arXiv subjects

Christopher Liaw

Publications and source records attributed to Christopher Liaw.

At least 19 recordsLinked to original sources

arXiv2026

From Compensation Design to Budget-Feasible Mechanisms: A Constant Approximation for Subadditive Valuations

Budget-feasible mechanism design is a classic framework introduced by Singer, but there is still a wide gap between existing upper and lower bounds. In this paper, we significantly advance the state of the art. First, without computational constraints, we show that there exists a universally truthful budget-feasible mechanism with the following approximation ratios: - $3$ for monotone submodular valuations and $e+1$ for nonmonotone submodular valuations, improving over $3.798$ and $9.742$, respectively. - $e+1$ for XOS valuations, improving over $28$. In large markets, our approximation can be improved deterministically to $e$. - $2e+1$ for subadditive valuations, improving over $33$. In large markets, our approximation can be improved deterministically to $2e$. Moreover, for subadditive valuations, we obtain a constant-approximation mechanism that runs in polynomial time using demand queries. This improves over the previous best approximation of $O(\log \log n)$, resolving a long-standing open problem going back to Dobzinski, Papadimitriou, and Singer, who conjectured that a constant approximation requires exponentially many demand queries. We obtain these results through a simple and unifying framework based on non-truthful indirect mechanisms, recently coined compensation design. In particular, through a potential argument, we establish constant price-of-stability bounds for compensation design based on marginal-contribution payment rules, which we then translate into truthful direct mechanisms. For subadditive valuations, the core of the argument is a new smoothing lemma showing that every subadditive function can be approximated within a factor of $2$ by a self-bounding function. This is also of independent interest, readily addressing an open question in multiwinner elections by showing the existence of a $2e$-approximate core even under subadditive valuations.

cs.GT↗

arXiv2026

Compensation Design

We introduce compensation design, the problem of designing payment rules that incentivize high-quality contributions in decentralized environments. Here, a budget-constrained principal with a monotone submodular value function aims to design a payment rule, while agents decide whether to opt in or out depending on their private cost. We show that a simple cost-oblivious and anonymous marginal-contribution payment rule guarantees that pure Nash equilibria always exist and attain a price of anarchy (PoA) of at most $2+o_{\lambda}(1)$ in the large-market regime ($\lambda \to 0$) where each individual cost is at most a $\lambda$ fraction of the budget. We further show that the factor $2$ is unavoidable among deterministic cost-oblivious rules. Surprisingly, we identify a counterexample showing that a payment rule based on the Shapley value may admit no pure Nash equilibria. We then extend our scope to coarse correlated equilibria. This is further motivated by our intractability result: although a pure Nash equilibrium always exists, computing one is PLS-complete. We establish that coarse correlated equilibria also attain a PoA bound of at most $2+o_\lambda(1)$, and this guarantee in fact extends even under the payment rule induced by the Shapley value. Moreover, we move beyond monotone submodular value functions and binary actions. First, for (monotone) XOS valuations, we show that no oracle-efficient payment rule can attain a PoA bound of $O(n^{1/2 - \epsilon})$. Second, for submodular but non-monotone valuations, we show that a broad class of natural payment rules fails to guarantee a bounded PoA. Finally, we extend compensation design to the setting where each agent has a combinatorial action set. We provide randomized payment rules with logarithmic PoA guarantees for subadditive values, and matching lower bounds that apply even in the single-agent additive-value setting.

cs.GT↗

arXiv2025

A Unified Approach to Submodular Maximization Under Noise

We consider the problem of maximizing a submodular function with access to a noisy value oracle for the function instead of an exact value oracle. Similar to prior work, we assume that the noisy oracle is persistent in that multiple calls to the oracle for a specific set always return the same value. In this model, Hassidim and Singer (2017) design a $(1-1/e)$-approximation algorithm for monotone submodular maximization subject to a cardinality constraint, and Huang et al (2022) design a $(1-1/e)/2$-approximation algorithm for monotone submodular maximization subject to any arbitrary matroid constraint. In this paper, we design a meta-algorithm that allows us to take any "robust" algorithm for exact submodular maximization as a black box and transform it into an algorithm for the noisy setting while retaining the approximation guarantee. By using the meta-algorithm with the measured continuous greedy algorithm, we obtain a $(1-1/e)$-approximation (resp. $1/e$-approximation) for monotone (resp. non-monotone) submodular maximization subject to a matroid constraint under noise. Furthermore, by using the meta-algorithm with the double greedy algorithm, we obtain a $1/2$-approximation for unconstrained (non-monotone) submodular maximization under noise.

