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A decision tree recursively splits a feature space R^d and then assigns class labels based on the resulting partition. Decision trees have been part of the basic machine-learning toolkit for decades. A large body of work considers heuristic algorithms that compute a decision tree from training data, usually aiming to minimize in particular the size of the resulting tree. In contrast, little is known about the complexity of the underlying computational problem of computing a minimum-size tree for the given training data. We study this problem with respect to the number d of dimensions of the feature space R^d, which contains n training examples. We show that it can be solved in O(n^(2d + 1)) time, but under reasonable complexity-theoretic assumptions it is not possible to achieve f(d) * n^o(d / log d) running time. The problem is solvable in (dR)^O(dR) * n^(1+o(1)) time, if there are exactly two classes and R is an upper bound on the number of tree leaves labeled with the first class.

AAAI 2023

The Influence of Dimensions on the Complexity of Computing Decision Trees

exact algorithm

np-hard problem

parameterized complexity

optimization

dynamic programming

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ECAI 2020

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The purpose of the AAAI conference series is to promote research in Artificial Intelligence (AI) and foster scientific exchange between researchers, practitioners, scientists, students, and engineers across the entirety of AI and its affiliated disciplines. AAAI-23 is the Thirty-Seventh AAAI Conference on Artificial Intelligence. The theme of this conference is to create collaborative bridges within and beyond AI. Like previous AAAI conferences, AAAI-23 will feature technical paper presentations, special tracks, invited speakers, workshops, tutorials, poster sessions, senior member presentations, competitions, and exhibit programs, and two new activities: a Bridge Program and a Lab Program. Many of these activities are tailored to the theme of bridges and all are selected according to the highest standards, with additional programs for students and young researchers. 
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The purpose of the AAAI conference series is to promote research in Artificial Intelligence (AI) and foster scientific exchange between researchers, practitioners, scientists, students, and engineers across the entirety of AI and its affiliated disciplines.

In reinforcement learning (RL), a reward function that aligns exactly with a task's true performance metric is often necessarily sparse. For example, a true task metric might encode a reward of 1 upon success and 0 otherwise. The sparsity of these true task metrics can make them hard to learn from, so in practice they are often replaced with alternative dense reward functions. These dense reward functions are typically designed by experts through an ad hoc process of trial and error. In this process, experts manually search for a reward function that improves performance with respect to the task metric while also enabling an RL algorithm to learn faster. This process raises the question of whether the same reward function is optimal for all algorithms, i.e., whether the reward function can be overfit to a particular algorithm. In this paper, we study the consequences of this wide yet unexamined practice of trial-and-error reward design. We first conduct computational experiments that confirm that reward functions can be overfit to learning algorithms and their hyperparameters. We then conduct a controlled observation study which emulates expert practitioners' typical experiences of reward design, in which we similarly find evidence of reward function overfitting. We also find that experts' typical approach to reward design---of adopting a myopic strategy and weighing the relative goodness of each state-action pair---leads to misdesign through invalid task specifications, since RL algorithms use cumulative reward rather than rewards for individual state-action pairs as an optimization target.

Code and supporting data for all experiments: https://github.com/serenabooth/reward-design-perils.

The Perils of Trial-and-Error Reward Design: Misdesign through Overfitting and Invalid Task Specifications

Class-incremental learning (CIL) aims to train a classification model while the number of classes increases phase-by-phase. An inherent challenge of CIL is the stability-plasticity tradeoff, i.e., CIL models should keep stable to retain old knowledge and keep plastic to absorb new knowledge. However, none of the existing CIL models can achieve the optimal tradeoff in different data-receiving settings—where typically the training-from-half (TFH) setting needs more stability, but the training-from-scratch (TFS) needs more plasticity. To this end, we design an online learning method that can adaptively optimize the tradeoff without knowing the setting as a priori. Specifically, we first introduce the key hyperparameters that influence the tradeoff, e.g., knowledge distillation (KD) loss weights, learning rates, and classifier types. Then, we formulate the hyperparameter optimization process as an online Markov Decision Process (MDP) problem and propose a specific algorithm to solve it. We apply local estimated rewards and a classic bandit algorithm Exp3 (Auer et al. 2002) to address the issues when applying online MDP methods to the CIL protocol. Our method consistently improves top-performing CIL methods in both TFH and TFS settings, e.g., boosting the average accuracy of TFH and TFS by 2.2 percentage points on ImageNet-Full, compared to the state-of-the-art (Liu et al. 2021b).

