arXiv subjects

Zakaria Mhammedi

Publications and source records attributed to Zakaria Mhammedi.

30 records · Page 2Linked to original sources

arXiv2022

Exploiting the Curvature of Feasible Sets for Faster Projection-Free Online Learning

In this paper, we develop new efficient projection-free algorithms for Online Convex Optimization (OCO). Online Gradient Descent (OGD) is an example of a classical OCO algorithm that guarantees the optimal $O(\sqrt{T})$ regret bound. However, OGD and other projection-based OCO algorithms need to perform a Euclidean projection onto the feasible set $\mathcal{C}\subset \mathbb{R}^d$ whenever their iterates step outside $\mathcal{C}$. For various sets of interests, this projection step can be computationally costly, especially when the ambient dimension is large. This has motivated the development of projection-free OCO algorithms that swap Euclidean projections for often much cheaper operations such as Linear Optimization (LO). However, state-of-the-art LO-based algorithms only achieve a suboptimal $O(T^{3/4})$ regret for general OCO. In this paper, we leverage recent results in parameter-free Online Learning, and develop an OCO algorithm that makes two calls to an LO Oracle per round and achieves the near-optimal $\widetilde{O}(\sqrt{T})$ regret whenever the feasible set is strongly convex. We also present an algorithm for general convex sets that makes $\widetilde O(d)$ expected number of calls to an LO Oracle per round and guarantees a $\widetilde O(T^{2/3})$ regret, improving on the previous best $O(T^{3/4})$. We achieve the latter by approximating any convex set $\mathcal{C}$ by a strongly convex one, where LO can be performed using $\widetilde {O}(d)$ expected number of calls to an LO Oracle for $\mathcal{C}$.

cs.LG↗

arXiv2022

Damped Online Newton Step for Portfolio Selection

We revisit the classic online portfolio selection problem, where at each round a learner selects a distribution over a set of portfolios to allocate its wealth. It is known that for this problem a logarithmic regret with respect to Cover's loss is achievable using the Universal Portfolio Selection algorithm, for example. However, all existing algorithms that achieve a logarithmic regret for this problem have per-round time and space complexities that scale polynomially with the total number of rounds, making them impractical. In this paper, we build on the recent work by Haipeng et al. 2018 and present the first practical online portfolio selection algorithm with a logarithmic regret and whose per-round time and space complexities depend only logarithmically on the horizon. Behind our approach are two key technical novelties of independent interest. We first show that the Damped Online Newton steps can approximate mirror descent iterates well, even when dealing with time-varying regularizers. Second, we present a new meta-algorithm that achieves an adaptive logarithmic regret (i.e. a logarithmic regret on any sub-interval) for mixable losses.

cs.LG↗

arXiv2022

Risk-Monotonicity in Statistical Learning

Acquisition of data is a difficult task in many applications of machine learning, and it is only natural that one hopes and expects the population risk to decrease (better performance) monotonically with increasing data points. It turns out, somewhat surprisingly, that this is not the case even for the most standard algorithms that minimize the empirical risk. Non-monotonic behavior of the risk and instability in training have manifested and appeared in the popular deep learning paradigm under the description of double descent. These problems highlight the current lack of understanding of learning algorithms and generalization. It is, therefore, crucial to pursue this concern and provide a characterization of such behavior. In this paper, we derive the first consistent and risk-monotonic (in high probability) algorithms for a general statistical learning setting under weak assumptions, consequently answering some questions posed by Viering et al. 2019 on how to avoid non-monotonic behavior of risk curves. We further show that risk monotonicity need not necessarily come at the price of worse excess risk rates. To achieve this, we derive new empirical Bernstein-like concentration inequalities of independent interest that hold for certain non-i.i.d.~processes such as Martingale Difference Sequences.

