arXiv subjects

Peter L. Bartlett

Publications and source records attributed to Peter L. Bartlett.

At least 55 records · Page 3Linked to original sources

arXiv2021

When does gradient descent with logistic loss find interpolating two-layer networks?

We study the training of finite-width two-layer smoothed ReLU networks for binary classification using the logistic loss. We show that gradient descent drives the training loss to zero if the initial loss is small enough. When the data satisfies certain cluster and separation conditions and the network is wide enough, we show that one step of gradient descent reduces the loss sufficiently that the first result applies.

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arXiv2021

When does gradient descent with logistic loss interpolate using deep networks with smoothed ReLU activations?

We establish conditions under which gradient descent applied to fixed-width deep networks drives the logistic loss to zero, and prove bounds on the rate of convergence. Our analysis applies for smoothed approximations to the ReLU, such as Swish and the Huberized ReLU, proposed in previous applied work. We provide two sufficient conditions for convergence. The first is simply a bound on the loss at initialization. The second is a data separation condition used in prior analyses.

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arXiv2021

Adversarial Examples in Multi-Layer Random ReLU Networks

We consider the phenomenon of adversarial examples in ReLU networks with independent gaussian parameters. For networks of constant depth and with a large range of widths (for instance, it suffices if the width of each layer is polynomial in that of any other layer), small perturbations of input vectors lead to large changes of outputs. This generalizes results of Daniely and Schacham (2020) for networks of rapidly decreasing width and of Bubeck et al (2021) for two-layer networks. The proof shows that adversarial examples arise in these networks because the functions that they compute are very close to linear. Bottleneck layers in the network play a key role: the minimal width up to some point in the network determines scales and sensitivities of mappings computed up to that point. The main result is for networks with constant depth, but we also show that some constraint on depth is necessary for a result of this kind, because there are suitably deep networks that, with constant probability, compute a function that is close to constant.

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arXiv2021

Preference learning along multiple criteria: A game-theoretic perspective

The literature on ranking from ordinal data is vast, and there are several ways to aggregate overall preferences from pairwise comparisons between objects. In particular, it is well known that any Nash equilibrium of the zero sum game induced by the preference matrix defines a natural solution concept (winning distribution over objects) known as a von Neumann winner. Many real-world problems, however, are inevitably multi-criteria, with different pairwise preferences governing the different criteria. In this work, we generalize the notion of a von Neumann winner to the multi-criteria setting by taking inspiration from Blackwell's approachability. Our framework allows for non-linear aggregation of preferences across criteria, and generalizes the linearization-based approach from multi-objective optimization. From a theoretical standpoint, we show that the Blackwell winner of a multi-criteria problem instance can be computed as the solution to a convex optimization problem. Furthermore, given random samples of pairwise comparisons, we show that a simple plug-in estimator achieves near-optimal minimax sample complexity. Finally, we showcase the practical utility of our framework in a user study on autonomous driving, where we find that the Blackwell winner outperforms the von Neumann winner for the overall preferences.

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arXiv2021

Agnostic learning with unknown utilities

Traditional learning approaches for classification implicitly assume that each mistake has the same cost. In many real-world problems though, the utility of a decision depends on the underlying context $x$ and decision $y$. However, directly incorporating these utilities into the learning objective is often infeasible since these can be quite complex and difficult for humans to specify. We formally study this as agnostic learning with unknown utilities: given a dataset $S = \{x_1, \ldots, x_n\}$ where each data point $x_i \sim \mathcal{D}$, the objective of the learner is to output a function $f$ in some class of decision functions $\mathcal{F}$ with small excess risk. This risk measures the performance of the output predictor $f$ with respect to the best predictor in the class $\mathcal{F}$ on the unknown underlying utility $u^*$. This utility $u^*$ is not assumed to have any specific structure. This raises an interesting question whether learning is even possible in our setup, given that obtaining a generalizable estimate of utility $u^*$ might not be possible from finitely many samples. Surprisingly, we show that estimating the utilities of only the sampled points~$S$ suffices to learn a decision function which generalizes well. We study mechanisms for eliciting information which allow a learner to estimate the utilities $u^*$ on the set $S$. We introduce a family of elicitation mechanisms by generalizing comparisons, called the $k$-comparison oracle, which enables the learner to ask for comparisons across $k$ different inputs $x$ at once. We show that the excess risk in our agnostic learning framework decreases at a rate of $O\left(\frac{1}{k} \right)$. This result brings out an interesting accuracy-elicitation trade-off -- as the order $k$ of the oracle increases, the comparative queries become harder to elicit from humans but allow for more accurate learning.

