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Tim Roughgarden

Publications and source records attributed to Tim Roughgarden.

At least 37 records · Page 2Linked to original sources
arXiv2023

Utilitarian Algorithm Configuration

We present the first nontrivial procedure for configuring heuristic algorithms to maximize the utility provided to their end users while also offering theoretical guarantees about performance. Existing procedures seek configurations that minimize expected runtime. However, very recent theoretical work argues that expected runtime minimization fails to capture algorithm designers' preferences. Here we show that the utilitarian objective also confers significant algorithmic benefits. Intuitively, this is because mean runtime is dominated by extremely long runs even when they are incredibly rare; indeed, even when an algorithm never gives rise to such long runs, configuration procedures that provably minimize mean runtime must perform a huge number of experiments to demonstrate this fact. In contrast, utility is bounded and monotonically decreasing in runtime, allowing for meaningful empirical bounds on a configuration's performance. This paper builds on this idea to describe effective and theoretically sound configuration procedures. We prove upper bounds on the runtime of these procedures that are similar to theoretical lower bounds, while also demonstrating their performance empirically.

cs.AI↗
arXiv2023

No-Regret Learning with Unbounded Losses: The Case of Logarithmic Pooling

For each of $T$ time steps, $m$ experts report probability distributions over $n$ outcomes; we wish to learn to aggregate these forecasts in a way that attains a no-regret guarantee. We focus on the fundamental and practical aggregation method known as logarithmic pooling -- a weighted average of log odds -- which is in a certain sense the optimal choice of pooling method if one is interested in minimizing log loss (as we take to be our loss function). We consider the problem of learning the best set of parameters (i.e. expert weights) in an online adversarial setting. We assume (by necessity) that the adversarial choices of outcomes and forecasts are consistent, in the sense that experts report calibrated forecasts. Imposing this constraint creates a (to our knowledge) novel semi-adversarial setting in which the adversary retains a large amount of flexibility. In this setting, we present an algorithm based on online mirror descent that learns expert weights in a way that attains $O(\sqrt{T} \log T)$ expected regret as compared with the best weights in hindsight.

cs.LG↗
arXiv2023

Transaction Fee Mechanism Design with Active Block Producers

The incentive-compatibility properties of blockchain transaction fee mechanisms have been investigated with *passive* block producers that are motivated purely by the net rewards earned at the consensus layer. This paper introduces a model of *active* block producers that have their own private valuations for blocks (representing, for example, additional value derived from the application layer). The block producer surplus in our model can be interpreted as one of the more common colloquial meanings of the term ``MEV.'' The main results of this paper show that transaction fee mechanism design is fundamentally more difficult with active block producers than with passive ones: with active block producers, no non-trivial or approximately welfare-maximizing transaction fee mechanism can be incentive-compatible for both users and block producers. These results can be interpreted as a mathematical justification for the current interest in augmenting transaction fee mechanisms with additional components such as order flow auctions, block producer competition, trusted hardware, or cryptographic techniques.

cs.GT↗
arXiv2023

From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation

This paper forges a strong connection between two seemingly unrelated forecasting problems: incentive-compatible forecast elicitation and forecast aggregation. Proper scoring rules are the well-known solution to the former problem. To each such rule $s$ we associate a corresponding method of aggregation, mapping expert forecasts and expert weights to a "consensus forecast," which we call *quasi-arithmetic (QA) pooling* with respect to $s$. We justify this correspondence in several ways: - QA pooling with respect to the two most well-studied scoring rules (quadratic and logarithmic) corresponds to the two most well-studied forecast aggregation methods (linear and logarithmic). - Given a scoring rule $s$ used for payment, a forecaster agent who sub-contracts several experts, paying them in proportion to their weights, is best off aggregating the experts' reports using QA pooling with respect to $s$, meaning this strategy maximizes its worst-case profit (over the possible outcomes). - The score of an aggregator who uses QA pooling is concave in the experts' weights. As a consequence, online gradient descent can be used to learn appropriate expert weights from repeated experiments with low regret. - The class of all QA pooling methods is characterized by a natural set of axioms (generalizing classical work by Kolmogorov on quasi-arithmetic means).

