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Register a new userMinimax Regret for Bandit Convex Optimisation of Ridge Functions
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Tor Lattimore
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Tor Lattimore
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We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form $f_t(x) = g_t(langle x, thetarangle)$ for convex $g_t : mathbb R to mathbb R$ and unknown $theta in mathbb R^d$ that is homogeneous over time. We provide a short information-theoretic proof that the minimax regret is at most $O(d sqrt{n} log(n operatorname{diam}(mathcal K)))$ where $n$ is the number of interactions, $d$ the dimension and $operatorname{diam}(mathcal K)$ is the diameter of the constraint set.
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Several learning problems involve solving min-max problems, e.g., empirical distributional robust learning or learning with non-standard aggregated losses. More specifically, these problems are convex-linear problems where the minimization is carried out over the model parameters $winmathcal{W}$ and the maximization over the empirical distribution $pinmathcal{K}$ of the training set indexes, where $mathcal{K}$ is the simplex or a subset of it. To design efficient methods, we let an online learning algorithm play against a (combinatorial) bandit algorithm. We argue that the efficiency of such approaches critically depends on the structure of $mathcal{K}$ and propose two properties of $mathcal{K}$ that facilitate designing efficient algorithms. We focus on a specific family of sets $mathcal{S}_{n,k}$ encompassing various learning applications and provide high-probability convergence guarantees to the minimax values.
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In this work, we study the problem of global optimization in univariate loss functions, where we analyze the regret of the popular lower bounding algorithms (e.g., Piyavskii-Shubert algorithm). For any given time $T$, instead of the widely available simple regret (which is the difference of the losses between the best estimation up to $T$ and the global optimizer), we study the cumulative regret up to that time. With a suitable lower bounding algorithm, we show that it is possible to achieve satisfactory cumulative regret bounds for different classes of functions. For Lipschitz continuous functions with the parameter $L$, we show that the cumulative regret is $O(Llog T)$. For Lipschitz smooth functions with the parameter $H$, we show that the cumulative regret is $O(H)$. We also analytically extend our results for a broader class of functions that covers both the Lipschitz continuous and smooth functions individually.
We consider the problem of inverse kinematics (IK), where one wants to find the parameters of a given kinematic skeleton that best explain a set of observed 3D joint locations. The kinematic skeleton has a tree structure, where each node is a joint that has an associated geometric transformation that is propagated to all its child nodes. The IK problem has various applications in vision and graphics, for example for tracking or reconstructing articulated objects, such as human hands or bodies. Most commonly, the IK problem is tackled using local optimisation methods. A major downside of these approaches is that, due to the non-convex nature of the problem, such methods are prone to converge to unwanted local optima and therefore require a good initialisation. In this paper we propose a convex optimisation approach for the IK problem based on semidefinite programming, which admits a polynomial-time algorithm that globally solves (a relaxation of) the IK problem. Experimentally, we demonstrate that the proposed method significantly outperforms local optimisation methods using different real-world skeletons.
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