We explore uncertainty quantification in large language models (LLMs), with the goal to identify when uncertainty in responses given a query is large. We simultaneously consider both epistemic and aleatoric uncertainties, where the former comes from the lack of knowledge about the ground truth (such as about facts or the language), and the latter comes from irreducible randomness (such as multiple possible answers). In particular, we derive an information-theoretic metric that allows to reliably detect when only epistemic uncertainty is large, in which case the output of the model is unreliable. This condition can be computed based solely on the output of the model obtained simply by some special iterative prompting based on the previous responses. Such quantification, for instance, allows to detect hallucinations (cases when epistemic uncertainty is high) in both single- and multi-answer responses. This is in contrast to many standard uncertainty quantification strategies (such as thresholding the log-likelihood of a response) where hallucinations in the multi-answer case cannot be detected. We conduct a series of experiments which demonstrate the advantage of our formulation. Further, our investigations shed some light on how the probabilities assigned to a given output by an LLM can be amplified by iterative prompting, which might be of independent interest.

We consider a contextual bandit problem with $S $ contexts and $K $ actions. In each round $t=1,2,\dots$ the learner observes a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into $r\le \min(S ,K)$ groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an $\epsilon$-optimal policy after using at most $\widetilde O(r (S +K )/\epsilon^2)$ samples with high probability and provide a matching $\widetilde\Omega(r (S +K )/\epsilon^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r ^3(S +K )T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O{\sqrt{\text{poly}(r)(S+K)T}}$ minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.

We study the non-stationary stochastic multi-armed bandit problem, where the reward statistics of each arm may change several times during the course of learning. The performance of a learning algorithm is evaluated in terms of its dynamic regret, which is defined as the difference between the expected cumulative reward of an agent choosing the optimal arm in every time step and the cumulative reward of the learning algorithm. One way to measure the hardness of such environments is to consider how many times the identity of the optimal arm can change. We propose a method that achieves, in $K$-armed bandit problems, a near-optimal $\widetilde O(\sqrt{K N(S+1)})$ dynamic regret, where $N$ is the time horizon of the problem and $S$ is the number of times the identity of the optimal arm changes, without prior knowledge of $S$. Previous works for this problem obtain regret bounds that scale with the number of changes (or the amount of change) in the reward functions, which can be much larger, or assume prior knowledge of $S$ to achieve similar bounds.

Learning with simulators is ubiquitous in mod-ern reinforcement learning (RL). The simulatorcan either correspond to a simplified version ofthe real environment (such as a physics simulation of a robot arm) or to the environment itself (such as in games like Atari and Go). Among algorithms that are provably sample-efficient in this setting, most make the unrealistic assumption that all possible environment states are known before learning begins, or perform global optimistic planning which is computationally inefficient. In this work, we focus on simulation-based RL under a more realistic local access protocol, where the state space is unknown and the simulator can only be queried at states that have previously been observed (initial states and those returned by previous queries). We propose an algorithm named CONFIDENT-LSVI based on the template of least-square value iteration. CONFIDENT-LSVI incrementally builds a coreset of important states and uses the simulator to revisit them. Assuming that the linear function class has low approximation error under the Bell-man optimality operator (a.k.a. low inherent Bell-man error), we bound the algorithm performance in terms of this error, and show that it is query-and computationally-efficient.

We study query and computationally efficient planning algorithms for discounted Markov decision processes (MDPs) with linear function approximation and a simulator. The agent is assumed to have local access to the simulator, meaning that the simulator can be queried only at states that have been encountered in previous steps. We propose two new algorithms for this setting, which we call confident Monte Carlo least-squares policy iteration (Confident MC-LSPI), and confident Monte Carlo Politex (Confident MC-Politex), respectively. The main novelty in our algorithms is that they gradually build a set of state-action pairs (“core set”) with which it can control the extrapolation errors. We show that our algorithms have polynomial query and computational cost in the dimension of the features, the effective planning horizon and the targeted sub-optimality, while the cost remains independent of the size of the state space. An interesting technical contribution of our work is the introduction of a novel proof technique that makes use of a virtual policy iteration algorithm. We use this method to leverage existing results on approximate policy iteration with $\ell_\infty$-bounded error to show that our algorithm can learn the optimal policy for the given initial state even only with local access to the simulator. We believe that this technique can be extended to broader settings beyond this work.

