems that typically arise in model-free empirical minimization. Our technique provides an answer to a conjecture of Greenshtein and Ritov [?] regarding the “persistence” rate for linear regression and allows us to prove an oracle inequality for the error of the regularized minimizer. It also demonstrates that empirical risk minimization gives optimal rates (up to log factors) of convex aggregation of a set of estimators of a regression function.

We study the problem of online learning Markov Decision Processes (MDPs) when both the transition distributions and loss functions are chosen by an adversary. We present an algorithm that, under a mixing assumption, achieves O(sqrt(Tlog|Π|)+log|Π|) regret with respect to a comparison set of policies Π. The regret is independent of the size of the state and action spaces. When expectations over sample paths can be computed efficiently and the comparison set Π has polynomial size, this algorithm is efficient. We also consider the episodic adversarial online shortest path problem. Here, in each episode an adversary may choose a weighted directed acyclic graph with an identified start and finish node. The goal of the learning algorithm is to choose a path that minimizes the loss while traversing from the start to finish node. At the end of each episode the loss function (given by weights on the edges) is revealed to the learning algorithm. The goal is to minimize regret with respect to a fixed policy for selecting paths. This problem is a special case of the online MDP problem. For randomly chosen graphs and adversarial losses, this problem can be efficiently solved. We show that it also can be efficiently solved for adversarial graphs and randomly chosen losses. When both graphs and losses are adversarially chosen, we present an efficient algorithm whose regret scales linearly with the number of distinct graphs. Finally, we show that designing efficient algorithms for the adversarial online shortest path problem (and hence for the adversarial MDP problem) is as hard as learning parity with noise, a notoriously difficult problem that has been used to design efficient cryptographic schemes.

We consider a popular problem in finance, option pricing, through the lens of an online learning game between Nature and an Investor. In the Black-Scholes option pricing model from 1973, the Investor can continuously hedge the risk of an option by trading the underlying asset, assuming that the asset's price fluctuates according to Geometric Brownian Motion (GBM). We consider a worst-case model, in which Nature chooses a sequence of price fluctuations under a cumulative quadratic volatility constraint, and the Investor can make a sequence of hedging decisions. Our main result is to show that the value of our proposed game, which is the “regret” of hedging strategy, converges to the Black-Scholes option price. We use significantly weaker assumptions than previous work---for instance, we allow large jumps in the asset price---and show that the Black-Scholes hedging strategy is near-optimal for the Investor even in this non-stochastic framework.

We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, which occurs if and only if the optimal strategy can be calculated without knowing the time horizon in advance. They put forward the question what families have exchangeable SNML strategies. We answer this question for one-dimensional exponential families: SNML is exchangeable only for three classes of natural exponential family distributions,namely the Gaussian, the gamma, and the Tweedie exponential family of order 3/2.

We present the Convex Polytope Machine (CPM), a novel non-linear learning algorithm for large-scale binary classification tasks. The CPM finds a large margin convex polytope separator which encloses one class. We develop a stochastic gradient descent based algorithm that is amenable to massive datasets, and augment it with a heuristic procedure to avoid sub-optimal local minima. Our experimental evaluations of the CPM on large-scale datasets from distinct domains (MNIST handwritten digit recognition, text topic, and web security) demonstrate that the CPM trains models faster, sometimes several orders of magnitude, than state-of-the-art similar approaches and kernel-SVM methods while achieving comparable or better classification performance. Our empirical results suggest that, unlike prior similar approaches, we do not need to control the number of sub-classifiers (sides of the polytope) to avoid overfitting.

We consider online prediction problems where the loss between the prediction and the outcome is measured by the squared Euclidean distance and its generalization, the squared Mahalanobis distance. We derive the minimax solutions for the case where the prediction and action spaces are the simplex (this setup is sometimes called the Brier game) and the _2 ball (this setup is related to Gaussian density estimation). We show that in both cases the value of each sub-game is a quadratic function of a simple statistic of the state, with coefficients that can be efficiently computed using an explicit recurrence relation. The resulting deterministic minimax strategy and randomized maximin strategy are linear functions of the statistic.

We consider the problem of controlling a Markov decision process (MDP) with a large state space, so as to minimize average cost. Since it is intractable to compete with the optimal policy for large scale problems, we pursue the more modest goal of competing with a low-dimensional family of policies. We use the dual linear programming formulation of the MDP average cost problem, in which the variable is a stationary distribution over state-action pairs, and we consider a neighborhood of a low-dimensional subset of the set of stationary distributions (defined in terms of state-action features) as the comparison class. We propose two techniques, one based on stochastic convex optimization, and one based on constraint sampling. In both cases, we give bounds that show that the performance of our algorithms approaches the best achievable by any policy in the comparison class. Most importantly, these results depend on the size of the comparison class, but not on the size of the state space. Preliminary experiments show the effectiveness of the proposed algorithms in a queuing application.

We consider the problem of controlling a Markov decision process (MDP) with a large state space, so as to minimize average cost. Since it is intractable to compete with the optimal policy for large scale problems, we pursue the more modest goal of competing with a low-dimensional family of policies. We use the dual linear programming formulation of the MDP average cost problem, in which the variable is a stationary distribution over state-action pairs, and we consider a neighborhood of a low-dimensional subset of the set of stationary distributions (defined in terms of state-action features) as the comparison class. We propose two techniques, one based on stochastic convex optimization, and one based on constraint sampling. In both cases, we give bounds that show that the performance of our algorithms approaches the best achievable by any policy in the comparison class. Most importantly, these results depend on the size of the comparison class, but not on the size of the state space. Preliminary experiments show the effectiveness of the proposed algorithms in a queuing application.