Backtracking is an inexact line search procedure that selects the first value in a sequence $x_0, x_0\beta, x_0\beta^2...$ that satisfies $g(x)\leq 0$ on $\mathbb{R}_+$ with $g(x)\leq 0$ iff $x\leq x^*$. This procedure is widely used in descent direction optimization algorithms with Armijo-type conditions. It both returns an estimate in $(\beta x^*,x^*]$ and enjoys an upper-bound $\lceil \log_{\beta} \epsilon/x_0 \rceil$ on the number of function evaluations to terminate, with $\epsilon$ a lower bound on $x^*$. The basic bracketing mechanism employed in several root-searching methods is adapted here for the purpose of performing inexact line searches, leading to a new class of inexact line search procedures. The traditional bisection algorithm for root-searching is transposed into a very simple method that completes the same inexact line search in at most $\lceil \log_2 \log_{\beta} \epsilon/x_0 \rceil$ function evaluations. A recent bracketing algorithm for root-searching which presents both minmax function evaluation cost (as the bisection algorithm) and superlinear convergence is also transposed, asymptotically requiring $\sim \log \log \log \epsilon/x_0 $ function evaluations for sufficiently smooth functions. Other bracketing algorithms for root-searching can be adapted in the same way. Numerical experiments suggest time savings of 50\% to 80\% in each call to the inexact search procedure.
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In bandits with distribution shifts, one aims to automatically detect an unknown number $L$ of changes in reward distribution, and restart exploration when necessary. While this problem remained open for many years, a recent breakthrough of Auer et al. (2018, 2019) provide the first adaptive procedure to guarantee an optimal (dynamic) regret $\sqrt{LT}$, for $T$ rounds, with no knowledge of $L$. However, not all distributional shifts are equally severe, e.g., suppose no best arm switches occur, then we cannot rule out that a regret $O(\sqrt{T})$ may remain possible; in other words, is it possible to achieve dynamic regret that optimally scales only with an unknown number of severe shifts? This unfortunately has remained elusive, despite various attempts (Auer et al., 2019, Foster et al., 2020). We resolve this problem in the case of two-armed bandits: we derive an adaptive procedure that guarantees a dynamic regret of order $\tilde{O}(\sqrt{\tilde{L} T})$, where $\tilde L \ll L$ captures an unknown number of severe best arm changes, i.e., with significant switches in rewards, and which last sufficiently long to actually require a restart. As a consequence, for any number $L$ of distributional shifts outside of these severe shifts, our procedure achieves regret just $\tilde{O}(\sqrt{T})\ll \tilde{O}(\sqrt{LT})$. Finally, we note that our notion of severe shift applies in both classical settings of stochastic switching bandits and of adversarial bandits.
We study episodic reinforcement learning in non-stationary linear (a.k.a. low-rank) Markov Decision Processes (MDPs), i.e, both the reward and transition kernel are linear with respect to a given feature map and are allowed to evolve either slowly or abruptly over time. For this problem setting, we propose OPT-WLSVI an optimistic model-free algorithm based on weighted least squares value iteration which uses exponential weights to smoothly forget data that are far in the past. We show that our algorithm, when competing against the best policy at each time, achieves a regret that is upper bounded by $\widetilde{\mathcal{O}}(d^{5/4}H^2 \Delta^{1/4} K^{3/4})$ where $d$ is the dimension of the feature space, $H$ is the planning horizon, $K$ is the number of episodes and $\Delta$ is a suitable measure of non-stationarity of the MDP. Moreover, we point out technical gaps in the study of forgetting strategies in non-stationary linear bandits setting made by previous works and we propose a fix to their regret analysis.
