Author name cluster

Grigoris Velegkas

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

18 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

On Agnostic PAC Learning in the Small Error Regime

Julian Asilis
Mikael Møller Høgsgaard
Grigoris Velegkas

Binary classification in the classic PAC model exhibits a curious phenomenon: Empirical Risk Minimization (ERM) learners are suboptimal in the realizable case yet optimal in the agnostic case. Roughly speaking, this owes itself to the fact that non-realizable distributions $\\mathcal{D}$ are more difficult to learn than realizable distributions -- even when one discounts a learner's error by $\\mathrm{err}(h^\\ast_\\mathcal{D})$, i. e. , the error of the best hypothesis in $\\mathcal{H}$. Thus, optimal agnostic learners are permitted to incur excess error on (easier-to-learn) distributions $\\mathcal{D}$ for which $\\tau = \\mathrm{err}(h^\\ast_\\mathcal{D})$ is small. Recent work of Hanneke, Larsen, and Zhivotovskiy (FOCS '24) addresses this shortcoming by including $\\tau$ itself as a parameter in the agnostic error term. In this more fine-grained model, they demonstrate tightness of the error lower bound $\\tau + \\Omega \\left(\\sqrt{\\frac{\\tau (d + \\log(1 / \\delta))}{m}} + \\frac{d + \\log(1 / \\delta)}{m} \\right)$ in a regime where $\\tau > d/m$, and leave open the question of whether there may be a higher lower bound when $\\tau \\approx d/m$, with $d$ denoting $\\mathrm{VC}(\\mathcal{H})$. In this work, we resolve this question by exhibiting a learner which achieves error $c \\cdot \\tau + O \\left(\\sqrt{\\frac{\\tau (d + \\log(1 / \\delta))}{m}} + \\frac{d + \\log(1 / \\delta)}{m} \\right)$ for a constant $c \\leq 2. 1$, matching the lower bound and demonstrating optimality when $\\tau =O( d/m)$. Further, our learner is computationally efficient and is based upon careful aggregations of ERM classifiers, making progress on two other questions of Hanneke, Larsen, and Zhivotovskiy (FOCS '24). We leave open the interesting question of whether our approach can be refined to lower the constant from 2. 1 to 1, which would completely settle the complexity of agnostic learning.

PDF Details

STOC Conference 2025 Conference Paper

On the Limits of Language Generation: Trade-Offs between Hallucination and Mode-Collapse

Alkis Kalavasis
Anay Mehrotra
Grigoris Velegkas

Specifying all desirable properties of a language model is challenging, but certain requirements seem essential for any good model. Given samples drawn from an unknown language, the trained model should (1) produce valid strings that have not been seen in the training data, and (2) be expressive enough to capture the full richness of the language. Otherwise, if the language model outputs invalid strings, it "hallucinates," and if it fails to capture the full range of the language, it suffers from "mode collapse." In this paper, we ask whether it is possible for a language model to meet both of these requirements. We investigate this question within a statistical setting of language generation, building on the seminal works of Gold (Inf. Control, 1967), Angluin (STOC, 1979), and Angluin (Tech. Report, 1988). In this setting, the language model is presented with randomly sampled strings from a distribution supported on an unknown language K, which is only known to belong to a possibly infinite collection of candidate languages. The goal of the model is to generate unseen strings from this target language. We say that the language model generates from K with consistency and breadth if, as the size of the training set increases, the set of strings it can output converges to the set of all unseen strings in K. Kleinberg and Mullainathan (NeurIPS, 2024) posed an open question of whether consistency and breadth in language generation are both possible. We answer this question negatively: for a large class of language models -- including next-token-prediction-based models -- this is impossible for most collections of candidate languages. This contrasts with the recent positive result of Kleinberg and Mullainathan, which demonstrated that consistent generation, without requiring breadth, is possible for any countable collection of candidate languages. Our finding highlights that generation with breadth is fundamentally different from generation without breadth. As a byproduct of our result, we also examine how many samples are required for generation with or without breadth, establishing near-tight bounds on the "learning curves" for generation in the statistical framework of Bousquet, Hanneke, Moran, van Handel, and Yehudayoff (STOC, 2021). Finally, our results also give some hope for consistent generation with breadth: it is achievable for any countable collection of languages when negative examples -- in the form of strings outside of K -- are available in addition to strings inside of K. This suggests that feedback in post-training, which encodes negative examples, can be crucial in reducing hallucinations while also limiting mode collapse.

