arXiv subjects

Pierre Gaillard

Publications and source records attributed to Pierre Gaillard.

49 records · Page 3Linked to original sources

arXiv2019

Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterations

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method solving the gossip problem that depends only on the spectral dimension of the network, that is, in the communication network set-up, the dimension of the space in which the agents live. This contrasts with previous work that required the spectral gap of the network as a parameter, or suffered from slow mixing. Our method shows an important improvement over existing algorithms in the non-asymptotic regime, i.e., when the values are far from being fully mixed in the network. Our approach stems from a polynomial-based point of view on gossip algorithms, as well as an approximation of the spectral measure of the graphs with a Jacobi measure. We show the power of the approach with simulations on various graphs, and with performance guarantees on graphs of known spectral dimension, such as grids and random percolation bonds. An extension of this work to distributed Laplacian solvers is discussed. As a side result, we also use the polynomial-based point of view to show the convergence of the message passing algorithm for gossip of Moallemi \& Van Roy on regular graphs. The explicit computation of the rate of the convergence shows that message passing has a slow rate of convergence on graphs with small spectral gap.

cs.MA↗

arXiv2019

Efficient online learning with kernels for adversarial large scale problems

We are interested in a framework of online learning with kernels for low-dimensional but large-scale and potentially adversarial datasets. We study the computational and theoretical performance of online variations of kernel Ridge regression. Despite its simplicity, the algorithm we study is the first to achieve the optimal regret for a wide range of kernels with a per-round complexity of order $n^α$ with $α< 2$. The algorithm we consider is based on approximating the kernel with the linear span of basis functions. Our contributions is two-fold: 1) For the Gaussian kernel, we propose to build the basis beforehand (independently of the data) through Taylor expansion. For $d$-dimensional inputs, we provide a (close to) optimal regret of order $O((\log n)^{d+1})$ with per-round time complexity and space complexity $O((\log n)^{2d})$. This makes the algorithm a suitable choice as soon as $n \gg e^d$ which is likely to happen in a scenario with small dimensional and large-scale dataset; 2) For general kernels with low effective dimension, the basis functions are updated sequentially in a data-adaptive fashion by sampling Nystr{ö}m points. In this case, our algorithm improves the computational trade-off known for online kernel regression.

stat.ML↗

arXiv2019

Target Tracking for Contextual Bandits: Application to Demand Side Management

We propose a contextual-bandit approach for demand side management by offering price incentives. More precisely, a target mean consumption is set at each round and the mean consumption is modeled as a complex function of the distribution of prices sent and of some contextual variables such as the temperature, weather, and so on. The performance of our strategies is measured in quadratic losses through a regret criterion. We offer $T^{2/3}$ upper bounds on this regret (up to poly-logarithmic terms)---and even faster rates under stronger assumptions---for strategies inspired by standard strategies for contextual bandits (like LinUCB, see Li et al., 2010). Simulations on a real data set gathered by UK Power Networks, in which price incentives were offered, show that our strategies are effective and may indeed manage demand response by suitably picking the price levels.

cs.LG↗

arXiv2019

Uniform regret bounds over $R^d$ for the sequential linear regression problem with the square loss

We consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in the aim set by Bartlett et al. (2015): obtain regret bounds that hold uniformly over all competitor vectors. When the feature sequence is known at the beginning of the game, they provided closed-form regret bounds of $2d B^2 \ln T + \mathcal{O}_T(1)$, where $T$ is the number of rounds and $B$ is a bound on the observations. Instead, we derive bounds with an optimal constant of $1$ in front of the $d B^2 \ln T$ term. In the case of sequentially revealed features, we also derive an asymptotic regret bound of $d B^2 \ln T$ for any individual sequence of features and bounded observations. All our algorithms are variants of the online non-linear ridge regression forecaster, either with a data-dependent regularization or with almost no regularization.

stat.ML↗

arXiv2018

Efficient online algorithms for fast-rate regret bounds under sparsity

We consider the online convex optimization problem. In the setting of arbitrary sequences and finite set of parameters, we establish a new fast-rate quantile regret bound. Then we investigate the optimization into the L1-ball by discretizing the parameter space. Our algorithm is projection free and we propose an efficient solution by restarting the algorithm on adaptive discretization grids. In the adversarial setting, we develop an algorithm that achieves several rates of convergence with different dependencies on the sparsity of the objective. In the i.i.d. setting, we establish new risk bounds that are adaptive to the sparsity of the problem and to the regularity of the risk (ranging from a rate 1 / $\sqrt T$ for general convex risk to 1 /T for strongly convex risk). These results generalize previous works on sparse online learning. They are obtained under a weak assumption on the risk (Łojasiewicz's assumption) that allows multiple optima which is crucial when dealing with degenerate situations.

math.ST↗

arXiv2017

Algorithmic Chaining and the Role of Partial Feedback in Online Nonparametric Learning

