Online convex optimization (OCO) is a widely used framework in online learning. In each round, the learner chooses a decision in some convex set and an adversary chooses a convex loss function, and then the learner suffers the loss associated with their chosen decision. However, in many of the motivating applications the loss of the learner depends not only on the current decision but on the entire history of decisions until that point. The OCO framework and existing generalizations thereof fail to capture this. In this work we introduce a generalization of the OCO framework, ``Online Convex Optimization with Unbounded Memory'', that captures long-term dependence on past decisions. We introduce the notion of $p$-effective memory capacity, $H_p$, that quantifies the maximum influence of past decisions on current losses. We prove a $O(\sqrt{H_1 T})$ policy regret bound and a stronger $O(\sqrt{H_p T})$ policy regret bound under mild additional assumptions. These bounds are optimal in terms of their dependence on the time horizon $T$. We show the broad applicability of our framework by using it to derive regret bounds, and to simplify existing regret bound derivations, for a variety of online learning problems including an online variant of performative prediction and online linear control.
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We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Combinatorica 15(3):435-454, 1995). Williamson et al.\ prove an approximation guarantee of two for connectivity augmentation problems where the connectivity requirements can be specified by so-called uncrossable functions. They state: ``Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar. This property characterizes uncrossable functions\dots\ A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems.'' Our main result proves an $O(1)$-approximation guarantee via the primal-dual method for a class of functions that generalizes the notion of an uncrossable function. We mention that the support of every optimal dual solution could be non-laminar for instances that can be handled by our methods. We present two applications of our main result: (1) An $O(1)$-approximation algorithm for augmenting the family of near-minimum cuts of a graph. (2) An $O(1)$-approximation algorithm for the model of $(p,2)$-Flexible Graph Connectivity. Keywords: { Primal-Dual Method, Network Design, $f$-Connectivity Problem, Near-Minimum Cuts, Approximation Algorithms, Flexible Graph Connectivity. }
We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and consider a particular Riemannian quotient geometry in a search space that has a total space equipped with the Euclidean metric. When the original objective f satisfies standard restricted strong convexity and smoothness properties, we characterize the global landscape of the factorized objective under the Riemannian quotient geometry. We show the entire search space can be divided into three regions: (R1) the region near the target parameter of interest, where the factorized objective is geodesically strongly convex and smooth; (R2) the region containing neighborhoods of all strict saddle points; (R3) the remaining regions, where the factorized objective has a large gradient. To our best knowledge, this is the first global landscape analysis of the Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our results provide a fully geometric explanation for the superior performance of vanilla gradient descent under the Burer-Monteiro factorization. When f satisfies a weaker restricted strict convexity property, we show there exists a neighborhood near local minimizers such that the factorized objective is geodesically convex. To prove our results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge. Our conclusions are also based on a result of independent interest stating that the geodesic ball centered at Y with a radius 1/3 of the least singular value of Y is a geodesically convex set under the Riemannian quotient geometry, which as a corollary, also implies a quantitative bound of the convexity radius in the Bures-Wasserstein space. The convexity radius obtained is sharp up to constants.
In this study, we investigate the generalization of LSTM, ReLU and GRU models on counting tasks over long sequences. Previous theoretical work has established that RNNs with ReLU activation and LSTMs have the capacity for counting with suitable configuration, while GRUs have limitations that prevent correct counting over longer sequences. Despite this and some positive empirical results for LSTMs on Dyck-1 languages, our experimental results show that LSTMs fail to learn correct counting behavior for sequences that are significantly longer than in the training data. ReLUs show much larger variance in behavior and in most cases worse generalization. The long sequence generalization is empirically related to validation loss, but reliable long sequence generalization seems not practically achievable through backpropagation with current techniques. We demonstrate different failure modes for LSTMs, GRUs and ReLUs. In particular, we observe that the saturation of activation functions in LSTMs and the correct weight setting for ReLUs to generalize counting behavior are not achieved in standard training regimens. In summary, learning generalizable counting behavior is still an open problem and we discuss potential approaches for further research.