Details

NeurIPS Conference 2025 Conference Paper

On Union-Closedness of Language Generation

Steve Hanneke
Amin Karbasi
Anay Mehrotra
Grigoris Velegkas

We investigate language generation in the limit – a model by Kleinberg and Mullainathan and extended by Li, Raman, and Tewari. While Kleinberg and Mullainathan proved generation is possible for all countable collections, Li, Raman, and Tewari defined a hierarchy of generation notions (uniform, non-uniform, and generatable) and explored their feasibility for uncountable collections. Our first set of results resolve two open questions of Li et al. by proving finite unions of generatable or non-uniformly generatable classes need not be generatable. These follow from a stronger result: there is non-uniformly generatable class and a uniformly generatable class whose union is non-generatable. This adds to the aspects along which language generation in the limit is different from traditional tasks in statistical learning theory like classification, which are closed under finite unions. In particular, it implies that given two generators for different collections, one cannot combine them to obtain a single "more powerful" generator, prohibiting this notion of boosting. Our construction also addresses a third of Li et al. 's open questions on whether there are uncountable classes that are non-uniformly generatable and do not satisfy the eventually unbounded closure (EUC) condition introduced by Li et al. Our approach utilizes carefully constructed classes along with a novel diagonalization argument that could be of independent interest in the growing area of language generation.

PDF Details

ICML Conference 2025 Conference Paper

Procurement Auctions via Approximately Optimal Submodular Optimization

Yuan Deng
Amin Karbasi
Vahab Mirrokni
Renato Paes Leme
Grigoris Velegkas
Song Zuo

We study the problem of procurement auctions, in which an auctioneer seeks to acquire services from a group of strategic sellers with private costs. The quality of the services is measured through some submodular function that is known to the auctioneer. Our goal is to design computationally efficient procurement auctions that (approximately) maximize the difference between the quality of the acquired services and the total cost of the sellers, in a way that is incentive compatible (IC) and individual rational (IR) for the sellers, and generates non-negative surplus (NAS) for the auctioneer. Our contribution is twofold: i) we provide an improved analysis of existing algorithms for non-positive submodular function maximization and ii) we design computationally efficient frameworks that transform submodular function optimization algorithms to mechanisms that are IC and IR for the sellers, NAS for the auctioneer, and approximation-preserving. Our frameworks are general and work both in the offline setting where the auctioneer can observe the bids and the services of all the sellers simultaneously, and in the online setting where the sellers arrive in an adversarial order and the auctioneer has to make an irrevocable decision whether to purchase their service or not. We further investigate whether it is possible to convert state-of-art submodular optimization algorithms into descending auctions. We focus on the adversarial setting, meaning that the schedule of the descending prices is determined by an adversary. We show that a submodular optimization algorithm satisfying bi-criteria $(1/2, 1)$-approximation in welfare can be effectively converted to a descending auction in this setting. We further establish a connection between descending auctions and online submodular optimization. Finally, we demonstrate the practical applications of our frameworks by instantiating them with different state-of-the-art submodular optimization algorithms and comparing their welfare performance through empirical experiments on publicly available datasets that consist of thousands of sellers.

Details

NeurIPS Conference 2024 Conference Paper

Injecting Undetectable Backdoors in Obfuscated Neural Networks and Language Models

Alkis Kalavasis
Amin Karbasi
Argyris Oikonomou
Katerina Sotiraki
Grigoris Velegkas
Manolis Zampetakis

As ML models become increasingly complex and integral to high-stakes domains such as finance and healthcare, they also become more susceptible to sophisticated adversarial attacks. We investigate the threat posed by undetectable backdoors, as defined in Goldwasser et al. [2022], in models developed by insidious external expert firms. When such backdoors exist, they allow the designer of the model to sell information on how to slightly perturb their input to change the outcome of the model. We develop a general strategy to plant backdoors to obfuscated neural networks, that satisfy the security properties of the celebrated notion of indistinguishability obfuscation. Applying obfuscation before releasing neural networks is a strategy that is well motivated to protect sensitive information of the external expert firm. Our method to plant backdoors ensures that even if the weights and architecture of the obfuscated model are accessible, the existence ofthe backdoor is still undetectable. Finally, we introduce the notion of undetectable backdoors to language models and extend our neural network backdoor attacks to such models based on the existence of steganographic functions.