We study two model selection settings in stochastic linear bandits (LB). In the first setting, which we refer to as feature selection, the expected reward of the LB problem is in the linear span of at least one of $M$ feature maps (models). In the second setting, the reward parameter of the LB problem is arbitrarily selected from $M$ models represented as (possibly) overlapping balls in $\mathbb R^d$. However, the agent only has access to misspecified models, i.e., estimates of the centers and radii of the balls. We refer to this setting as parameter selection. For each setting, we develop and analyze a computationally efficient algorithm that is based on a reduction from bandits to full-information problems. This allows us to obtain regret bounds that are not worse (up to a $\sqrt{\log M}$ factor) than the case where the true model is known. This is the best reported dependence on the number of models $M$ in these settings. Finally, we empirically show the effectiveness of our algorithms using synthetic and real-world experiments.

Model-free reinforcement learning algorithms combined with value function approximation have recently achieved impressive performance in a variety of application domains. However, the theoretical understanding of such algorithms is limited, and existing results are largely focused on episodic or discounted Markov decision processes (MDPs). In this work, we present adaptive approximate policy iteration (AAPI), a learning scheme which enjoys a O(T^{2/3}) regret bound for undiscounted, continuing learning in uniformly ergodic MDPs. This is an improvement over the best existing bound of O(T^{3/4}) for the average-reward case with function approximation. Our algorithm and analysis rely on online learning techniques, where value functions are treated as losses. The main technical novelty is the use of a data-dependent adaptive learning rate coupled with a so-called optimistic prediction of upcoming losses. In addition to theoretical guarantees, we demonstrate the advantages of our approach empirically on several environments.

We consider the problem of local planning in fixed-horizon Markov Decision Processes (MDPs) with a generative model under the assumption that the optimal value function lies close to the span of a feature map. The generative model provides a restricted, “local” access to the MDP: The planner can ask for random transitions from previously returned states and arbitrary actions, and the features are also only accessible for the states that are encountered in this process. As opposed to previous work (e.g. Lattimore et al. (2020)) where linear realizability of all policies was assumed, we consider the significantly relaxed assumption of a single linearly realizable (deterministic) policy. A recent lower bound by Weisz et al. (2020) established that the related problem when the action-value function of the optimal policy is linearly realizable requires an exponential number of queries, either in $H$ (the horizon of the MDP) or $d$ (the dimension of the feature mapping). Their construction crucially relies on having an exponentially large action set. In contrast, in this work, we establish that $\poly(H,d)$ planning is possible with state value function realizability whenever the action set has a constant size. In particular, we present the TensorPlan algorithm which uses $\poly((dH/\delta)^A)$ simulator queries to find a $\delta$-optimal policy relative to any deterministic policy for which the value function is linearly realizable with some bounded parameter (with a known bound). This is the first algorithm to give a polynomial query complexity guarantee using only linear-realizability of a single competing value function. Whether the computation cost is similarly bounded remains an interesting open question. We also extend the upper bound to the near-realizable case and to the infinite-horizon discounted MDP setup. The upper bounds are complemented by a lower bound which states that in the infinite-horizon episodic setting, planners that achieve constant suboptimality need exponentially many queries, either in the dimension or the number of actions.

In this work, we study algorithms for learning in infinite-horizon undiscounted Markov decision processes (MDPs) with function approximation. We first show that the regret analysis of the Politex algorithm (a version of regularized policy iteration) can be sharpened from $O(T^{3/4})$ to $O(\sqrt{T})$ under nearly identical assumptions, and instantiate the bound with linear function approximation. Our result provides the first high-probability $O(\sqrt{T})$ regret bound for a computationally efficient algorithm in this setting. The exact implementation of Politex with neural network function approximation is inefficient in terms of memory and computation. Since our analysis suggests that we need to approximate the average of the action-value functions of past policies well, we propose a simple efficient implementation where we train a single Q-function on a replay buffer with past data. We show that this often leads to superior performance over other implementation choices, especially in terms of wall-clock time. Our work also provides a novel theoretical justification for using experience replay within policy iteration algorithms.