cs.LG↗

arXiv2021

Efficient Projection-Free Online Convex Optimization with Membership Oracle

In constrained convex optimization, existing methods based on the ellipsoid or cutting plane method do not scale well with the dimension of the ambient space. Alternative approaches such as Projected Gradient Descent only provide a computational benefit for simple convex sets such as Euclidean balls, where Euclidean projections can be performed efficiently. For other sets, the cost of the projections can be too high. To circumvent these issues, alternative methods based on the famous Frank-Wolfe algorithm have been studied and used. Such methods use a Linear Optimization Oracle at each iteration instead of Euclidean projections; the former can often be performed efficiently. Such methods have also been extended to the online and stochastic optimization settings. However, the Frank-Wolfe algorithm and its variants do not achieve the optimal performance, in terms of regret or rate, for general convex sets. What is more, the Linear Optimization Oracle they use can still be computationally expensive in some cases. In this paper, we move away from Frank-Wolfe style algorithms and present a new reduction that turns any algorithm A defined on a Euclidean ball (where projections are cheap) to an algorithm on a constrained set C contained within the ball, without sacrificing the performance of the original algorithm A by much. Our reduction requires O(T log T) calls to a Membership Oracle on C after T rounds, and no linear optimization on C is needed. Using our reduction, we recover optimal regret bounds [resp. rates], in terms of the number of iterations, in online [resp. stochastic] convex optimization. Our guarantees are also useful in the offline convex optimization setting when the dimension of the ambient space is large.

math.OC↗

arXiv2020

Learning the Linear Quadratic Regulator from Nonlinear Observations

We introduce a new problem setting for continuous control called the LQR with Rich Observations, or RichLQR. In our setting, the environment is summarized by a low-dimensional continuous latent state with linear dynamics and quadratic costs, but the agent operates on high-dimensional, nonlinear observations such as images from a camera. To enable sample-efficient learning, we assume that the learner has access to a class of decoder functions (e.g., neural networks) that is flexible enough to capture the mapping from observations to latent states. We introduce a new algorithm, RichID, which learns a near-optimal policy for the RichLQR with sample complexity scaling only with the dimension of the latent state space and the capacity of the decoder function class. RichID is oracle-efficient and accesses the decoder class only through calls to a least-squares regression oracle. Our results constitute the first provable sample complexity guarantee for continuous control with an unknown nonlinearity in the system model and general function approximation.

cs.LG↗

arXiv2020

Lipschitz and Comparator-Norm Adaptivity in Online Learning

We study Online Convex Optimization in the unbounded setting where neither predictions nor gradient are constrained. The goal is to simultaneously adapt to both the sequence of gradients and the comparator. We first develop parameter-free and scale-free algorithms for a simplified setting with hints. We present two versions: the first adapts to the squared norms of both comparator and gradients separately using $O(d)$ time per round, the second adapts to their squared inner products (which measure variance only in the comparator direction) in time $O(d^3)$ per round. We then generalize two prior reductions to the unbounded setting; one to not need hints, and a second to deal with the range ratio problem (which already arises in prior work). We discuss their optimality in light of prior and new lower bounds. We apply our methods to obtain sharper regret bounds for scale-invariant online prediction with linear models.

cs.LG↗

arXiv2020

PAC-Bayesian Bound for the Conditional Value at Risk

Conditional Value at Risk (CVaR) is a family of "coherent risk measures" which generalize the traditional mathematical expectation. Widely used in mathematical finance, it is garnering increasing interest in machine learning, e.g., as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the CVaR of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical CVaR is small. We achieve this by reducing the problem of estimating CVaR to that of merely estimating an expectation. This then enables us, as a by-product, to obtain concentration inequalities for CVaR even when the random variable in question is unbounded.

cs.LG↗

arXiv2019

PAC-Bayes Un-Expected Bernstein Inequality

We present a new PAC-Bayesian generalization bound. Standard bounds contain a $\sqrt{L_n \cdot \KL/n}$ complexity term which dominates unless $L_n$, the empirical error of the learning algorithm's randomized predictions, vanishes. We manage to replace $L_n$ by a term which vanishes in many more situations, essentially whenever the employed learning algorithm is sufficiently stable on the dataset at hand. Our new bound consistently beats state-of-the-art bounds both on a toy example and on UCI datasets (with large enough $n$). Theoretically, unlike existing bounds, our new bound can be expected to converge to $0$ faster whenever a Bernstein/Tsybakov condition holds, thus connecting PAC-Bayesian generalization and {\em excess risk\/} bounds---for the latter it has long been known that faster convergence can be obtained under Bernstein conditions. Our main technical tool is a new concentration inequality which is like Bernstein's but with $X^2$ taken outside its expectation.