PDF Details DOI

NeurIPS Conference 2024 Conference Paper

On the Computational Landscape of Replicable Learning

Alkis Kalavasis
Amin Karbasi
Grigoris Velegkas
Felix Zhou

We study computational aspects of algorithmic replicability, a notion of stability introduced by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. Motivated by a recent line of work that established strong statistical connections betweenreplicability and other notions of learnability such as online learning, private learning, and SQ learning, we aim tounderstand better the computational connections between replicability and these learning paradigms. Our first result shows that there is a concept class that is efficiently replicably PAC learnable, but, under standardcryptographic assumptions, no efficient online learner exists for this class. Subsequently, we design an efficientreplicable learner for PAC learning parities when the marginal distribution is far from uniform, making progress on aquestion posed by Impagliazzo et al. [STOC, 2022]. To obtain this result, we design a replicable lifting framework inspired byBlanc, Lange, Malik, and Tan [STOC, 2023], that transforms in a black-box manner efficient replicable PAC learners under theuniform marginal distribution over the Boolean hypercube to replicable PAC learners under any marginal distribution, with sample and time complexity that depends on a certain measure of the complexity of the distribution. Finally, we show that any pure DP learner can be transformed in a black-box manner to a replicable learner, with time complexity polynomial in the confidence and accuracy parameters, but exponential in the representation dimension of the underlying hypothesis class.

PDF Details DOI

NeurIPS Conference 2024 Conference Paper

Randomized Truthful Auctions with Learning Agents

Gagan Aggarwal
Anupam Gupta
Andres Perlroth
Grigoris Velegkas

We study a setting where agents use no-regret learning algorithms to participate in repeated auctions. Recently, Kolumbus and Nisan [2022a] showed, rather surprisingly, that when bidders participate in second-price auctions using no-regret bidding algorithms, no matter how large the number of interactions $T$ is, the runner-up bidder may not converge to bidding truthfully. Our first result shows that this holds forall deterministictruthful auctions. We also show that the ratio of the learning rates of different bidders can qualitatively affect the convergence of the bidders. Next, we consider the problem of revenue maximization in this environment. In the setting with fully rational bidders, the seminal result of Myerson [1981] showed that revenue can be maximized by using a second-price auction with reserves. We show that, in stark contrast, in our setting with learning bidders, randomized auctions can have strictly better revenue guarantees than second-price auctions with reserves, when $T$ is large enough. To do this, we provide a black-box transformation from any truthful auction $A$ to an auction $A'$ such that: i) all mean-based no-regret learners that participate in $A'$ converge to bidding truthfully, ii) the distance between the allocation rule and the payment rule between $A, A'$ is negligible. Finally, we study revenue maximization in the non-asymptotic regime. We define a notion of auctioneer regret that compares the revenue generated to the revenue of a second price auction with truthful bids. When the auctioneer has to use the same auction throughout the interaction, we show an (almost) tight regret bound of $\tilde{\Theta}(T^{3/4})$. Then, we consider the case where the auctioneer can use different auctions throughout the interaction, but in a way that is oblivious to the bids. For this setting, we show an (almost) tight bound of $\tilde{\Theta}(\sqrt{T})$.