We study bandit model selection in stochastic environments. Our approach relies on a master algorithm that selects between candidate base algorithms. We develop a master-base algorithm abstraction that can work with general classes of base algorithms and different type of adversarial master algorithms. Our methods rely on a novel and generic smoothing transformation for bandit algorithms that permits us to obtain optimal $O(\sqrt{T})$ model selection guarantees for stochastic contextual bandit problems as long as the optimal base algorithm satisfies a high probability regret guarantee. We show through a lower bound that even when one of the base algorithms has $O(\log T)$ regret, in general it is impossible to get better than $\Omega(\sqrt{T})$ regret in model selection, even asymptotically. Using our techniques, we address model selection in a variety of problems such as misspecified linear contextual bandits \citep{lattimore2019learning}, linear bandit with unknown dimension \citep{Foster-Krishnamurthy-Luo-2019} and reinforcement learning with unknown feature maps. Our algorithm requires the knowledge of the optimal base regret to adjust the master learning rate. We show that without such prior knowledge any master can suffer a regret larger than the optimal base regret.

The success of many graph-based machine learning tasks highly depends on an appropriate representation learned from the graph data. Most work has focused on learning node embeddings that preserve proximity as opposed to structural role-based embeddings that preserve the structural similarity among nodes. These methods fail to capture higher-order structural dependencies and connectivity patterns that are crucial for structural role-based applications such as visitor stitching from web logs. In this work, we formulate higher-order network representation learning and describe a general framework called HONE for learning such structural node embeddings from networks via the subgraph patterns (network motifs, graphlet orbits/positions) in a nodes neighborhood. A general diffusion mechanism is introduced in HONE along with a space-efficient approach that avoids explicit construction of the k-step motif-based matrices using a k-step linear operator. Furthermore, HONE is shown to be fast and efficient with a worst-case time complexity that is nearly-linear in the number of edges. The experiments demonstrate the effectiveness of HONE for a number of important tasks including link prediction and visitor stitching from large web log data.

Model-free approaches for reinforcement learning (RL) and continuous control find policies based only on past states and rewards, without fitting a model of the system dynamics. They are appealing as they are general purpose and easy to implement; however, they also come with fewer theoretical guarantees than model-based RL. In this work, we present a new model-free algorithm for controlling linear quadratic (LQ) systems, and show that its regret scales as $O(T^{\xi+2/3})$ for any small $\xi>0$ if time horizon satisfies $T>C^{1/\xi}$ for a constant $C$. The algorithm is based on a reduction of control of Markov decision processes to an expert prediction problem. In practice, it corresponds to a variant of policy iteration with forced exploration, where the policy in each phase is greedy with respect to the average of all previous value functions. This is the first model-free algorithm for adaptive control of LQ systems that provably achieves sublinear regret and has a polynomial computation cost. Empirically, our algorithm dramatically outperforms standard policy iteration, but performs worse than a model-based approach.

We address the problem of finding an optimal policy in a Markov decision process (MDP) under a restricted policy class defined by the convex hull of a set of base policies. This problem is of great interest in applications in which a number of reasonably good (or safe) policies are already known and we are interested in optimizing in their convex hull. We first prove that solving this problem is NP-hard. We then propose an efficient algorithm that finds a policy whose performance is almost as good as that of the best convex combination of the base policies, under the assumption that the occupancy measures of the base policies have a large overlap. The running time of the proposed algorithm is linear in the number of states and polynomial in the number of base policies. A distinct advantage of the proposed algorithm is that, apart from the computation of the occupancy measures of the base policies, it does not need to interact with the environment during the optimization process. This is especially important (i) in problems that due to concerns such as safety, we are restricted in interacting with the environment only through the (safe) base policies, and (ii) in complex systems where estimating the value of a policy can be a time consuming process.