Online Hyperparameter Optimization for Class-Incremental Learning

We provide a complete characterization for the computational complexity of finding approximate equilibria in two-action graphical games. We consider the two most well-studied approximation notions: $\varepsilon$-Nash equilibria ($\varepsilon$-NE) and $\varepsilon$-well-supported Nash equilibria ($\varepsilon$-WSNE), where $\varepsilon \in [0,1]$. We prove that computing an $\varepsilon$-NE is PPAD-complete for any constant $\varepsilon < 1/2$, while a very simple algorithm (namely, letting all players mix uniformly between their two actions) yields a $1/2$-NE. On the other hand, we show that computing an $\varepsilon$-WSNE is PPAD-complete for any constant $\varepsilon < 1$, while a $1$-WSNE is trivial to achieve, because any strategy profile is a $1$-WSNE. All of our lower bounds immediately also apply to graphical games with more than two actions per player.

Tight Inapproximability for Graphical Games

Understanding mathematical questions effectively is a crucial task, which can benefit many applications, such as difficulty estimation. Researchers have drawn much attention to designing pre-training models for question representations due to the scarcity of human annotations (e.g., labeling difficulty). However, unlike general free-format texts (e.g., user comments), mathematical questions are generally designed with explicit purposes and mathematical logic, and usually consist of more complex content, such as formulas, and related mathematical knowledge (e.g., Function). Therefore, the problem of holistically representing mathematical questions remains underexplored. To this end, in this paper, we propose a novel contrastive pre-training approach for mathematical question representations, namely QuesCo, which attempts to bring questions with more similar purposes closer. Specifically, we first design two-level question augmentations, including content-level and structure-level, which generate literally diverse question pairs with similar purposes. Then, to fully exploit hierarchical information of knowledge concepts, we propose a knowledge hierarchy-aware rank strategy (KHAR), which ranks the similarities between questions in a fine-grained manner. Next, we adopt a ranking contrastive learning task to optimize our model based on the augmented and ranked questions. We conduct extensive experiments on two real-world mathematical datasets. The experimental results demonstrate the effectiveness of our model.

Towards a Holistic Understanding of Mathematical Questions with Contrastive Pre-training

Transformer models are widely used in AI applications such as Natural Language Processing (NLP), Computer Vision (CV), etc. However, enormous computation workload be-comes an obstacle to train large transformer models efficiently. Recently, some methods focus on reducing the computation workload during the training by skipping some layers. How-ever, these methods use simple probability distribution and coarse-grained probability calculation, which significantly affect the model accuracy. To address the issue, in this paper we propose a novel method to accelerate training—Sensitivity-Based Layer Dropping (SBLD). SBLD uses lay-er-wise sensitivity data to switch on/off transformer layers in proper order to keep high accuracy. Besides, we adjust the probability of skipping transformer layers with a scheduler to accelerate training speed and get faster convergence. Our results show that SBLD solves the accuracy drop issue com-pared with prior layer dropping methods. Our SBLD method can decrease end-to-end training time by 19.67% during training of GPT-3 Medium model, the same time increasing the accuracy by 1.65% w.r.t. baseline. Furthermore, for SwinV2-L model the obtained Top-1 and Top-5 accuracies are also higher vs. the baseline. Thus, the proposed method is efficient and practical to improve the large transformer model training.

Acceleration of Large Transformer Model Training by Sensitivity-Based Layer Dropping

Behavioral Cloning (BC) aims at learning a policy that mimics the behavior demonstrated by an expert. The current theoretical understanding of BC is limited to the case of finite actions. In this paper, we study BC with the goal of providing theoretical guarantees on the performance of the imitator policy in the case of continuous actions. We start by deriving a novel bound on the performance gap based on Wasserstein distance, applicable for continuous-action experts, holding under the assumption that the value function is Lipschitz continuous. Since this latter condition is hardy fulfilled in practice, even for Lipschitz Markov Decision Processes and policies, we propose a relaxed setting, proving that value function is always H\"older continuous. This result is of independent interest and allows obtaining in BC a general bound for the performance of the imitator policy. Finally, we analyze noise injection, a common practice in which the expert's action is executed in the environment after the application of a noise kernel. We show that this practice allows deriving stronger performance guarantees, at the price of a bias due to the noise addition.