We investigate contextual online learning with nonparametric (Lipschitz) comparison classes under different assumptions on losses and feedback information. For full information feedback and Lipschitz losses, we design the first explicit algorithm achieving the minimax regret rate (up to log factors). In a partial feedback model motivated by second-price auctions, we obtain algorithms for Lipschitz and semi-Lipschitz losses with regret bounds improving on the known bounds for standard bandit feedback. Our analysis combines novel results for contextual second-price auctions with a novel algorithmic approach based on chaining. When the context space is Euclidean, our chaining approach is efficient and delivers an even better regret bound.

stat.ML↗

arXiv2016

Sparse Accelerated Exponential Weights

We consider the stochastic optimization problem where a convex function is minimized observing recursively the gradients. We introduce SAEW, a new procedure that accelerates exponential weights procedures with the slow rate $1/\sqrt{T}$ to procedures achieving the fast rate $1/T$. Under the strong convexity of the risk, we achieve the optimal rate of convergence for approximating sparse parameters in $\mathbb{R}^d$. The acceleration is achieved by using successive averaging steps in an online fashion. The procedure also produces sparse estimators thanks to additional hard threshold steps.

math.ST↗

arXiv2015

A Chaining Algorithm for Online Nonparametric Regression

We consider the problem of online nonparametric regression with arbitrary deterministic sequences. Using ideas from the chaining technique, we design an algorithm that achieves a Dudley-type regret bound similar to the one obtained in a non-constructive fashion by Rakhlin and Sridharan (2014). Our regret bound is expressed in terms of the metric entropy in the sup norm, which yields optimal guarantees when the metric and sequential entropies are of the same order of magnitude. In particular our algorithm is the first one that achieves optimal rates for online regression over H{ö}lder balls. In addition we show for this example how to adapt our chaining algorithm to get a reasonable computational efficiency with similar regret guarantees (up to a log factor).

stat.ML↗

arXiv2015

Multi-parametric solutions to the NLS equation

The structure of the solutions to the one dimensional focusing nonlin-ear Schr{ö}dinger equation (NLS) for the order N in terms of quasi rational functions is given here. We first give the proof that the solutions can be expressed as a ratio of two wronskians of order 2N and then two determinants by an exponential depending on t with 2N -- 2 parameters. It also is proved that for the order N , the solutions can be written as the product of an exponential depending on t by a quotient of two polynomials of degree N (N + 1) in x and t. The solutions depend on 2N -- 2 parameters and give when all these parameters are equal to 0, the analogue of the famous Peregrine breather PN. It is fundamental to note that in this representation at order N , all these solutions can be seen as deformations with 2N -- 2 parameters of the famous Peregrine breather PN. With this method, we already built Peregrine breathers until order N = 10, and their deformations depending on 2N -- 2 parameters.

math-ph↗

arXiv2014

A consistent deterministic regression tree for non-parametric prediction of time series

We study online prediction of bounded stationary ergodic processes. To do so, we consider the setting of prediction of individual sequences and build a deterministic regression tree that performs asymptotically as well as the best L-Lipschitz constant predictors. Then, we show why the obtained regret bound entails the asymptotical optimality with respect to the class of bounded stationary ergodic processes.

math.ST↗

arXiv2014

A Second-order Bound with Excess Losses

We study online aggregation of the predictions of experts, and first show new second-order regret bounds in the standard setting, which are obtained via a version of the Prod algorithm (and also a version of the polynomially weighted average algorithm) with multiple learning rates. These bounds are in terms of excess losses, the differences between the instantaneous losses suffered by the algorithm and the ones of a given expert. We then demonstrate the interest of these bounds in the context of experts that report their confidences as a number in the interval [0,1] using a generic reduction to the standard setting. We conclude by two other applications in the standard setting, which improve the known bounds in case of small excess losses and show a bounded regret against i.i.d. sequences of losses.

stat.ML↗

arXiv2012

Mirror Descent Meets Fixed Share (and feels no regret)

Mirror descent with an entropic regularizer is known to achieve shifting regret bounds that are logarithmic in the dimension. This is done using either a carefully designed projection or by a weight sharing technique. Via a novel unified analysis, we show that these two approaches deliver essentially equivalent bounds on a notion of regret generalizing shifting, adaptive, discounted, and other related regrets. Our analysis also captures and extends the generalized weight sharing technique of Bousquet and Warmuth, and can be refined in several ways, including improvements for small losses and adaptive tuning of parameters.

cs.LG↗

arXiv2012

Forecasting electricity consumption by aggregating specialized experts

We consider the setting of sequential prediction of arbitrary sequences based on specialized experts. We first provide a review of the relevant literature and present two theoretical contributions: a general analysis of the specialist aggregation rule of Freund et al. (1997) and an adaptation of fixed-share rules of Herbster and Warmuth (1998) in this setting. We then apply these rules to the sequential short-term (one-day-ahead) forecasting of electricity consumption; to do so, we consider two data sets, a Slovakian one and a French one, respectively concerned with hourly and half-hourly predictions. We follow a general methodology to perform the stated empirical studies and detail in particular tuning issues of the learning parameters. The introduced aggregation rules demonstrate an improved accuracy on the data sets at hand; the improvements lie in a reduced mean squared error but also in a more robust behavior with respect to large occasional errors.

stat.ML↗