We study sample complexity of optimizing “hill-climbing friendly” functions defined on a graph under noisy observations. We define a notion of convexity, and we show that a variant of best-arm identification can find a near-optimal solution after a small number of queries that is independent of the size of the graph. For functions that have local minima and are nearly convex, we show a sample complexity for the classical simulated annealing under noisy observations. We show effectiveness of the greedy algorithm with restarts and the simulated annealing on problems of graph-based nearest neighbor classification as well as a web advertising application.

Upper Confidence Bound (UCB) method is arguably the most celebrated one used in online decision making with partial information feedback. Existing techniques for constructing confidence bounds are typically built upon various concentration inequalities, which thus lead to over-exploration. In this paper, we propose a non-parametric and data-dependent UCB algorithm based on the multiplier bootstrap. To improve its finite sample performance, we further incorporate second-order correction into the above construction. In theory, we derive both problem-dependent and problem-independent regret bounds for multi-armed bandits under a much weaker tail assumption than the standard sub-Gaussianity. Numerical results demonstrate significant regret reductions by our method, in comparison with several baselines in a range of multi-armed and linear bandit problems.

We study the effects of approximate inference on the performance of Thompson sampling in the $k$-armed bandit problems. Thompson sampling is a successful algorithm for online decision-making but requires posterior inference, which often must be approximated in practice. We show that even small constant inference error (in $\alpha$-divergence) can lead to poor performance (linear regret) due to under-exploration (for $\alpha<1$) or over-exploration (for $\alpha>0$) by the approximation. While for $\alpha > 0$ this is unavoidable, for $\alpha \leq 0$ the regret can be improved by adding a small amount of forced exploration even when the inference error is a large constant.

We study bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: $\backslash$emph{\{}Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards?{\}} First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.

Many web systems rank and present a list of items to users, from recommender systems to search and advertising. An important problem in practice is to evaluate new ranking policies offline and optimize them before they are deployed. We address this problem by proposing evaluation algorithms for estimating the expected number of clicks on ranked lists from historical logged data. The existing algorithms are not guaranteed to be statistically efficient in our problem because the number of recommended lists can grow exponentially with their length. To overcome this challenge, we use models of user interaction with the list of items, the so-called click models, to construct estimators that learn statistically efficiently. We analyze our estimators and prove that they are more efficient than the estimators that do not use the structure of the click model, under the assumption that the click model holds. We evaluate our estimators in a series of experiments on a real-world dataset and show that they consistently outperform prior estimators.

We propose a practical non-episodic PSRL algorithm that unlike recent state-of-the-art PSRL algorithms uses a deterministic, model-independent episode switching schedule. Our algorithm termed deterministic schedule PSRL (DS-PSRL) is efficient in terms of time, sample, and space complexity. We prove a Bayesian regret bound under mild assumptions. Our result is more generally applicable to multiple parameters and continuous state action problems. We compare our algorithm with state-of-the-art PSRL algorithms on standard discrete and continuous problems from the literature. Finally, we show how the assumptions of our algorithm satisfy a sensible parameterization for a large class of problems in sequential recommendations.

We propose the Hit-and-Run algorithm for planning and sampling problems in non- convex spaces. For sampling, we show the first analysis of the Hit-and-Run algorithm in non-convex spaces and show that it mixes fast as long as certain smoothness conditions are satisfied. In particular, our analysis reveals an intriguing connection between fast mixing and the existence of smooth measure-preserving mappings from a convex space to the non-convex space. For planning, we show advantages of Hit-and- Run compared to state-of-the-art planning methods such as Rapidly-Exploring Random Trees.

We study minimax strategies for the online prediction problem with expert advice. It has been conjectured that a simple adversary strategy, called COMB, is near optimal in this game for any number of experts. Our results and new insights make progress in this direction by showing that, up to a small additive term, COMB is minimax optimal in the finite-time three expert problem. In addition, we provide for this setting a new near minimax optimal COMB-based learner. Prior to this work, in this problem, learners obtaining the optimal multiplicative constant in their regret rate were known only when $K=2$ or $K\rightarrow\infty$. We characterize, when $K=3$, the regret of the game scaling as $\sqrt{8/(9\pi)T}\pm \log(T)^2$ which gives for the first time the optimal constant in the leading ($\sqrt{T}$) term of the regret.