cs.DS↗

arXiv2025

Risk-Averse and Optimistic Advertiser Incentive Compatibility in Auto-bidding

The rise of auto-bidding has created challenges for ensuring advertiser incentive compatibility, particularly when advertisers delegate bidding to agents with high-level constraints. One challenge in defining incentive compatibility is the multiplicity of equilibria. After advertisers submit reports, it is unclear what the result will be and one only has knowledge of a range of possible results. Nevertheless, Alimohammadi et al. proposed a notion of Auto-bidding Incentive Compatibility (AIC) which serves to highlight that auctions may not incentivize truthful reporting of constraints. However, their definition of AIC is very stringent as it requires that the worst-case outcome of an advertiser's truthful report is at least as good as the best-case outcome of any of the advertiser's possible deviations. Indeed, they show both First-Price Auction and Second-Price Auction are not AIC. Moreover, the AIC definition precludes having ordinal preferences on the possible constraints that the advertiser can report. In this paper, we introduce two refined and relaxed concepts: Risk-Averse Auto-bidding Incentive Compatibility (RAIC) and Optimistic Auto-bidding Incentive Compatibility (OAIC). RAIC (OAIC) stipulates that truthful reporting is preferred if its least (most) favorable equilibrium outcome is no worse than the least (most) favorable equilibrium outcome from any misreport. This distinction allows for a clearer modeling of ordinal preferences for advertisers with differing attitudes towards equilibrium uncertainty. We demonstrate that SPA satisfies both RAIC and OAIC. Furthermore, we show that SPA also meets these conditions for two advertisers when they are assumed to employ uniform bidding. These findings provide new insights into the incentive properties of SPA in auto-bidding environments, particularly when considering advertisers' perspectives on equilibrium selection.

cs.GT↗

arXiv2025

Equilibria and Learning in Modular Marketplaces

We envision a marketplace where diverse entities offer specialized "modules" through APIs, allowing users to compose the outputs of these modules for complex tasks within a given budget. This paper studies the market design problem in such an ecosystem, where module owners strategically set prices for their APIs (to maximize their profit) and a central platform orchestrates the aggregation of module outputs at query-time. One can also think about this as a first-price procurement auction with budgets. The first observation is that if the platform's algorithm is to find the optimal set of modules then this could result in a poor outcome, in the sense that there are price equilibria which provide arbitrarily low value for the user. We show that under a suitable version of the "bang-per-buck" algorithm for the knapsack problem, an $\varepsilon$-approximate equilibrium always exists, for any arbitrary $\varepsilon > 0$. Further, our first main result shows that with this algorithm any such equilibrium provides a constant approximation to the optimal value that the buyer could get under various constraints including (i) a budget constraint and (ii) a budget and a matroid constraint. Finally, we demonstrate that these efficient equilibria can be learned through decentralized price adjustments by module owners using no-regret learning algorithms.

cs.GT↗

arXiv2024

Auto-bidding and Auctions in Online Advertising: A Survey

In this survey, we summarize recent developments in research fueled by the growing adoption of automated bidding strategies in online advertising. We explore the challenges and opportunities that have arisen as markets embrace this autobidding and cover a range of topics in this area, including bidding algorithms, equilibrium analysis and efficiency of common auction formats, and optimal auction design.

cs.GT↗

arXiv2024

Agnostic Private Density Estimation for GMMs via List Global Stability

We consider the problem of private density estimation for mixtures of unrestricted high dimensional Gaussians in the agnostic setting. We prove the first upper bound on the sample complexity of this problem. Previously, private learnability of high dimensional GMMs was only known in the realizable setting [Afzali et al., 2024]. To prove our result, we exploit the notion of $\textit{list global stability}$ [Ghazi et al., 2021b,a] that was originally introduced in the context of private supervised learning. We define an agnostic variant of this definition, showing that its existence is sufficient for agnostic private density estimation. We then construct an agnostic list globally stable learner for GMMs.

