arXiv subjects

H. Brendan McMahan

Publications and source records attributed to H. Brendan McMahan.

50 records · Page 3Linked to original sources

arXiv2017

On the Protection of Private Information in Machine Learning Systems: Two Recent Approaches

The recent, remarkable growth of machine learning has led to intense interest in the privacy of the data on which machine learning relies, and to new techniques for preserving privacy. However, older ideas about privacy may well remain valid and useful. This note reviews two recent works on privacy in the light of the wisdom of some of the early literature, in particular the principles distilled by Saltzer and Schroeder in the 1970s.

stat.ML↗

arXiv2016

Practical Secure Aggregation for Federated Learning on User-Held Data

Secure Aggregation protocols allow a collection of mutually distrust parties, each holding a private value, to collaboratively compute the sum of those values without revealing the values themselves. We consider training a deep neural network in the Federated Learning model, using distributed stochastic gradient descent across user-held training data on mobile devices, wherein Secure Aggregation protects each user's model gradient. We design a novel, communication-efficient Secure Aggregation protocol for high-dimensional data that tolerates up to 1/3 users failing to complete the protocol. For 16-bit input values, our protocol offers 1.73x communication expansion for $2^{10}$ users and $2^{20}$-dimensional vectors, and 1.98x expansion for $2^{14}$ users and $2^{24}$ dimensional vectors.

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arXiv2016

Deep Learning with Differential Privacy

Machine learning techniques based on neural networks are achieving remarkable results in a wide variety of domains. Often, the training of models requires large, representative datasets, which may be crowdsourced and contain sensitive information. The models should not expose private information in these datasets. Addressing this goal, we develop new algorithmic techniques for learning and a refined analysis of privacy costs within the framework of differential privacy. Our implementation and experiments demonstrate that we can train deep neural networks with non-convex objectives, under a modest privacy budget, and at a manageable cost in software complexity, training efficiency, and model quality.

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arXiv2016

Federated Optimization: Distributed Machine Learning for On-Device Intelligence

We introduce a new and increasingly relevant setting for distributed optimization in machine learning, where the data defining the optimization are unevenly distributed over an extremely large number of nodes. The goal is to train a high-quality centralized model. We refer to this setting as Federated Optimization. In this setting, communication efficiency is of the utmost importance and minimizing the number of rounds of communication is the principal goal. A motivating example arises when we keep the training data locally on users' mobile devices instead of logging it to a data center for training. In federated optimziation, the devices are used as compute nodes performing computation on their local data in order to update a global model. We suppose that we have extremely large number of devices in the network --- as many as the number of users of a given service, each of which has only a tiny fraction of the total data available. In particular, we expect the number of data points available locally to be much smaller than the number of devices. Additionally, since different users generate data with different patterns, it is reasonable to assume that no device has a representative sample of the overall distribution. We show that existing algorithms are not suitable for this setting, and propose a new algorithm which shows encouraging experimental results for sparse convex problems. This work also sets a path for future research needed in the context of \federated optimization.

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arXiv2015

A Survey of Algorithms and Analysis for Adaptive Online Learning

We present tools for the analysis of Follow-The-Regularized-Leader (FTRL), Dual Averaging, and Mirror Descent algorithms when the regularizer (equivalently, prox-function or learning rate schedule) is chosen adaptively based on the data. Adaptivity can be used to prove regret bounds that hold on every round, and also allows for data-dependent regret bounds as in AdaGrad-style algorithms (e.g., Online Gradient Descent with adaptive per-coordinate learning rates). We present results from a large number of prior works in a unified manner, using a modular and tight analysis that isolates the key arguments in easily re-usable lemmas. This approach strengthens pre-viously known FTRL analysis techniques to produce bounds as tight as those achieved by potential functions or primal-dual analysis. Further, we prove a general and exact equivalence between an arbitrary adaptive Mirror Descent algorithm and a correspond- ing FTRL update, which allows us to analyze any Mirror Descent algorithm in the same framework. The key to bridging the gap between Dual Averaging and Mirror Descent algorithms lies in an analysis of the FTRL-Proximal algorithm family. Our regret bounds are proved in the most general form, holding for arbitrary norms and non-smooth regularizers with time-varying weight.

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arXiv2014

Unconstrained Online Linear Learning in Hilbert Spaces: Minimax Algorithms and Normal Approximations

We study algorithms for online linear optimization in Hilbert spaces, focusing on the case where the player is unconstrained. We develop a novel characterization of a large class of minimax algorithms, recovering, and even improving, several previous results as immediate corollaries. Moreover, using our tools, we develop an algorithm that provides a regret bound of $\mathcal{O}\Big(U \sqrt{T \log(U \sqrt{T} \log^2 T +1)}\Big)$, where $U$ is the $L_2$ norm of an arbitrary comparator and both $T$ and $U$ are unknown to the player. This bound is optimal up to $\sqrt{\log \log T}$ terms. When $T$ is known, we derive an algorithm with an optimal regret bound (up to constant factors). For both the known and unknown $T$ case, a Normal approximation to the conditional value of the game proves to be the key analysis tool.

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arXiv2013

Large-Scale Learning with Less RAM via Randomization

We reduce the memory footprint of popular large-scale online learning methods by projecting our weight vector onto a coarse discrete set using randomized rounding. Compared to standard 32-bit float encodings, this reduces RAM usage by more than 50% during training and by up to 95% when making predictions from a fixed model, with almost no loss in accuracy. We also show that randomized counting can be used to implement per-coordinate learning rates, improving model quality with little additional RAM. We prove these memory-saving methods achieve regret guarantees similar to their exact variants. Empirical evaluation confirms excellent performance, dominating standard approaches across memory versus accuracy tradeoffs.