Safety is a desirable property that can immensely increase the applicability of learning algorithms in real-world decision-making problems. It is much easier for a company to deploy an algorithm that is safe, i.e., guaranteed to perform at least as well as a baseline. In this paper, we study the issue of safety in contextual linear bandits that have application in many different fields including personalized ad recommendation in online marketing. We formulate a notion of safety for this class of algorithms. We develop a safe contextual linear bandit algorithm, called conservative linear UCB (CLUCB), that simultaneously minimizes its regret and satisfies the safety constraint, i.e., maintains its performance above a fixed percentage of the performance of a baseline strategy, uniformly over time. We prove an upper-bound on the regret of CLUCB and show that it can be decomposed into two terms: 1) an upper-bound for the regret of the standard linear UCB algorithm that grows with the time horizon and 2) a constant term that accounts for the loss of being conservative in order to satisfy the safety constraint. We empirically show that our algorithm is safe and validate our theoretical analysis.

We introduce a simple, efficient method that improves stochastic policies for Markov decision processes. The computational complexity is the same as that of the value estimation problem. We prove that when the value estimation error is small, this method gives an improvement in performance that increases with certain variance properties of the initial policy and transition dynamics. Performance in numerical experiments compares favorably with previous policy improvement algorithms.

We study average and total cost Markov decision problems with large state spaces. Since the computational and statistical costs of finding the optimal policy scale with the size of the state space, we focus on searching for near-optimality in a low-dimensional family of policies. In particular, we show that for problems with a Kullback-Leibler divergence cost function, we can reduce policy optimization to a convex optimization and solve it approximately using a stochastic subgradient algorithm. We show that the performance of the resulting policy is close to the best in the low-dimensional family. We demonstrate the efficacy of our approach by controlling the important crowdsourcing application of budget allocation in crowd labeling.

We consider an adversarial formulation of the problem ofpredicting a time series with square loss. The aim is to predictan arbitrary sequence of vectors almost as well as the bestsmooth comparator sequence in retrospect. Our approach allowsnatural measures of smoothness such as the squared norm ofincrements. More generally, we consider a linear time seriesmodel and penalize the comparator sequence through the energy ofthe implied driving noise terms. We derive the minimax strategyfor all problems of this type and show that it can be implementedefficiently. The optimal predictions are linear in the previousobservations. We obtain an explicit expression for the regret interms of the parameters defining the problem. For typical,simple definitions of smoothness, the computation of the optimalpredictions involves only sparse matrices. In the case ofnorm-constrained data, where the smoothness is defined in termsof the squared norm of the comparator's increments, we show thatthe regret grows as $T/\sqrt{\lambda_T}$, where $T$ is the lengthof the game and $\lambda_T$ is an increasing limit on comparatorsmoothness.

We study linear control problems with quadratic losses and adversarially chosen tracking targets. We present an efficient algorithm for this problem and show that, under standard conditions on the linear system, its regret with respect to an optimal linear policy grows as O(\log^2 T), where T is the number of rounds of the game. We also study a problem with adversarially chosen transition dynamics; we present an exponentially-weighted average algorithm for this problem, and we give regret bounds that grow as O(\sqrt T).

We consider the problem of controlling a Markov decision process (MDP) with a large state space, so as to minimize average cost. Since it is intractable to compete with the optimal policy for large scale problems, we pursue the more modest goal of competing with a low-dimensional family of policies. We use the dual linear programming formulation of the MDP average cost problem, in which the variable is a stationary distribution over state-action pairs, and we consider a neighborhood of a low-dimensional subset of the set of stationary distributions (defined in terms of state-action features) as the comparison class. We propose two techniques, one based on stochastic convex optimization, and one based on constraint sampling. In both cases, we give bounds that show that the performance of our algorithms approaches the best achievable by any policy in the comparison class. Most importantly, these results depend on the size of the comparison class, but not on the size of the state space. Preliminary experiments show the effectiveness of the proposed algorithms in a queuing application.