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arXiv2021

Infinite-Horizon Offline Reinforcement Learning with Linear Function Approximation: Curse of Dimensionality and Algorithm

In this paper, we investigate the sample complexity of policy evaluation in infinite-horizon offline reinforcement learning (also known as the off-policy evaluation problem) with linear function approximation. We identify a hard regime $dγ^{2}>1$, where $d$ is the dimension of the feature vector and $γ$ is the discount rate. In this regime, for any $q\in[γ^{2},1]$, we can construct a hard instance such that the smallest eigenvalue of its feature covariance matrix is $q/d$ and it requires $Ω\left(\frac{d}{γ^{2}\left(q-γ^{2}\right)\varepsilon^{2}}\exp\left(Θ\left(dγ^{2}\right)\right)\right)$ samples to approximate the value function up to an additive error $\varepsilon$. Note that the lower bound of the sample complexity is exponential in $d$. If $q=γ^{2}$, even infinite data cannot suffice. Under the low distribution shift assumption, we show that there is an algorithm that needs at most $O\left(\max\left\{ \frac{\left\Vert θ^π\right\Vert _{2}^{4}}{\varepsilon^{4}}\log\frac{d}δ,\frac{1}{\varepsilon^{2}}\left(d+\log\frac{1}δ\right)\right\} \right)$ samples ($θ^π$ is the parameter of the policy in linear function approximation) and guarantees approximation to the value function up to an additive error of $\varepsilon$ with probability at least $1-δ$.

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arXiv2021

Deep learning: a statistical viewpoint

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

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arXiv2021

Failures of model-dependent generalization bounds for least-norm interpolation

We consider bounds on the generalization performance of the least-norm linear regressor, in the over-parameterized regime where it can interpolate the data. We describe a sense in which any generalization bound of a type that is commonly proved in statistical learning theory must sometimes be very loose when applied to analyze the least-norm interpolant. In particular, for a variety of natural joint distributions on training examples, any valid generalization bound that depends only on the output of the learning algorithm, the number of training examples, and the confidence parameter, and that satisfies a mild condition (substantially weaker than monotonicity in sample size), must sometimes be very loose -- it can be bounded below by a constant when the true excess risk goes to zero.

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arXiv2020

Optimal Mean Estimation without a Variance

We study the problem of heavy-tailed mean estimation in settings where the variance of the data-generating distribution does not exist. Concretely, given a sample $\mathbf{X} = \{X_i\}_{i = 1}^n$ from a distribution $\mathcal{D}$ over $\mathbb{R}^d$ with mean $μ$ which satisfies the following \emph{weak-moment} assumption for some ${α\in [0, 1]}$: \begin{equation*} \forall \|v\| = 1: \mathbb{E}_{X \thicksim \mathcal{D}}[\lvert \langle X - μ, v\rangle \rvert^{1 + α}] \leq 1, \end{equation*} and given a target failure probability, $δ$, our goal is to design an estimator which attains the smallest possible confidence interval as a function of $n,d,δ$. For the specific case of $α= 1$, foundational work of Lugosi and Mendelson exhibits an estimator achieving subgaussian confidence intervals, and subsequent work has led to computationally efficient versions of this estimator. Here, we study the case of general $α$, and establish the following information-theoretic lower bound on the optimal attainable confidence interval: \begin{equation*} Ω\left(\sqrt{\frac{d}{n}} + \left(\frac{d}{n}\right)^{\fracα{(1 + α)}} + \left(\frac{\log 1 / δ}{n}\right)^{\fracα{(1 + α)}}\right). \end{equation*} Moreover, we devise a computationally-efficient estimator which achieves this lower bound.