We consider a multi-agent episodic MDP setup where an agent (leader) takes action at each step of the episode followed by another agent (follower). The state evolution and rewards depend on the joint action pair of the leader and the follower. Such type of interactions can find applications in many domains such as smart grids, mechanism design, security, and policymaking. We are interested in how to learn policies for both the players with provable performance guarantee under a bandit feedback setting. We focus on a setup where both the leader and followers are {\em non-myopic}, i.e., they both seek to maximize their rewards over the entire episode and consider a linear MDP which can model continuous state-space which is very common in many RL applications. We propose a {\em model-free} RL algorithm and show that $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret bounds can be achieved for both the leader and the follower, where $d$ is the dimension of the feature mapping, $H$ is the length of the episode, and $T$ is the total number of steps under the bandit feedback information setup. Thus, our result holds even when the number of states becomes infinite. The algorithm relies on {\em novel} adaptation of the LSVI-UCB algorithm. Specifically, we replace the standard greedy policy (as the best response) with the soft-max policy for both the leader and the follower. This turns out to be key in establishing uniform concentration bound for the value functions. To the best of our knowledge, this is the first sub-linear regret bound guarantee for the Markov games with non-myopic followers with function approximation.
Deep learning has had tremendous success at learning low-dimensional representations of high-dimensional data. This success would be impossible if there was no hidden low-dimensional structure in data of interest; this existence is posited by the manifold hypothesis, which states that the data lies on an unknown manifold of low intrinsic dimension. In this paper, we argue that this hypothesis does not properly capture the low-dimensional structure typically present in image data. Assuming that data lies on a single manifold implies intrinsic dimension is identical across the entire data space, and does not allow for subregions of this space to have a different number of factors of variation. To address this deficiency, we put forth the union of manifolds hypothesis, which states that data lies on a disjoint union of manifolds of varying intrinsic dimensions. We empirically verify this hypothesis on commonly-used image datasets, finding that indeed, observed data lies on a disconnected set and that intrinsic dimension is not constant. We also provide insights into the implications the union of manifolds hypothesis has for deep learning, both supervised and unsupervised, showing that designing models with an inductive bias for this structure improves performance across classification and generative modelling tasks.
Deep neural networks are usually trained with stochastic gradient descent (SGD), which minimizes objective function using very rough approximations of gradient, only averaging to the real gradient. Standard approaches like momentum or ADAM only consider a single direction, and do not try to model distance from extremum - neglecting valuable information from calculated sequence of gradients, often stagnating in some suboptimal plateau. Second order methods could exploit these missed opportunities, however, beside suffering from very large cost and numerical instabilities, many of them attract to suboptimal points like saddles due to negligence of signs of curvatures (as eigenvalues of Hessian). Saddle-free Newton method is a rare example of addressing this issue - changes saddle attraction into repulsion, and was shown to provide essential improvement for final value this way. However, it neglects noise while modelling second order behavior, focuses on Krylov subspace for numerical reasons, and requires costly eigendecomposion. Maintaining SFN advantages, there are proposed inexpensive ways for exploiting these opportunities. Second order behavior is linear dependence of first derivative - we can optimally estimate it from sequence of noisy gradients with least square linear regression, in online setting here: with weakening weights of old gradients. Statistically relevant subspace is suggested by PCA of recent noisy gradients - in online setting it can be made by slowly rotating considered directions toward new gradients, gradually replacing old directions with recent statistically relevant. Eigendecomposition can be also performed online: with regularly performed step of QR method to maintain diagonal Hessian. Outside the second order modeled subspace we can simultaneously perform gradient descent.
We study the problem of zero-order optimization of a strongly convex function. The goal is to find the minimizer of the function by a sequential exploration of its values, under measurement noise. We study the impact of higher order smoothness properties of the function on the optimization error and on the cumulative regret. To solve this problem we consider a randomized approximation of the projected gradient descent algorithm. The gradient is estimated by a randomized procedure involving two function evaluations and a smoothing kernel. We derive upper bounds for this algorithm both in the constrained and unconstrained settings and prove minimax lower bounds for any sequential search method. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters. Based on this algorithm, we also propose an estimator of the minimum value of the function achieving almost sharp oracle behavior. We compare our results with the state-of-the-art, highlighting a number of key improvements.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.
Learning similarity functions between image pairs with deep neural networks yields highly correlated activations of embeddings. In this work, we show how to improve the robustness of such embeddings by exploiting the independence within ensembles. To this end, we divide the last embedding layer of a deep network into an embedding ensemble and formulate training this ensemble as an online gradient boosting problem. Each learner receives a reweighted training sample from the previous learners. Further, we propose two loss functions which increase the diversity in our ensemble. These loss functions can be applied either for weight initialization or during training. Together, our contributions leverage large embedding sizes more effectively by significantly reducing correlation of the embedding and consequently increase retrieval accuracy of the embedding. Our method works with any differentiable loss function and does not introduce any additional parameters during test time. We evaluate our metric learning method on image retrieval tasks and show that it improves over state-of-the-art methods on the CUB 200-2011, Cars-196, Stanford Online Products, In-Shop Clothes Retrieval and VehicleID datasets.
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