PDF Details DOI

ICML Conference 2024 Conference Paper

Replicable Learning of Large-Margin Halfspaces

Alkis Kalavasis
Amin Karbasi
Kasper Green Larsen
Grigoris Velegkas
Felix Zhou 0002

We provide an efficient replicable algorithm for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell (STOC, 2022). We design the first dimension-independent replicable algorithm for this task which runs in polynomial time, is proper, and has strictly improved sample complexity compared to the one achieved by Impagliazzo et al. (STOC, 2022) with respect to all the relevant parameters. Moreover, our algorithm has sample complexity that is optimal with respect to the accuracy parameter $\epsilon$. Departing from the requirement of polynomial time algorithms, using the DP-to-Replicability reduction of Bun et al. (STOC 2023), we show how to obtain a replicable algorithm for large-margin halfspaces with improved sample complexity with respect to the margin parameter $\tau$, but running time doubly exponential in $1/\tau^2$ and worse sample complexity dependence on $\epsilon$ than our previous algorithm. We then design an improved algorithm with better sample complexity than both of our previous algorithms and running time exponential in $1/\tau^{2}. $

Details

NeurIPS Conference 2024 Conference Paper

Universal Rates for Active Learning

Steve Hanneke
Amin Karbasi
Shay Moran
Grigoris Velegkas

In this work we study the problem of actively learning binary classifiers from a given concept class, i. e. , learning by utilizing unlabeled data and submitting targeted queries about their labels to a domain expert. We evaluate the quality of our solutions by considering the learning curves they induce, i. e. , the rate of decrease of the misclassification probability as the number of label queries increases. The majority of the literature on active learning has focused on obtaining uniform guarantees on the error rate which are only able to explain the upper envelope of the learning curves over families of different data-generating distributions. We diverge from this line of work and we focus on the distribution-dependent framework of universal learning whose goal is to obtain guarantees that hold for any fixed distribution, but do not apply uniformly over all the distributions. We provide a complete characterization of the optimal learning rates that are achievable by algorithms that have to specify the number of unlabeled examples they use ahead of their execution. Moreover, we identify combinatorial complexity measures that give rise to each case of our tetrachotomic characterization. This resolves an open question that was posed by Balcan et al. (2010). As a byproduct of our main result, we develop an active learning algorithm for partial concept classes that achieves exponential learning rates in the uniform setting.

PDF Details DOI

NeurIPS Conference 2023 Conference Paper

Optimal Learners for Realizable Regression: PAC Learning and Online Learning

Idan Attias
Steve Hanneke
Alkis Kalavasis
Amin Karbasi
Grigoris Velegkas

In this work, we aim to characterize the statistical complexity of realizable regression both in the PAC learning setting and the online learning setting. Previous work had established the sufficiency of finiteness of the fat shattering dimension for PAC learnability and the necessity of finiteness of the scaled Natarajan dimension, but little progress had been made towards a more complete characterization since the work of Simon 1997 (SICOMP '97). To this end, we first introduce a minimax instance optimal learner for realizable regression and propose a novel dimension that both qualitatively and quantitatively characterizes which classes of real-valued predictors are learnable. We then identify a combinatorial dimension related to the graph dimension that characterizes ERM learnability in the realizable setting. Finally, we establish a necessary condition for learnability based on a combinatorial dimension related to the DS dimension, and conjecture that it may also be sufficient in this context. Additionally, in the context of online learning we provide a dimension that characterizes the minimax instance optimal cumulative loss up to a constant factor and design an optimal online learner for realizable regression, thus resolving an open question raised by Daskalakis and Golowich in STOC '22.

PDF Details

NeurIPS Conference 2023 Conference Paper

Replicability in Reinforcement Learning

Amin Karbasi
Grigoris Velegkas
Lin Yang
Felix Zhou

We initiate the mathematical study of replicability as an algorithmic property in the context of reinforcement learning (RL). We focus on the fundamental setting of discounted tabular MDPs with access to a generative model. Inspired by Impagliazzo et al. [2022], we say that an RL algorithm is replicable if, with high probability, it outputs the exact same policy after two executions on i. i. d. samples drawn from the generator when its internal randomness is the same. We first provide an efficient $\rho$-replicable algorithm for $(\varepsilon, \delta)$-optimal policy estimation with sample and time complexity $\widetilde O\left(\frac{N^3\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$, where $N$ is the number of state-action pairs. Next, for the subclass of deterministic algorithms, we provide a lower bound of order $\Omega\left(\frac{N^3}{(1-\gamma)^3\cdot\varepsilon^2\cdot\rho^2}\right)$. Then, we study a relaxed version of replicability proposed by Kalavasis et al. [2023] called TV indistinguishability. We design a computationally efficient TV indistinguishable algorithm for policy estimation whose sample complexity is $\widetilde O\left(\frac{N^2\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$. At the cost of $\exp(N)$ running time, we transform these TV indistinguishable algorithms to $\rho$-replicable ones without increasing their sample complexity. Finally, we introduce the notion of approximate-replicability where we only require that two outputted policies are close under an appropriate statistical divergence (e. g. , Renyi) and show an improved sample complexity of $\widetilde O\left(\frac{N\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$.