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arXiv2020

Stochastic Gradient and Langevin Processes

We prove quantitative convergence rates at which discrete Langevin-like processes converge to the invariant distribution of a related stochastic differential equation. We study the setup where the additive noise can be non-Gaussian and state-dependent and the potential function can be non-convex. We show that the key properties of these processes depend on the potential function and the second moment of the additive noise. We apply our theoretical findings to studying the convergence of Stochastic Gradient Descent (SGD) for non-convex problems and corroborate them with experiments using SGD to train deep neural networks on the CIFAR-10 dataset.

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arXiv2020

Self-Distillation Amplifies Regularization in Hilbert Space

Knowledge distillation introduced in the deep learning context is a method to transfer knowledge from one architecture to another. In particular, when the architectures are identical, this is called self-distillation. The idea is to feed in predictions of the trained model as new target values for retraining (and iterate this loop possibly a few times). It has been empirically observed that the self-distilled model often achieves higher accuracy on held out data. Why this happens, however, has been a mystery: the self-distillation dynamics does not receive any new information about the task and solely evolves by looping over training. To the best of our knowledge, there is no rigorous understanding of this phenomenon. This work provides the first theoretical analysis of self-distillation. We focus on fitting a nonlinear function to training data, where the model space is Hilbert space and fitting is subject to $\ell_2$ regularization in this function space. We show that self-distillation iterations modify regularization by progressively limiting the number of basis functions that can be used to represent the solution. This implies (as we also verify empirically) that while a few rounds of self-distillation may reduce over-fitting, further rounds may lead to under-fitting and thus worse performance.

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arXiv2020

OSOM: A simultaneously optimal algorithm for multi-armed and linear contextual bandits

We consider the stochastic linear (multi-armed) contextual bandit problem with the possibility of hidden simple multi-armed bandit structure in which the rewards are independent of the contextual information. Algorithms that are designed solely for one of the regimes are known to be sub-optimal for the alternate regime. We design a single computationally efficient algorithm that simultaneously obtains problem-dependent optimal regret rates in the simple multi-armed bandit regime and minimax optimal regret rates in the linear contextual bandit regime, without knowing a priori which of the two models generates the rewards. These results are proved under the condition of stochasticity of contextual information over multiple rounds. Our results should be viewed as a step towards principled data-dependent policy class selection for contextual bandits.

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arXiv2020

Optimal Robust Linear Regression in Nearly Linear Time

We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = \langle X,w^* \rangle + ε$ (with $X \in \mathbb{R}^d$ and $ε$ independent), in which an $η$ fraction of the samples have been adversarially corrupted. We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $\mathbb{E} [XX^\top]$ has bounded condition number and $ε$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $ε$ is sub-Gaussian. In both settings, our estimators: (a) Achieve optimal sample complexities and recovery guarantees up to log factors and (b) Run in near linear time ($\tilde{O}(nd / η^6)$). Prior to our work, polynomial time algorithms achieving near optimal sample complexities were only known in the setting where $X$ is Gaussian with identity covariance and $ε$ is Gaussian, and no linear time estimators were known for robust linear regression in any setting. Our estimators and their analysis leverage recent developments in the construction of faster algorithms for robust mean estimation to improve runtimes, and refined concentration of measure arguments alongside Gaussian rounding techniques to improve statistical sample complexities.