stat.ML↗

arXiv2024

Improved Online Learning Algorithms for CTR Prediction in Ad Auctions

In this work, we investigate the online learning problem of revenue maximization in ad auctions, where the seller needs to learn the click-through rates (CTRs) of each ad candidate and charge the price of the winner through a pay-per-click manner. We focus on two models of the advertisers' strategic behaviors. First, we assume that the advertiser is completely myopic; i.e.~in each round, they aim to maximize their utility only for the current round. In this setting, we develop an online mechanism based on upper-confidence bounds that achieves a tight $O(\sqrt{T})$ regret in the worst-case and negative regret when the values are static across all the auctions and there is a gap between the highest expected value (i.e.~value multiplied by their CTR) and second highest expected value ad. Next, we assume that the advertiser is non-myopic and cares about their long term utility. This setting is much more complex since an advertiser is incentivized to influence the mechanism by bidding strategically in earlier rounds. In this setting, we provide an algorithm to achieve negative regret for the static valuation setting (with a positive gap), which is in sharp contrast with the prior work that shows $O(T^{2/3})$ regret when the valuation is generated by adversary.

cs.IR↗

arXiv2023

Efficiency of Non-Truthful Auctions in Auto-bidding with Budget Constraints

We study the efficiency of non-truthful auctions for auto-bidders with both return on spend (ROS) and budget constraints. The efficiency of a mechanism is measured by the price of anarchy (PoA), which is the worst case ratio between the liquid welfare of any equilibrium and the optimal (possibly randomized) allocation. Our first main result is that the first-price auction (FPA) is optimal, among deterministic mechanisms, in this setting. Without any assumptions, the PoA of FPA is $n$ which we prove is tight for any deterministic mechanism. However, under a mild assumption that a bidder's value for any query does not exceed their total budget, we show that the PoA is at most $2$. This bound is also tight as it matches the optimal PoA without a budget constraint. We next analyze two randomized mechanisms: randomized FPA (rFPA) and "quasi-proportional" FPA. We prove two results that highlight the efficacy of randomization in this setting. First, we show that the PoA of rFPA for two bidders is at most $1.8$ without requiring any assumptions. This extends prior work which focused only on an ROS constraint. Second, we show that quasi-proportional FPA has a PoA of $2$ for any number of bidders, without any assumptions. Both of these bypass lower bounds in the deterministic setting. Finally, we study the setting where bidders are assumed to bid uniformly. We show that uniform bidding can be detrimental for efficiency in deterministic mechanisms while being beneficial for randomized mechanisms, which is in stark contrast with the settings without budget constraints.

cs.GT↗

arXiv2023

Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples

We study the problem of estimating mixtures of Gaussians under the constraint of differential privacy (DP). Our main result is that $\text{poly}(k,d,1/\alpha,1/\varepsilon,\log(1/\delta))$ samples are sufficient to estimate a mixture of $k$ Gaussians in $\mathbb{R}^d$ up to total variation distance $\alpha$ while satisfying $(\varepsilon, \delta)$-DP. This is the first finite sample complexity upper bound for the problem that does not make any structural assumptions on the GMMs. To solve the problem, we devise a new framework which may be useful for other tasks. On a high level, we show that if a class of distributions (such as Gaussians) is (1) list decodable and (2) admits a "locally small'' cover (Bun et al., 2021) with respect to total variation distance, then the class of its mixtures is privately learnable. The proof circumvents a known barrier indicating that, unlike Gaussians, GMMs do not admit a locally small cover (Aden-Ali et al., 2021b).

stat.ML↗

arXiv2023

The Power of Two-sided Recruitment in Two-sided Markets

We consider the problem of maximizing the gains from trade (GFT) in two-sided markets. The seminal impossibility result by Myerson and Satterthwaite shows that even for bilateral trade, there is no individually rational (IR), Bayesian incentive compatible (BIC) and budget balanced (BB) mechanism that can achieve the full GFT. Moreover, the optimal BIC, IR and BB mechanism that maximizes the GFT is known to be complex and heavily depends on the prior. In this paper, we pursue a Bulow-Klemperer-style question, i.e., does augmentation allow for prior-independent mechanisms to compete against the optimal mechanism? Our first main result shows that in the double auction setting with $m$ i.i.d. buyers and $n$ i.i.d. sellers, by augmenting $O(1)$ buyers and sellers to the market, the GFT of a simple, dominant strategy incentive compatible (DSIC), and prior-independent mechanism in the augmented market is at least the optimal in the original market, when the buyers' distribution first-order stochastically dominates the sellers' distribution. Next, we go beyond the i.i.d. setting and study the power of two-sided recruitment in more general markets. Our second main result is that for any $\epsilon > 0$ and any set of $O(1/\epsilon)$ buyers and sellers where the buyers' value exceeds the sellers' value with constant probability, if we add these additional agents into any market with arbitrary correlations, the Trade Reduction mechanism obtains a $(1-\epsilon)$-approximation of the GFT of the augmented market. Importantly, the newly recruited agents are agnostic to the original market.

