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Francesco Orabona

Publications and source records attributed to Francesco Orabona.

At least 73 records · Page 4Linked to original sources
arXiv2019

On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes

Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze in the convex and non-convex settings a generalized version of the AdaGrad stepsizes. We show sufficient conditions for these stepsizes to achieve almost sure asymptotic convergence of the gradients to zero, proving the first guarantee for generalized AdaGrad stepsizes in the non-convex setting. Moreover, we show that these stepsizes allow to automatically adapt to the level of noise of the stochastic gradients in both the convex and non-convex settings, interpolating between $O(1/T)$ and $O(1/\sqrt{T})$, up to logarithmic terms.

stat.ML↗
arXiv2018

Black-Box Reductions for Parameter-free Online Learning in Banach Spaces

We introduce several new black-box reductions that significantly improve the design of adaptive and parameter-free online learning algorithms by simplifying analysis, improving regret guarantees, and sometimes even improving runtime. We reduce parameter-free online learning to online exp-concave optimization, we reduce optimization in a Banach space to one-dimensional optimization, and we reduce optimization over a constrained domain to unconstrained optimization. All of our reductions run as fast as online gradient descent. We use our new techniques to improve upon the previously best regret bounds for parameter-free learning, and do so for arbitrary norms.

cs.LG↗
arXiv2018

Efficient Online Bandit Multiclass Learning with $\tilde{O}(\sqrt{T})$ Regret

We present an efficient second-order algorithm with $\tilde{O}(\frac{1}η\sqrt{T})$ regret for the bandit online multiclass problem. The regret bound holds simultaneously with respect to a family of loss functions parameterized by $η$, for a range of $η$ restricted by the norm of the competitor. The family of loss functions ranges from hinge loss ($η=0$) to squared hinge loss ($η=1$). This provides a solution to the open problem of (J. Abernethy and A. Rakhlin. An efficient bandit algorithm for $\sqrt{T}$-regret in online multiclass prediction? In COLT, 2009). We test our algorithm experimentally, showing that it also performs favorably against earlier algorithms.

cs.LG↗
arXiv2017

Online Learning for Changing Environments using Coin Betting

A key challenge in online learning is that classical algorithms can be slow to adapt to changing environments. Recent studies have proposed "meta" algorithms that convert any online learning algorithm to one that is adaptive to changing environments, where the adaptivity is analyzed in a quantity called the strongly-adaptive regret. This paper describes a new meta algorithm that has a strongly-adaptive regret bound that is a factor of $\sqrt{\log(T)}$ better than other algorithms with the same time complexity, where $T$ is the time horizon. We also extend our algorithm to achieve a first-order (i.e., dependent on the observed losses) strongly-adaptive regret bound for the first time, to our knowledge. At its heart is a new parameter-free algorithm for the learning with expert advice (LEA) problem in which experts sometimes do not output advice for consecutive time steps (i.e., \emph{sleeping} experts). This algorithm is derived by a reduction from optimal algorithms for the so-called coin betting problem. Empirical results show that our algorithm outperforms state-of-the-art methods in both learning with expert advice and metric learning scenarios.

stat.ML↗
arXiv2017

Training Deep Networks without Learning Rates Through Coin Betting

Deep learning methods achieve state-of-the-art performance in many application scenarios. Yet, these methods require a significant amount of hyperparameters tuning in order to achieve the best results. In particular, tuning the learning rates in the stochastic optimization process is still one of the main bottlenecks. In this paper, we propose a new stochastic gradient descent procedure for deep networks that does not require any learning rate setting. Contrary to previous methods, we do not adapt the learning rates nor we make use of the assumed curvature of the objective function. Instead, we reduce the optimization process to a game of betting on a coin and propose a learning-rate-free optimal algorithm for this scenario. Theoretical convergence is proven for convex and quasi-convex functions and empirical evidence shows the advantage of our algorithm over popular stochastic gradient algorithms.

cs.LG↗
arXiv2017

Improved Strongly Adaptive Online Learning using Coin Betting

This paper describes a new parameter-free online learning algorithm for changing environments. In comparing against algorithms with the same time complexity as ours, we obtain a strongly adaptive regret bound that is a factor of at least $\sqrt{\log(T)}$ better, where $T$ is the time horizon. Empirical results show that our algorithm outperforms state-of-the-art methods in learning with expert advice and metric learning scenarios.