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arXiv2020

Sharp convergence rates for Langevin dynamics in the nonconvex setting

We study the problem of sampling from a distribution $p^*(x) \propto \exp\left(-U(x)\right)$, where the function $U$ is $L$-smooth everywhere and $m$-strongly convex outside a ball of radius $R$, but potentially nonconvex inside this ball. We study both overdamped and underdamped Langevin MCMC and establish upper bounds on the number of steps required to obtain a sample from a distribution that is within $ε$ of $p^*$ in $1$-Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the iteration complexity is $\tilde{\mathcal{O}}\left(e^{cLR^2}d/ε^2\right)$, where $d$ is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the iteration complexity is $\tilde{\mathcal{O}}\left(e^{cLR^2}\sqrt{d}/ε\right)$ for an explicit positive constant $c$. Surprisingly, the iteration complexity for both these algorithms is only polynomial in the dimension $d$ and the target accuracy $ε$. It is exponential, however, in the problem parameter $LR^2$, which is a measure of non-log-concavity of the target distribution.

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arXiv2020

On Thompson Sampling with Langevin Algorithms

Thompson sampling for multi-armed bandit problems is known to enjoy favorable performance in both theory and practice. However, it suffers from a significant limitation computationally, arising from the need for samples from posterior distributions at every iteration. We propose two Markov Chain Monte Carlo (MCMC) methods tailored to Thompson sampling to address this issue. We construct quickly converging Langevin algorithms to generate approximate samples that have accuracy guarantees, and we leverage novel posterior concentration rates to analyze the regret of the resulting approximate Thompson sampling algorithm. Further, we specify the necessary hyperparameters for the MCMC procedure to guarantee optimal instance-dependent frequentist regret while having low computational complexity. In particular, our algorithms take advantage of both posterior concentration and a sample reuse mechanism to ensure that only a constant number of iterations and a constant amount of data is needed in each round. The resulting approximate Thompson sampling algorithm has logarithmic regret and its computational complexity does not scale with the time horizon of the algorithm.

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arXiv2020

High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\left(\frac{d^{1/4}}{ \varepsilon^{1/2}} \right)$ steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with $α$-th order smoothness, we prove that the mixing time scales as $O \left(\frac{d^{1/4}}{\varepsilon^{1/2}} + \frac{d^{1/2}}{\varepsilon^{1/(α- 1)}} \right)$.

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arXiv2020

Derivative-Free Methods for Policy Optimization: Guarantees for Linear Quadratic Systems

We study derivative-free methods for policy optimization over the class of linear policies. We focus on characterizing the convergence rate of these methods when applied to linear-quadratic systems, and study various settings of driving noise and reward feedback. We show that these methods provably converge to within any pre-specified tolerance of the optimal policy with a number of zero-order evaluations that is an explicit polynomial of the error tolerance, dimension, and curvature properties of the problem. Our analysis reveals some interesting differences between the settings of additive driving noise and random initialization, as well as the settings of one-point and two-point reward feedback. Our theory is corroborated by extensive simulations of derivative-free methods on these systems. Along the way, we derive convergence rates for stochastic zero-order optimization algorithms when applied to a certain class of non-convex problems.

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arXiv2020

Greedy Convex Ensemble

We consider learning a convex combination of basis models, and present some new theoretical and empirical results that demonstrate the effectiveness of a greedy approach. Theoretically, we first consider whether we can use linear, instead of convex, combinations, and obtain generalization results similar to existing ones for learning from a convex hull. We obtain a negative result that even the linear hull of very simple basis functions can have unbounded capacity, and is thus prone to overfitting; on the other hand, convex hulls are still rich but have bounded capacities. Secondly, we obtain a generalization bound for a general class of Lipschitz loss functions. Empirically, we first discuss how a convex combination can be greedily learned with early stopping, and how a convex combination can be non-greedily learned when the number of basis models is known a priori. Our experiments suggest that the greedy scheme is competitive with or better than several baselines, including boosting and random forests. The greedy algorithm requires little effort in hyper-parameter tuning, and also seems able to adapt to the underlying complexity of the problem. Our code is available at https://github.com/tan1889/gce.

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