We study the problem of solving Packing Integer Programs (PIPs) in the online setting, where columns in $[0,1]^d$ of the constraint matrix are revealed sequentially, and the goal is to pick a subset of the columns that sum to at most $B$ in each coordinate while maximizing the objective. Excellent results are known in the secretary setting, where the columns are adversarially chosen, but presented in a uniformly random order. However, these existing algorithms are susceptible to adversarial attacks: they try to "learn" characteristics of a good solution, but tend to over-fit to the model, and hence a small number of adversarial corruptions can cause the algorithm to fail. In this paper, we give the first robust algorithms for Packing Integer Programs, specifically in the recently proposed Byzantine Secretary framework. Our techniques are based on a two-level use of online learning, to robustly learn an approximation to the optimal value, and then to use this robust estimate to pick a good solution. These techniques are general and we use them to design robust algorithms for PIPs in the prophet model as well, specifically in the Prophet-with-Augmentations framework. We also improve known results in the Byzantine Secretary framework: we make the non-constructive results algorithmic and improve the existing bounds for single-item and matroid constraints.
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.
For deploying a deep learning model into production, it needs to be both accurate and compact to meet the latency and memory constraints. This usually results in a network that is deep (to ensure performance) and yet thin (to improve computational efficiency). In this paper, we propose an efficient method to train a deep thin network with a theoretic guarantee. Our method is motivated by model compression. It consists of three stages. In the first stage, we sufficiently widen the deep thin network and train it until convergence. In the second stage, we use this well-trained deep wide network to warm up (or initialize) the original deep thin network. This is achieved by letting the thin network imitate the immediate outputs of the wide network from layer to layer. In the last stage, we further fine tune this well initialized deep thin network. The theoretical guarantee is established by using mean field analysis, which shows the advantage of layerwise imitation over traditional training deep thin networks from scratch by backpropagation. We also conduct large-scale empirical experiments to validate our approach. By training with our method, ResNet50 can outperform ResNet101, and BERT_BASE can be comparable with BERT_LARGE, where both the latter models are trained via the standard training procedures as in the literature.
This paper proposes a model-free Reinforcement Learning (RL) algorithm to synthesise policies for an unknown Markov Decision Process (MDP), such that a linear time property is satisfied. We convert the given property into a Limit Deterministic Buchi Automaton (LDBA), then construct a synchronized MDP between the automaton and the original MDP. According to the resulting LDBA, a reward function is then defined over the state-action pairs of the product MDP. With this reward function, our algorithm synthesises a policy whose traces satisfies the linear time property: as such, the policy synthesis procedure is "constrained" by the given specification. Additionally, we show that the RL procedure sets up an online value iteration method to calculate the maximum probability of satisfying the given property, at any given state of the MDP - a convergence proof for the procedure is provided. Finally, the performance of the algorithm is evaluated via a set of numerical examples. We observe an improvement of one order of magnitude in the number of iterations required for the synthesis compared to existing approaches.
For neural networks (NNs) with rectified linear unit (ReLU) or binary activation functions, we show that their training can be accomplished in a reduced parameter space. Specifically, the weights in each neuron can be trained on the unit sphere, as opposed to the entire space, and the threshold can be trained in a bounded interval, as opposed to the real line. We show that the NNs in the reduced parameter space are mathematically equivalent to the standard NNs with parameters in the whole space. The reduced parameter space shall facilitate the optimization procedure for the network training, as the search space becomes (much) smaller. We demonstrate the improved training performance using numerical examples.
In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.
Model initialisation is an important component of object tracking. Tracking algorithms are generally provided with the first frame of a sequence and a bounding box (BB) indicating the location of the object. This BB may contain a large number of background pixels in addition to the object and can lead to parts-based tracking algorithms initialising their object models in background regions of the BB. In this paper, we tackle this as a missing labels problem, marking pixels sufficiently away from the BB as belonging to the background and learning the labels of the unknown pixels. Three techniques, One-Class SVM (OC-SVM), Sampled-Based Background Model (SBBM) (a novel background model based on pixel samples), and Learning Based Digital Matting (LBDM), are adapted to the problem. These are evaluated with leave-one-video-out cross-validation on the VOT2016 tracking benchmark. Our evaluation shows both OC-SVMs and SBBM are capable of providing a good level of segmentation accuracy but are too parameter-dependent to be used in real-world scenarios. We show that LBDM achieves significantly increased performance with parameters selected by cross validation and we show that it is robust to parameter variation.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
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