While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the \textsc{LeadingOnes} and \textsc{Jump} benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $\sigma$ into another one $\tau$, but also the precise cycle structure of $\sigma \tau^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $\Theta(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{\Theta(m)}$ on jump functions with jump size~$m$.%
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We study the hidden-action principal-agent problem in an online setting. In each round, the principal posts a contract that specifies the payment to the agent based on each outcome. The agent then makes a strategic choice of action that maximizes her own utility, but the action is not directly observable by the principal. The principal observes the outcome and receives utility from the agent's choice of action. Based on past observations, the principal dynamically adjusts the contracts with the goal of maximizing her utility. We introduce an online learning algorithm and provide an upper bound on its Stackelberg regret. We show that when the contract space is $[0,1]^m$, the Stackelberg regret is upper bounded by $\widetilde O(\sqrt{m} \cdot T^{1-C/m})$, and lower bounded by $\Omega(T^{1-1/(m+2)})$. This result shows that exponential-in-$m$ samples are both sufficient and necessary to learn a near-optimal contract, resolving an open problem on the hardness of online contract design. When contracts are restricted to some subset $\mathcal{F} \subset [0,1]^m$, we define an intrinsic dimension of $\mathcal{F}$ that depends on the covering number of the spherical code in the space and bound the regret in terms of this intrinsic dimension. When $\mathcal{F}$ is the family of linear contracts, the Stackelberg regret grows exactly as $\Theta(T^{2/3})$. The contract design problem is challenging because the utility function is discontinuous. Bounding the discretization error in this setting has been an open problem. In this paper, we identify a limited set of directions in which the utility function is continuous, allowing us to design a new discretization method and bound its error. This approach enables the first upper bound with no restrictions on the contract and action space.
Multitask learning assumes that models capable of learning from multiple tasks can achieve better quality and efficiency via knowledge transfer, a key feature of human learning. Though, state of the art ML models rely on high customization for each task and leverage size and data scale rather than scaling the number of tasks. Also, continual learning, that adds the temporal aspect to multitask, is often focused to the study of common pitfalls such as catastrophic forgetting instead of being studied at a large scale as a critical component to build the next generation artificial intelligence.We propose an evolutionary method capable of generating large scale multitask models that support the dynamic addition of new tasks. The generated multitask models are sparsely activated and integrates a task-based routing that guarantees bounded compute cost and fewer added parameters per task as the model expands.The proposed method relies on a knowledge compartmentalization technique to achieve immunity against catastrophic forgetting and other common pitfalls such as gradient interference and negative transfer. We demonstrate empirically that the proposed method can jointly solve and achieve competitive results on 69public image classification tasks, for example improving the state of the art on a competitive benchmark such as cifar10 by achieving a 15% relative error reduction compared to the best model trained on public data.
The design and operation of systems are conventionally viewed as a sequential decision-making process that is informed by data from physical experiments and simulations. However, the integration of these high-dimensional and heterogeneous data sources requires the consideration of the impact of a decision on a system's remaining life cycle. Consequently, this introduces a degree of complexity that in most cases can only be solved through an integrated decision-making approach. In this perspective paper, we use the digital twin concept to formulate an integrated perspective for the design of systems. Specifically, we show how the digital twin concept enables the integration of system design decisions and operational decisions during each stage of a system's life cycle. This perspective has two advantages: (i) improved system performance as more effective decisions can be made, and (ii) improved data efficiency as it provides a framework to utilize data from multiple sources and design instances. From the formal definition, we identify a set of eight capabilities that are vital constructs to bring about the potential, as defined in this paper, that the digital twin concept holds for the design of systems. Subsequently, by comparing these capabilities with the available literature on digital twins, we identify a set of research questions and forecast what their broader impact might be. By conceptualizing the potential that the digital twin concept holds for the design of systems, we hope to contribute to the convergence of definitions, problem formulations, research gaps, and value propositions in this burgeoning field. Addressing the research questions, associated with the digital twin-inspired formulation for the design of systems, will bring about more advanced systems that can meet some of the societies' grand challenges.
Two central paradigms have emerged in the reinforcement learning (RL) community: online RL and offline RL. In the online RL setting, the agent has no prior knowledge of the environment, and must interact with it in order to find an $\epsilon$-optimal policy. In the offline RL setting, the learner instead has access to a fixed dataset to learn from, but is unable to otherwise interact with the environment, and must obtain the best policy it can from this offline data. Practical scenarios often motivate an intermediate setting: if we have some set of offline data and, in addition, may also interact with the environment, how can we best use the offline data to minimize the number of online interactions necessary to learn an $\epsilon$-optimal policy? In this work, we consider this setting, which we call the \textsf{FineTuneRL} setting, for MDPs with linear structure. We characterize the necessary number of online samples needed in this setting given access to some offline dataset, and develop an algorithm, \textsc{FTPedel}, which is provably optimal. We show through an explicit example that combining offline data with online interactions can lead to a provable improvement over either purely offline or purely online RL. Finally, our results illustrate the distinction between \emph{verifiable} learning, the typical setting considered in online RL, and \emph{unverifiable} learning, the setting often considered in offline RL, and show that there is a formal separation between these regimes.