PDF Details

ICLR Conference 2023 Conference Paper

Replicable Bandits

Hossein Esfandiari
Alkis Kalavasis
Amin Karbasi
Andreas Krause 0001
Vahab Mirrokni
Grigoris Velegkas

In this paper, we introduce the notion of replicable policies in the context of stochastic bandits, one of the canonical problems in interactive learning. A policy in the bandit environment is called replicable if it pulls, with high probability, the exact same sequence of arms in two different and independent executions (i.e., under independent reward realizations). We show that not only do replicable policies exist, but also they achieve almost the same optimal (non-replicable) regret bounds in terms of the time horizon. More specifically, in the stochastic multi-armed bandits setting, we develop a policy with an optimal problem-dependent regret bound whose dependence on the replicability parameter is also optimal. Similarly, for stochastic linear bandits (with finitely and infinitely many arms) we develop replicable policies that achieve the best-known problem-independent regret bounds with an optimal dependency on the replicability parameter. Our results show that even though randomization is crucial for the exploration-exploitation trade-off, an optimal balance can still be achieved while pulling the exact same arms in two different rounds of executions.

Details

NeurIPS Conference 2023 Conference Paper

Replicable Clustering

Hossein Esfandiari
Amin Karbasi
Vahab Mirrokni
Grigoris Velegkas
Felix Zhou

We design replicable algorithms in the context of statistical clustering under the recently introduced notion of replicability from Impagliazzo et al. [2022]. According to this definition, a clustering algorithm is replicable if, with high probability, its output induces the exact same partition of the sample space after two executions on different inputs drawn from the same distribution, when its internal randomness is shared across the executions. We propose such algorithms for the statistical $k$-medians, statistical $k$-means, and statistical $k$-centers problems by utilizing approximation routines for their combinatorial counterparts in a black-box manner. In particular, we demonstrate a replicable $O(1)$-approximation algorithm for statistical Euclidean $k$-medians ($k$-means) with $\operatorname{poly}(d)$ sample complexity. We also describe an $O(1)$-approximation algorithm with an additional $O(1)$-additive error for statistical Euclidean $k$-centers, albeit with $\exp(d)$ sample complexity. In addition, we provide experiments on synthetic distributions in 2D using the $k$-means++ implementation from sklearn as a black-box that validate our theoretical results.

PDF Details

ICML Conference 2023 Conference Paper

Statistical Indistinguishability of Learning Algorithms

Alkis Kalavasis
Amin Karbasi
Shay Moran
Grigoris Velegkas

When two different parties use the same learning rule on their own data, how can we test whether the distributions of the two outcomes are similar? In this paper, we study the similarity of outcomes of learning rules through the lens of the Total Variation (TV) distance of distributions. We say that a learning rule is TV indistinguishable if the expected TV distance between the posterior distributions of its outputs, executed on two training data sets drawn independently from the same distribution, is small. We first investigate the learnability of hypothesis classes using TV indistinguishable learners. Our main results are information-theoretic equivalences between TV indistinguishability and existing algorithmic stability notions such as replicability and approximate differential privacy. Then, we provide statistical amplification and boosting algorithms for TV indistinguishable learners.