stat.ML↗
arXiv2017

High Dimensional Inference with Random Maximum A-Posteriori Perturbations

This paper presents a new approach, called perturb-max, for high-dimensional statistical inference that is based on applying random perturbations followed by optimization. This framework injects randomness to maximum a-posteriori (MAP) predictors by randomly perturbing the potential function for the input. A classic result from extreme value statistics asserts that perturb-max operations generate unbiased samples from the Gibbs distribution using high-dimensional perturbations. Unfortunately, the computational cost of generating so many high-dimensional random variables can be prohibitive. However, when the perturbations are of low dimension, sampling the perturb-max prediction is as efficient as MAP optimization. This paper shows that the expected value of perturb-max inference with low dimensional perturbations can be used sequentially to generate unbiased samples from the Gibbs distribution. Furthermore the expected value of the maximal perturbations is a natural bound on the entropy of such perturb-max models. A measure concentration result for perturb-max values shows that the deviation of their sampled average from its expectation decays exponentially in the number of samples, allowing effective approximation of the expectation.

cs.LG↗
arXiv2016

Scale-Free Online Learning

We design and analyze algorithms for online linear optimization that have optimal regret and at the same time do not need to know any upper or lower bounds on the norm of the loss vectors. Our algorithms are instances of the Follow the Regularized Leader (FTRL) and Mirror Descent (MD) meta-algorithms. We achieve adaptiveness to the norms of the loss vectors by scale invariance, i.e., our algorithms make exactly the same decisions if the sequence of loss vectors is multiplied by any positive constant. The algorithm based on FTRL works for any decision set, bounded or unbounded. For unbounded decisions sets, this is the first adaptive algorithm for online linear optimization with a non-vacuous regret bound. In contrast, we show lower bounds on scale-free algorithms based on MD on unbounded domains.

cs.LG↗
arXiv2016

Coin Betting and Parameter-Free Online Learning

In the recent years, a number of parameter-free algorithms have been developed for online linear optimization over Hilbert spaces and for learning with expert advice. These algorithms achieve optimal regret bounds that depend on the unknown competitors, without having to tune the learning rates with oracle choices. We present a new intuitive framework to design parameter-free algorithms for \emph{both} online linear optimization over Hilbert spaces and for learning with expert advice, based on reductions to betting on outcomes of adversarial coins. We instantiate it using a betting algorithm based on the Krichevsky-Trofimov estimator. The resulting algorithms are simple, with no parameters to be tuned, and they improve or match previous results in terms of regret guarantee and per-round complexity.

cs.LG↗
arXiv2016

Scalable Greedy Algorithms for Transfer Learning

In this paper we consider the binary transfer learning problem, focusing on how to select and combine sources from a large pool to yield a good performance on a target task. Constraining our scenario to real world, we do not assume the direct access to the source data, but rather we employ the source hypotheses trained from them. We propose an efficient algorithm that selects relevant source hypotheses and feature dimensions simultaneously, building on the literature on the best subset selection problem. Our algorithm achieves state-of-the-art results on three computer vision datasets, substantially outperforming both transfer learning and popular feature selection baselines in a small-sample setting. We also present a randomized variant that achieves the same results with the computational cost independent from the number of source hypotheses and feature dimensions. Also, we theoretically prove that, under reasonable assumptions on the source hypotheses, our algorithm can learn effectively from few examples.

cs.CV↗
arXiv2016

Solving Ridge Regression using Sketched Preconditioned SVRG

We develop a novel preconditioning method for ridge regression, based on recent linear sketching methods. By equipping Stochastic Variance Reduced Gradient (SVRG) with this preconditioning process, we obtain a significant speed-up relative to fast stochastic methods such as SVRG, SDCA and SAG.

cs.LG↗
arXiv2015

Optimal Non-Asymptotic Lower Bound on the Minimax Regret of Learning with Expert Advice

We prove non-asymptotic lower bounds on the expectation of the maximum of $d$ independent Gaussian variables and the expectation of the maximum of $d$ independent symmetric random walks. Both lower bounds recover the optimal leading constant in the limit. A simple application of the lower bound for random walks is an (asymptotically optimal) non-asymptotic lower bound on the minimax regret of online learning with expert advice.

stat.ML↗
arXiv2015

The ABACOC Algorithm: a Novel Approach for Nonparametric Classification of Data Streams