Since three decades binary decision diagrams, representing efficiently Boolean functions, are widely used, in many distinct contexts like model verification, machine learning. The most famous variant, called reduced ordered binary decision diagram (ROBDD for short), can be viewed as the result of a compaction procedure on the full decision tree. In this paper we aim at computing the exact distribution of the Boolean functions in $k$~variables according to the ROBDD size, where the ROBDD size is equal to the size of the underlying directed acyclic graph (DAG) structure. Recall the number of Boolean functions is equal to $2^{2^k}$, which is of double exponential growth; hence a combinatorial explosion is to be expected. The maximal size of a ROBDD with $k$ variables is $M_k \sim 2^k / k$ and thus, the support of the ROBDD size distribution is also of length $M_k$, making $M_k$ a natural complexity unit for our problem. In this paper, we develop the first polynomial algorithm to derive the distribution of the Boolean functions with respect to their ROBDD sizes. The algorithm is essentially quartic in $M_k$ for the time complexity and quadratic for the space complexity. The main obstacle is to take into account dependencies inside the DAG structure, and we propose a new combinatorial counting procedure reminiscent of the inclusion-exclusion principle. As a by-product, we present an efficient polynomial unranking algorithm for ROBDDs, which in turn yields a uniform random sampler over the set of ROBDDs of a given size or of a given profile. This is a great improvement to the uniform sampler over the set of all Boolean functions in $k$ variables. Indeed, due to the Shannon effect, the uniform distribution over Boolean functions is heavily biased to extremely complex functions, with near maximal ROBDD size, thus preventing to sample small ROBDDs
In large-scale recommender systems, retrieving top N relevant candidates accurately with resource constrain is crucial. To evaluate the performance of such retrieval models, Recall@N, the frequency of positive samples being retrieved in the top N ranking, is widely used. However, most of the conventional loss functions for retrieval models such as softmax cross-entropy and pairwise comparison methods do not directly optimize Recall@N. Moreover, those conventional loss functions cannot be customized for the specific retrieval size N required by each application and thus may lead to sub-optimal performance. In this paper, we proposed the Customizable Recall@N Optimization Loss (CROLoss), a loss function that can directly optimize the Recall@N metrics and is customizable for different choices of N. This proposed CROLoss formulation defines a more generalized loss function space, covering most of the conventional loss functions as special cases. Furthermore, we develop the Lambda method, a gradient-based method that invites more flexibility and can further boost the system performance. We evaluate the proposed CROLoss on two public benchmark datasets. The results show that CROLoss achieves SOTA results over conventional loss functions for both datasets with various choices of retrieval size N. CROLoss has been deployed onto our online E-commerce advertising platform, where a fourteen-day online A/B test demonstrated that CROLoss contributes to a significant business revenue growth of 4.75%.
In many applications, it is of interest to identify a parsimonious set of features, or panel, from multiple candidates that achieves a desired level of performance in predicting a response. This task is often complicated in practice by missing data arising from the sampling design or other random mechanisms. Most recent work on variable selection in missing data contexts relies in some part on a finite-dimensional statistical model, e.g., a generalized or penalized linear model. In cases where this model is misspecified, the selected variables may not all be truly scientifically relevant and can result in panels with suboptimal classification performance. To address this limitation, we propose a nonparametric variable selection algorithm combined with multiple imputation to develop flexible panels in the presence of missing-at-random data. We outline strategies based on the proposed algorithm that achieve control of commonly used error rates. Through simulations, we show that our proposal has good operating characteristics and results in panels with higher classification and variable selection performance compared to several existing penalized regression approaches in cases where a generalized linear model is misspecified. Finally, we use the proposed method to develop biomarker panels for separating pancreatic cysts with differing malignancy potential in a setting where complicated missingness in the biomarkers arose due to limited specimen volumes.
In the Set Cover problem, we are given a set system with each set having a weight, and we want to find a collection of sets that cover the universe, whilst having low total weight. There are several approaches known (based on greedy approaches, relax-and-round, and dual-fitting) that achieve a $H_k \approx \ln k + O(1)$ approximation for this problem, where the size of each set is bounded by $k$. Moreover, getting a $\ln k - O(\ln \ln k)$ approximation is hard. Where does the truth lie? Can we close the gap between the upper and lower bounds? An improvement would be particularly interesting for small values of $k$, which are often used in reductions between Set Cover and other combinatorial optimization problems. We consider a non-oblivious local-search approach: to the best of our knowledge this gives the first $H_k$-approximation for Set Cover using an approach based on local-search. Our proof fits in one page, and gives a integrality gap result as well. Refining our approach by considering larger moves and an optimized potential function gives an $(H_k - \Omega(\log^2 k)/k)$-approximation, improving on the previous bound of $(H_k - \Omega(1/k^8))$ (\emph{R.\ Hassin and A.\ Levin, SICOMP '05}) based on a modified greedy algorithm.
The (extended) Binary Value Principle (eBVP: $\sum_{i=1}^n x_i2^{i-1} = -k$ for $k>0$ and $x^2_i=x_i$) has received a lot of attention recently, several lower bounds have been proved for it (Alekseev et al 2020, Alekseev 2021, Part and Tzameret 2021). Also it has been shown (Alekseev et al 2020) that the probabilistically verifiable Ideal Proof System (IPS) (Grochow and Pitassi 2018) together with eBVP polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with the algebraic version of Tseitin's extension rule (Ext-PC). Contrary to IPS, this is a Cook--Reckhow proof system. We show that in this context eBVP still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule (Grigoriev and Hirsch 2003), which is absolutely unclear in the context of ordinary Polynomial Calculus. On the other hand, we demonstrate that eBVP probably does not help in proving exponential lower bounds for Boolean tautologies: we show that an Ext-PC (even with the Square Root Rule) derivation of any such tautology from eBVP must be of exponential size.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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