cs.GT↗

arXiv2023

Polynomial Time and Private Learning of Unbounded Gaussian Mixture Models

We study the problem of privately estimating the parameters of $d$-dimensional Gaussian Mixture Models (GMMs) with $k$ components. For this, we develop a technique to reduce the problem to its non-private counterpart. This allows us to privatize existing non-private algorithms in a blackbox manner, while incurring only a small overhead in the sample complexity and running time. As the main application of our framework, we develop an $(\varepsilon, δ)$-differentially private algorithm to learn GMMs using the non-private algorithm of Moitra and Valiant [MV10] as a blackbox. Consequently, this gives the first sample complexity upper bound and first polynomial time algorithm for privately learning GMMs without any boundedness assumptions on the parameters. As part of our analysis, we prove a tight (up to a constant factor) lower bound on the total variation distance of high-dimensional Gaussians which can be of independent interest.

stat.ML↗

arXiv2023

User Response in Ad Auctions: An MDP Formulation of Long-Term Revenue Optimization

We propose a new Markov Decision Process (MDP) model for ad auctions to capture the user response to the quality of ads, with the objective of maximizing the long-term discounted revenue. By incorporating user response, our model takes into consideration all three parties involved in the auction (advertiser, auctioneer, and user). The state of the user is modeled as a user-specific click-through rate (CTR) with the CTR changing in the next round according to the set of ads shown to the user in the current round. We characterize the optimal mechanism for this MDP as a Myerson's auction with a notion of modified virtual value, which relies on the value distribution of the advertiser, the current user state, and the future impact of showing the ad to the user. Leveraging this characterization, we design a sample-efficient and computationally-efficient algorithm which outputs an approximately optimal policy that requires only sample access to the true MDP and the value distributions of the bidders. Finally, we propose a simple mechanism built upon second price auctions with personalized reserve prices and show it can achieve a constant-factor approximation to the optimal long term discounted revenue.

cs.GT↗

arXiv2022

Continuous Prediction with Experts' Advice

Prediction with experts' advice is one of the most fundamental problems in online learning and captures many of its technical challenges. A recent line of work has looked at online learning through the lens of differential equations and continuous-time analysis. This viewpoint has yielded optimal results for several problems in online learning. In this paper, we employ continuous-time stochastic calculus in order to study the discrete-time experts' problem. We use these tools to design a continuous-time, parameter-free algorithm with improved guarantees for the quantile regret. We then develop an analogous discrete-time algorithm with a very similar analysis and identical quantile regret bounds. Finally, we design an anytime continuous-time algorithm with regret matching the optimal fixed-time rate when the gains are independent Brownian Motions; in many settings, this is the most difficult case. This gives some evidence that, even with adversarial gains, the optimal anytime and fixed-time regrets may coincide.

cs.LG↗

arXiv2022

Efficiency of non-truthful auctions under auto-bidding

Auto-bidding is now widely adopted as an interface between advertisers and internet advertising as it allows advertisers to specify high-level goals, such as maximizing value subject to a value-per-spend constraint. Prior research has mostly focused on auctions which are truthful (such as SPA) since uniform bidding is optimal in such auctions, which makes it manageable to reason about equilibria. A tantalizing question is whether one can obtain more efficient outcomes by leaving the realm of truthful auctions. This is the first paper to study non-truthful auctions in the prior-free auto-bidding setting. Our first result is that non-truthfulness provides no benefit when one considers deterministic auctions. Any deterministic mechanism has a price of anarchy (PoA) of at least $2$, even for $2$ bidders; this matches what can be achieved by deterministic truthful mechanisms. In particular, we prove that the first price auction has PoA of exactly $2$. For our second result, we construct a randomized non-truthful auction that achieves a PoA of $1.8$ for $2$ bidders. This is the best-known PoA for this problem. The previously best-known PoA for this problem was $1.9$ and was achieved with a truthful mechanism. Moreover, we demonstrate the benefit of non-truthfulness in this setting by showing that the truthful version of this randomized auction also has a PoA of $1.9$. Finally, we show that no auction (even randomized, non-truthful) can improve upon a PoA bound of $2$ as the number of advertisers grow to infinity.