Details

NeurIPS Conference 2022 Conference Paper

Multiclass Learnability Beyond the PAC Framework: Universal Rates and Partial Concept Classes

Alkis Kalavasis
Grigoris Velegkas
Amin Karbasi

In this paper we study the problem of multiclass classification with a bounded number of different labels $k$, in the realizable setting. We extend the traditional PAC model to a) distribution-dependent learning rates, and b) learning rates under data-dependent assumptions. First, we consider the universal learning setting (Bousquet, Hanneke, Moran, van Handel and Yehudayoff, STOC'21), for which we provide a complete characterization of the achievable learning rates that holds for every fixed distribution. In particular, we show the following trichotomy: for any concept class, the optimal learning rate is either exponential, linear or arbitrarily slow. Additionally, we provide complexity measures of the underlying hypothesis class that characterize when these rates occur. Second, we consider the problem of multiclass classification with structured data (such as data lying on a low dimensional manifold or satisfying margin conditions), a setting which is captured by partial concept classes (Alon, Hanneke, Holzman and Moran, FOCS'21). Partial concepts are functions that can be undefined in certain parts of the input space. We extend the traditional PAC learnability of total concept classes to partial concept classes in the multiclass setting and investigate differences between partial and total concepts.

PDF Details

NeurIPS Conference 2022 Conference Paper

Reinforcement Learning with Logarithmic Regret and Policy Switches

Grigoris Velegkas
Zhuoran Yang
Amin Karbasi

In this paper, we study the problem of regret minimization for episodic Reinforcement Learning (RL) both in the model-free and the model-based setting. We focus on learning with general function classes and general model classes, and we derive results that scale with the eluder dimension of these classes. In contrast to the existing body of work that mainly establishes instance-independent regret guarantees, we focus on the instance-dependent setting and show that the regret scales logarithmically with the horizon $T$, provided that there is a gap between the best and the second best action in every state. In addition, we show that such a logarithmic regret bound is realizable by algorithms with $O(\log T)$ switching cost (also known as adaptivity complexity). In other words, these algorithms rarely switch their policy during the course of their execution. Finally, we complement our results with lower bounds which show that even in the tabular setting, we cannot hope for regret guarantees lower than $O(\log T)$.

PDF Details

NeurIPS Conference 2022 Conference Paper

Universal Rates for Interactive Learning

Steve Hanneke
Amin Karbasi
Shay Moran
Grigoris Velegkas

Consider the task of learning an unknown concept from a given concept class; to what extent does interacting with a domain expert accelerate the learning process? It is common to measure the effectiveness of learning algorithms by plotting the "learning curve", that is, the decay of the error rate as a function of the algorithm's resources (examples, queries, etc). Thus, the overarching question in this work is whether (and which kind of) interaction accelerates the learning curve. Previous work in interactive learning focused on uniform bounds on the learning rates which only capture the upper envelope of the learning curves over families of data distributions. We thus formalize our overarching question within the distribution dependent framework of universal learning, which aims to understand the performance of learning algorithms on every data distribution, but without requiring a single upper bound which applies uniformly to all distributions. Our main result reveals a fundamental trichotomy of interactive learning rates, thus providing a complete characterization of universal interactive learning. As a corollary we deduce a strong affirmative answer to our overarching question, showing that interaction is beneficial. Remarkably, we show that in important cases such benefits are realized with label queries, that is, by active learning algorithms. On the other hand, our lower bounds apply to arbitrary binary queries and, hence, they hold in any interactive learning setting.

PDF Details

SODA Conference 2021 Conference Paper

An Efficient ∊ -BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization

Yang Cai 0001
Argyris Oikonomou
Grigoris Velegkas
Mingfei Zhao

We consider the black-box reduction from multidimensional revenue maximization to virtual welfare maximization. Cai et al. [12, 13, 14, 15] show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible ( ∊ -BIC). We provide two new polynomial time transformations that convert any ∊ -BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss. • Our first transformation applies to any mechanism design setting with downward-closed outcome space and only requires sample access to the agents' type distributions. • Our second transformation applies to the fully general outcome space, removing the downward-closed assumption, but requires full access to the agents' type distributions. Both transformations only require query access to the original ∊ -BIC mechanism. Other ∊ -BIC to BIC transformations for revenue exist in the literature [23, 36, 18] but all require exponential time to run in both of the settings we consider. As an application of our transformations, we improve the reduction by Cai et al. [12, 13, 14, 15] to generate an exactly BIC mechanism.

Details