cs.LG↗

arXiv2019

Lipschitz Adaptivity with Multiple Learning Rates in Online Learning

We aim to design adaptive online learning algorithms that take advantage of any special structure that might be present in the learning task at hand, with as little manual tuning by the user as possible. A fundamental obstacle that comes up in the design of such adaptive algorithms is to calibrate a so-called step-size or learning rate hyperparameter depending on variance, gradient norms, etc. A recent technique promises to overcome this difficulty by maintaining multiple learning rates in parallel. This technique has been applied in the MetaGrad algorithm for online convex optimization and the Squint algorithm for prediction with expert advice. However, in both cases the user still has to provide in advance a Lipschitz hyperparameter that bounds the norm of the gradients. Although this hyperparameter is typically not available in advance, tuning it correctly is crucial: if it is set too small, the methods may fail completely; but if it is taken too large, performance deteriorates significantly. In the present work we remove this Lipschitz hyperparameter by designing new versions of MetaGrad and Squint that adapt to its optimal value automatically. We achieve this by dynamically updating the set of active learning rates. For MetaGrad, we further improve the computational efficiency of handling constraints on the domain of prediction, and we remove the need to specify the number of rounds in advance.

cs.LG↗

arXiv2018

Constant Regret, Generalized Mixability, and Mirror Descent

We consider the setting of prediction with expert advice; a learner makes predictions by aggregating those of a group of experts. Under this setting, and for the right choice of loss function and "mixing" algorithm, it is possible for the learner to achieve a constant regret regardless of the number of prediction rounds. For example, a constant regret can be achieved for \emph{mixable} losses using the \emph{aggregating algorithm}. The \emph{Generalized Aggregating Algorithm} (GAA) is a name for a family of algorithms parameterized by convex functions on simplices (entropies), which reduce to the aggregating algorithm when using the \emph{Shannon entropy} $\operatorname{S}$. For a given entropy $Φ$, losses for which a constant regret is possible using the \textsc{GAA} are called $Φ$-mixable. Which losses are $Φ$-mixable was previously left as an open question. We fully characterize $Φ$-mixability and answer other open questions posed by \cite{Reid2015}. We show that the Shannon entropy $\operatorname{S}$ is fundamental in nature when it comes to mixability; any $Φ$-mixable loss is necessarily $\operatorname{S}$-mixable, and the lowest worst-case regret of the \textsc{GAA} is achieved using the Shannon entropy. Finally, by leveraging the connection between the \emph{mirror descent algorithm} and the update step of the GAA, we suggest a new \emph{adaptive} generalized aggregating algorithm and analyze its performance in terms of the regret bound.

cs.LG↗

arXiv2017

Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections

The problem of learning long-term dependencies in sequences using Recurrent Neural Networks (RNNs) is still a major challenge. Recent methods have been suggested to solve this problem by constraining the transition matrix to be unitary during training which ensures that its norm is equal to one and prevents exploding gradients. These methods either have limited expressiveness or scale poorly with the size of the network when compared with the simple RNN case, especially when using stochastic gradient descent with a small mini-batch size. Our contributions are as follows; we first show that constraining the transition matrix to be unitary is a special case of an orthogonal constraint. Then we present a new parametrisation of the transition matrix which allows efficient training of an RNN while ensuring that the matrix is always orthogonal. Our results show that the orthogonal constraint on the transition matrix applied through our parametrisation gives similar benefits to the unitary constraint, without the time complexity limitations.

cs.LG↗

arXiv2017

Adversarial Generation of Real-time Feedback with Neural Networks for Simulation-based Training

Simulation-based training (SBT) is gaining popularity as a low-cost and convenient training technique in a vast range of applications. However, for a SBT platform to be fully utilized as an effective training tool, it is essential that feedback on performance is provided automatically in real-time during training. It is the aim of this paper to develop an efficient and effective feedback generation method for the provision of real-time feedback in SBT. Existing methods either have low effectiveness in improving novice skills or suffer from low efficiency, resulting in their inability to be used in real-time. In this paper, we propose a neural network based method to generate feedback using the adversarial technique. The proposed method utilizes a bounded adversarial update to minimize a L1 regularized loss via back-propagation. We empirically show that the proposed method can be used to generate simple, yet effective feedback. Also, it was observed to have high effectiveness and efficiency when compared to existing methods, thus making it a promising option for real-time feedback generation in SBT.

cs.LG↗