cs.GT↗

arXiv2022

Private and polynomial time algorithms for learning Gaussians and beyond

We present a fairly general framework for reducing $(\varepsilon, δ)$ differentially private (DP) statistical estimation to its non-private counterpart. As the main application of this framework, we give a polynomial time and $(\varepsilon,δ)$-DP algorithm for learning (unrestricted) Gaussian distributions in $\mathbb{R}^d$. The sample complexity of our approach for learning the Gaussian up to total variation distance $α$ is $\widetilde{O}(d^2/α^2 + d^2\sqrt{\ln(1/δ)}/α\varepsilon + d\ln(1/δ) / α\varepsilon)$ matching (up to logarithmic factors) the best known information-theoretic (non-efficient) sample complexity upper bound due to Aden-Ali, Ashtiani, and Kamath (ALT'21). In an independent work, Kamath, Mouzakis, Singhal, Steinke, and Ullman (arXiv:2111.04609) proved a similar result using a different approach and with $O(d^{5/2})$ sample complexity dependence on $d$. As another application of our framework, we provide the first polynomial time $(\varepsilon, δ)$-DP algorithm for robust learning of (unrestricted) Gaussians with sample complexity $\widetilde{O}(d^{3.5})$. In another independent work, Kothari, Manurangsi, and Velingker (arXiv:2112.03548) also provided a polynomial time $(\varepsilon, δ)$-DP algorithm for robust learning of Gaussians with sample complexity $\widetilde{O}(d^8)$.

stat.ML↗

arXiv2021

Optimal anytime regret with two experts

We consider the classical problem of prediction with expert advice. In the fixed-time setting, where the time horizon is known in advance, algorithms that achieve the optimal regret are known when there are two, three, or four experts or when the number of experts is large. Much less is known about the problem in the anytime setting, where the time horizon is not known in advance. No minimax optimal algorithm was previously known in the anytime setting, regardless of the number of experts. Even for the case of two experts, Luo and Schapire have left open the problem of determining the optimal algorithm. We design the first minimax optimal algorithm for minimizing regret in the anytime setting. We consider the case of two experts, and prove that the optimal regret is $γ\sqrt{t} / 2$ at all time steps $t$, where $γ$ is a natural constant that arose 35 years ago in studying fundamental properties of Brownian motion. The algorithm is designed by considering a continuous analogue of the regret problem, which is solved using ideas from stochastic calculus.

cs.LG↗

arXiv2021

Privately Learning Mixtures of Axis-Aligned Gaussians

We consider the problem of learning mixtures of Gaussians under the constraint of approximate differential privacy. We prove that $\widetilde{O}(k^2 d \log^{3/2}(1/δ) / α^2 \varepsilon)$ samples are sufficient to learn a mixture of $k$ axis-aligned Gaussians in $\mathbb{R}^d$ to within total variation distance $α$ while satisfying $(\varepsilon, δ)$-differential privacy. This is the first result for privately learning mixtures of unbounded axis-aligned (or even unbounded univariate) Gaussians. If the covariance matrices of each of the Gaussians is the identity matrix, we show that $\widetilde{O}(kd/α^2 + kd \log(1/δ) / α\varepsilon)$ samples are sufficient. Recently, the "local covering" technique of Bun, Kamath, Steinke, and Wu has been successfully used for privately learning high-dimensional Gaussians with a known covariance matrix and extended to privately learning general high-dimensional Gaussians by Aden-Ali, Ashtiani, and Kamath. Given these positive results, this approach has been proposed as a promising direction for privately learning mixtures of Gaussians. Unfortunately, we show that this is not possible. We design a new technique for privately learning mixture distributions. A class of distributions $\mathcal{F}$ is said to be list-decodable if there is an algorithm that, given "heavily corrupted" samples from $f\in \mathcal{F}$, outputs a list of distributions, $\widehat{\mathcal{F}}$, such that one of the distributions in $\widehat{\mathcal{F}}$ approximates $f$. We show that if $\mathcal{F}$ is privately list-decodable, then we can privately learn mixtures of distributions in $\mathcal{F}$. Finally, we show axis-aligned Gaussian distributions are privately list-decodable, thereby proving mixtures of such distributions are privately learnable.

cs.LG↗