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arXiv2013

Minimax Optimal Algorithms for Unconstrained Linear Optimization

We design and analyze minimax-optimal algorithms for online linear optimization games where the player's choice is unconstrained. The player strives to minimize regret, the difference between his loss and the loss of a post-hoc benchmark strategy. The standard benchmark is the loss of the best strategy chosen from a bounded comparator set. When the the comparison set and the adversary's gradients satisfy L_infinity bounds, we give the value of the game in closed form and prove it approaches sqrt(2T/pi) as T -> infinity. Interesting algorithms result when we consider soft constraints on the comparator, rather than restricting it to a bounded set. As a warmup, we analyze the game with a quadratic penalty. The value of this game is exactly T/2, and this value is achieved by perhaps the simplest online algorithm of all: unprojected gradient descent with a constant learning rate. We then derive a minimax-optimal algorithm for a much softer penalty function. This algorithm achieves good bounds under the standard notion of regret for any comparator point, without needing to specify the comparator set in advance. The value of this game converges to sqrt{e} as T ->infinity; we give a closed-form for the exact value as a function of T. The resulting algorithm is natural in unconstrained investment or betting scenarios, since it guarantees at worst constant loss, while allowing for exponential reward against an "easy" adversary.

cs.LG↗

arXiv2012

On Calibrated Predictions for Auction Selection Mechanisms

Calibration is a basic property for prediction systems, and algorithms for achieving it are well-studied in both statistics and machine learning. In many applications, however, the predictions are used to make decisions that select which observations are made. This makes calibration difficult, as adjusting predictions to achieve calibration changes future data. We focus on click-through-rate (CTR) prediction for search ad auctions. Here, CTR predictions are used by an auction that determines which ads are shown, and we want to maximize the value generated by the auction. We show that certain natural notions of calibration can be impossible to achieve, depending on the details of the auction. We also show that it can be impossible to maximize auction efficiency while using calibrated predictions. Finally, we give conditions under which calibration is achievable and simultaneously maximizes auction efficiency: roughly speaking, bids and queries must not contain information about CTRs that is not already captured by the predictions.

cs.GT↗

arXiv2012

No-Regret Algorithms for Unconstrained Online Convex Optimization

Some of the most compelling applications of online convex optimization, including online prediction and classification, are unconstrained: the natural feasible set is R^n. Existing algorithms fail to achieve sub-linear regret in this setting unless constraints on the comparator point x^* are known in advance. We present algorithms that, without such prior knowledge, offer near-optimal regret bounds with respect to any choice of x^*. In particular, regret with respect to x^* = 0 is constant. We then prove lower bounds showing that our guarantees are near-optimal in this setting.

cs.LG↗

arXiv2011

A Unified View of Regularized Dual Averaging and Mirror Descent with Implicit Updates

We study three families of online convex optimization algorithms: follow-the-proximally-regularized-leader (FTRL-Proximal), regularized dual averaging (RDA), and composite-objective mirror descent. We first prove equivalence theorems that show all of these algorithms are instantiations of a general FTRL update. This provides theoretical insight on previous experimental observations. In particular, even though the FOBOS composite mirror descent algorithm handles L1 regularization explicitly, it has been observed that RDA is even more effective at producing sparsity. Our results demonstrate that FOBOS uses subgradient approximations to the L1 penalty from previous rounds, leading to less sparsity than RDA, which handles the cumulative penalty in closed form. The FTRL-Proximal algorithm can be seen as a hybrid of these two, and outperforms both on a large, real-world dataset. Our second contribution is a unified analysis which produces regret bounds that match (up to logarithmic terms) or improve the best previously known bounds. This analysis also extends these algorithms in two important ways: we support a more general type of composite objective and we analyze implicit updates, which replace the subgradient approximation of the current loss function with an exact optimization.

cs.LG↗

arXiv2010

Adaptive Bound Optimization for Online Convex Optimization

We introduce a new online convex optimization algorithm that adaptively chooses its regularization function based on the loss functions observed so far. This is in contrast to previous algorithms that use a fixed regularization function such as L2-squared, and modify it only via a single time-dependent parameter. Our algorithm's regret bounds are worst-case optimal, and for certain realistic classes of loss functions they are much better than existing bounds. These bounds are problem-dependent, which means they can exploit the structure of the actual problem instance. Critically, however, our algorithm does not need to know this structure in advance. Rather, we prove competitive guarantees that show the algorithm provides a bound within a constant factor of the best possible bound (of a certain functional form) in hindsight.

cs.LG↗

arXiv2010

Less Regret via Online Conditioning

We analyze and evaluate an online gradient descent algorithm with adaptive per-coordinate adjustment of learning rates. Our algorithm can be thought of as an online version of batch gradient descent with a diagonal preconditioner. This approach leads to regret bounds that are stronger than those of standard online gradient descent for general online convex optimization problems. Experimentally, we show that our algorithm is competitive with state-of-the-art algorithms for large scale machine learning problems.

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arXiv2004

Online convex optimization in the bandit setting: gradient descent without a gradient

We consider a the general online convex optimization framework introduced by Zinkevich. In this setting, there is a sequence of convex functions. Each period, we must choose a signle point (from some feasible set) and pay a cost equal to the value of the next function on our chosen point. Zinkevich shows that, if the each function is revealed after the choice is made, then one can achieve vanishingly small regret relative the best single decision chosen in hindsight. We extend this to the bandit setting where we do not find out the entire functions but rather just their value at our chosen point. We show how to get vanishingly small regret in this setting. Our approach uses a simple approximation of the gradient that is computed from evaluating a function at a single (random) point. We show that this estimate is sufficient to mimic Zinkevich's gradient descent online analysis, with access to the gradient (only being able to evaluate the function at a single point).

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