Stream mining poses unique challenges to machine learning: predictive models are required to be scalable, incrementally trainable, must remain bounded in size (even when the data stream is arbitrarily long), and be nonparametric in order to achieve high accuracy even in complex and dynamic environments. Moreover, the learning system must be parameterless ---traditional tuning methods are problematic in streaming settings--- and avoid requiring prior knowledge of the number of distinct class labels occurring in the stream. In this paper, we introduce a new algorithmic approach for nonparametric learning in data streams. Our approach addresses all above mentioned challenges by learning a model that covers the input space using simple local classifiers. The distribution of these classifiers dynamically adapts to the local (unknown) complexity of the classification problem, thus achieving a good balance between model complexity and predictive accuracy. We design four variants of our approach of increasing adaptivity. By means of an extensive empirical evaluation against standard nonparametric baselines, we show state-of-the-art results in terms of accuracy versus model size. For the variant that imposes a strict bound on the model size, we show better performance against all other methods measured at the same model size value. Our empirical analysis is complemented by a theoretical performance guarantee which does not rely on any stochastic assumption on the source generating the stream.

stat.ML↗
arXiv2015

Fast Rates by Transferring from Auxiliary Hypotheses

In this work we consider the learning setting where, in addition to the training set, the learner receives a collection of auxiliary hypotheses originating from other tasks. We focus on a broad class of ERM-based linear algorithms that can be instantiated with any non-negative smooth loss function and any strongly convex regularizer. We establish generalization and excess risk bounds, showing that, if the algorithm is fed with a good combination of source hypotheses, generalization happens at the fast rate $\mathcal{O}(1/m)$ instead of the usual $\mathcal{O}(1/\sqrt{m})$. On the other hand, if the source hypotheses combination is a misfit for the target task, we recover the usual learning rate. As a byproduct of our study, we also prove a new bound on the Rademacher complexity of the smooth loss class under weaker assumptions compared to previous works.

cs.LG↗
arXiv2015

Scale-Free Algorithms for Online Linear Optimization

We design algorithms for online linear optimization that have optimal regret and at the same time do not need to know any upper or lower bounds on the norm of the loss vectors. We achieve adaptiveness to norms of loss vectors by scale invariance, i.e., our algorithms make exactly the same decisions if the sequence of loss vectors is multiplied by any positive constant. Our algorithms work for any decision set, bounded or unbounded. For unbounded decisions sets, these are the first truly adaptive algorithms for online linear optimization.

cs.LG↗
arXiv2014

A Generalized Online Mirror Descent with Applications to Classification and Regression

Online learning algorithms are fast, memory-efficient, easy to implement, and applicable to many prediction problems, including classification, regression, and ranking. Several online algorithms were proposed in the past few decades, some based on additive updates, like the Perceptron, and some on multiplicative updates, like Winnow. A unifying perspective on the design and the analysis of online algorithms is provided by online mirror descent, a general prediction strategy from which most first-order algorithms can be obtained as special cases. We generalize online mirror descent to time-varying regularizers with generic updates. Unlike standard mirror descent, our more general formulation also captures second order algorithms, algorithms for composite losses and algorithms for adaptive filtering. Moreover, we recover, and sometimes improve, known regret bounds as special cases of our analysis using specific regularizers. Finally, we show the power of our approach by deriving a new second order algorithm with a regret bound invariant with respect to arbitrary rescalings of individual features.

cs.LG↗
arXiv2014

Simultaneous Model Selection and Optimization through Parameter-free Stochastic Learning

Stochastic gradient descent algorithms for training linear and kernel predictors are gaining more and more importance, thanks to their scalability. While various methods have been proposed to speed up their convergence, the model selection phase is often ignored. In fact, in theoretical works most of the time assumptions are made, for example, on the prior knowledge of the norm of the optimal solution, while in the practical world validation methods remain the only viable approach. In this paper, we propose a new kernel-based stochastic gradient descent algorithm that performs model selection while training, with no parameters to tune, nor any form of cross-validation. The algorithm builds on recent advancement in online learning theory for unconstrained settings, to estimate over time the right regularization in a data-dependent way. Optimal rates of convergence are proved under standard smoothness assumptions on the target function, using the range space of the fractional integral operator associated with the kernel.

cs.LG↗