We study two problems of online learning under restricted information access. In the first problem, \emphprediction with limited advice, we consider a game of prediction with expert advice, where on each round of the game we query the advice of a subset of M out of N experts. We present an algorithm that achieves O(\sqrt(N/M)T\ln N) regret on T rounds of this game. The second problem, the \emphmultiarmed bandit with paid observations, is a variant of the adversarial N-armed bandit game, where on round t of the game we can observe the reward of any number of arms, but each observation has a cost c. We present an algorithm that achieves O((cN\ln N)^1/3 T^2/3 + \sqrtT \ln N) regret on T rounds of this game in the worst case. Furthermore, we present a number of refinements that treat arm- and time-dependent observation costs and achieve lower regret under benign conditions. We present lower bounds that show that, apart from the logarithmic factors, the worst-case regret bounds cannot be improved.

We study the problem of online learning Markov Decision Processes (MDPs) when both the transition distributions and loss functions are chosen by an adversary. We present an algorithm that, under a mixing assumption, achieves $O(\sqrt{T\log|\Pi|}+\log|\Pi|)$ regret with respect to a comparison set of policies $\Pi$. The regret is independent of the size of the state and action spaces. When expectations over sample paths can be computed efficiently and the comparison set $\Pi$ has polynomial size, this algorithm is efficient. We also consider the episodic adversarial online shortest path problem. Here, in each episode an adversary may choose a weighted directed acyclic graph with an identified start and finish node. The goal of the learning algorithm is to choose a path that minimizes the loss while traversing from the start to finish node. At the end of each episode the loss function (given by weights on the edges) is revealed to the learning algorithm. The goal is to minimize regret with respect to a fixed policy for selecting paths. This problem is a special case of the online MDP problem. For randomly chosen graphs and adversarial losses, this problem can be efficiently solved. We show that it also can be efficiently solved for adversarial graphs and randomly chosen losses. When both graphs and losses are adversarially chosen, we present an efficient algorithm whose regret scales linearly with the number of distinct graphs. Finally, we show that designing efficient algorithms for the adversarial online shortest path problem (and hence for the adversarial MDP problem) is as hard as learning parity with noise, a notoriously difficult problem that has been used to design efficient cryptographic schemes.

We improve the theoretical analysis and empirical performance of algorithms for the stochastic multi-armed bandit problem and the linear stochastic multi-armed bandit problem. In particular, we show that a simple modification of Auer’s UCB algorithm (Auer, 2002) achieves with high probability constant regret. More importantly, we modify and, consequently, improve the analysis of the algorithm for the for linear stochastic bandit problem studied by Auer (2002), Dani et al. (2008), Rusmevichientong and Tsitsiklis (2010), Li et al. (2010). Our modification improves the regret bound by a logarithmic factor, though experiments show a vast improvement. In both cases, the improvement stems from the construction of smaller confidence sets. For their construction we use a novel tail inequality for vector-valued martingales.

We consider the problem of anytime planning in continuous state and action spaces with non-linear deterministic dynamics. We review the existing approaches to this problem and find no algorithms that both quickly find feasible solutions and also eventually approach optimal solutions with additional time. The state-of-the-art solution to this problem is the rapidly-exploring random tree (RRT) algorithm that quickly finds a feasible solution. However, the RRT algorithm does not return better results with additional time. We introduce RRT++, an anytime extension of the basic RRT algorithm. We show that the new algorithm has desirable theoretical properties and experimentally show that it efficiently finds near optimal solutions.

An anytime algorithm is capable of returning a response to the given task at essentially any time; typically the quality of the response improves as the time increases. Here, we consider the challenge of learning when we should terminate such algorithms on each of a sequence of iid tasks, to optimize the expected average reward per unit time. We provide a system for addressing this challenge, which combines the global optimizer Cross-Entropy method with local gradient ascent. This paper theoretically investigates how far the estimated gradient is from the true gradient, then empirically demonstrates that this system is effective by applying it to a toy problem, as well as on a real-world face detection task.