Tight Performance Guarantees of Imitator Policies with Continuous Actions

To aggregate rankings into a social ranking, one can use scoring systems such as Plurality, Veto, and Borda. We distinguish three types of methods: ranking by score, ranking by repeatedly choosing a winner that we delete and rank at the top, and ranking by repeatedly choosing a loser that we delete and rank at the bottom. The latter method captures the frequently studied voting rules Single Transferable Vote (aka Instant Runoff Voting), Coombs, and Baldwin. In an experimental analysis, we show that the three types of methods produce different rankings in practice. We also provide evidence that sequentially selecting winners is most suitable to detect the "true" ranking of candidates. For different rules in our classes, we then study the (parameterized) computational complexity of deciding in which positions a given candidate can appear in the chosen ranking. As part of our analysis, we also consider the Winner Determination problem for STV, Coombs, and Baldwin and determine their complexity when there are few voters or candidates. 

Rank Aggregation Using Scoring Rules

While a growing body of literature has been studying new Graph Neural Networks (GNNs) that work on both homophilic and heterophilic graphs, little has been done on adapting classical GNNs to less-homophilic graphs. Although the ability to handle less-homophilic graphs is restricted, classical GNNs still stand out in several nice properties such as efficiency, simplicity, and explainability. In this work, we propose a novel graph restructuring method that can be integrated into any type of GNNs, including classical GNNs, to leverage the benefits of existing GNNs while alleviating their limitations. Our contribution is threefold: a) learning the weight of pseudo-eigenvectors for an adaptive spectral clustering that aligns well with known node labels, b) proposing a new density-aware homophilic metric that is robust to label imbalance, and c) reconstructing the adjacency matrix based on the result of adaptive spectral clustering to maximize the homophilic scores. The experimental results show that our graph restructuring method can significantly boost the performance of six classical GNNs by an average of 25% on less-homophilic graphs. The boosted performance is comparable to state-of-the-art methods.

Restructuring Graph for Higher Homophily via Adaptive Spectral Clustering

Recently, improving the robustness of policies across different environments attracts increasing attention in the reinforcement learning (RL) community. Existing robust RL methods mostly aim to achieve the max-min robustness by optimizing the policy’s performance in the worst-case environment. However, in practice, a user that uses an RL policy may have different preferences over its performance across environments. Clearly, the aforementioned max-min robustness is oftentimes too conservative to satisfy user preference. Therefore, in this paper, we integrate user preference into policy learning in robust RL, and propose a novel User-Oriented Robust RL (UOR-RL) framework. Specifically, we define a new User-Oriented Robustness (UOR) metric for RL, which allocates different weights to the environments according to user preference and generalizes the max-min robustness metric. To optimize the UOR metric, we develop two different UOR-RL training algorithms for the scenarios with or without a priori known environment distribution, respectively. Theoretically, we prove that our UOR-RL training algorithms converge to near-optimal policies even with inaccurate or completely no knowledge about the environment distribution. Furthermore, we carry out extensive experimental evaluations in 6 MuJoCo tasks. The experimental results demonstrate that UOR-RL is comparable to the state-of-the-art baselines under the average-case and worst-case performance metrics, and more importantly establishes new state-of-the-art performance under the UOR metric.

User-Oriented Robust Reinforcement Learning

The mobile communication enabled by cellular networks is the one of the main foundations of our modern society. Optimizing the performance of cellular networks and providing massive connectivity with improved coverage and user experience has a considerable social and economic impact on our daily life. This performance relies heavily on the configuration of the network parameters. However, with the massive increase in both the size and complexity of cellular networks, network management, especially parameter configuration, is becoming complicated. The current practice, which relies largely on experts' prior knowledge, is not adequate and will require lots of domain experts and high maintenance costs. In this work, we propose a learning-based framework for handover parameter configuration. The key challenge, in this case, is to tackle the complicated dependencies between neighboring cells and jointly optimize the whole network. Our framework addresses this challenge in two ways. First, we introduce a novel approach to imitate how the network responds to different network states and parameter values, called auto-grouping graph convolutional network (AG-GCN). During the parameter configuration stage, instead of solving the global optimization problem, we design a local multi-objective optimization strategy where each cell considers several local performance metrics to balance its own performance and its neighbors. We evaluate our proposed algorithm via a simulator constructed using real network data. We demonstrate that the handover parameters our model can find, achieve better average network throughput compared to those recommended by experts as well as alternative baselines, which can bring better network quality and stability. It has the potential to massively reduce costs arising from human expert intervention and maintenance.

The Influence of Dimensions on the Complexity of Computing Decision Trees

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