cs.GT↗
arXiv2023

When Bidders Are DAOs

In a typical decentralized autonomous organization (DAO), people organize themselves into a group that is programmatically managed. DAOs can act as bidders in auctions, with a DAO's bid treated by the auctioneer as if it had been submitted by an individual, without regard to the internal structure of the DAO. We study auctions in which the bidders are DAOs. More precisely, we consider the design of two-level auctions in which the "participants" are groups of bidders rather than individuals. Bidders form DAOs to pool resources, but must then also negotiate the terms by which the DAO's winnings are shared. We model the outcome of a DAO's negotiations by an aggregation function (which aggregates DAO members' bids into a single group bid), and a budget-balanced cost-sharing mechanism (that determines DAO members' access to the DAO's allocation and distributes the total payment demanded from the DAO to its members). We pursue two-level mechanisms that are incentive-compatible (with truthful bidding a dominant strategy for members of each DAO) and approximately welfare-optimal. We prove that, even in the case of a single-item auction, incentive-compatible welfare maximization is not possible: No matter what the outer mechanism and the cost-sharing mechanisms used by DAOs, the welfare of the resulting two-level mechanism can be a $\approx \ln n$ factor less than optimal. We complement this lower bound with a natural two-level mechanism that achieves a matching approximate welfare guarantee. Our upper bound also extends to multi-item auctions where individuals have additive valuations. Finally, we show that our positive results cannot be extended much further: Even in multi-item settings with unit-demand bidders, truthful two-level mechanisms form a highly restricted class and as a consequence cannot guarantee any non-trivial approximation of the maximum social welfare.

cs.GT↗
arXiv2023

Formalizing Preferences Over Runtime Distributions

When trying to solve a computational problem, we are often faced with a choice between algorithms that are guaranteed to return the right answer but differ in their runtime distributions (e.g., SAT solvers, sorting algorithms). This paper aims to lay theoretical foundations for such choices by formalizing preferences over runtime distributions. It might seem that we should simply prefer the algorithm that minimizes expected runtime. However, such preferences would be driven by exactly how slow our algorithm is on bad inputs, whereas in practice we are typically willing to cut off occasional, sufficiently long runs before they finish. We propose a principled alternative, taking a utility-theoretic approach to characterize the scoring functions that describe preferences over algorithms. These functions depend on the way our value for solving our problem decreases with time and on the distribution from which captimes are drawn. We describe examples of realistic utility functions and show how to leverage a maximum-entropy approach for modeling underspecified captime distributions. Finally, we show how to efficiently estimate an algorithm's expected utility from runtime samples.

cs.AI↗
arXiv2023

Complexity-Approximation Trade-offs in Exchange Mechanisms: AMMs vs. LOBs

This paper presents a general framework for the design and analysis of exchange mechanisms between two assets that unifies and enables comparisons between the two dominant paradigms for exchange, constant function market markers (CFMMs) and limit order books (LOBs). In our framework, each liquidity provider (LP) submits to the exchange a downward-sloping demand curve, specifying the quantity of the risky asset it wishes to hold at each price; the exchange buys and sells the risky asset so as to satisfy the aggregate submitted demand. In general, such a mechanism is budget-balanced and enables price discovery. Different exchange mechanisms correspond to different restrictions on the set of acceptable demand curves. The primary goal of this paper is to formalize an approximation-complexity trade-off that pervades the design of exchange mechanisms. For example, CFMMs give up expressiveness in favor of simplicity: the aggregate demand curve of the LPs can be described using constant space, but most demand curves cannot be well approximated by any function in the corresponding single-dimensional family. LOBs, intuitively, make the opposite trade-off: any downward-slowing demand curve can be well approximated by a collection of limit orders, but the space needed to describe the state of a LOB can be large. This paper introduces a general measure of {\em exchange complexity}, defined by the minimal set of basis functions that generate, through their conical hull, all of the demand functions allowed by an exchange. With this complexity measure in place, we investigate the design of {\em optimally expressive} exchange mechanisms, meaning the lowest complexity mechanisms that allow for arbitrary downward-sloping demand curves to be well approximated. As a case study, we interpret the complexity-approximation trade-offs in the widely-used Uniswap v3 AMM through the lens of our framework.

math.FA↗
arXiv2023

Byzantine Generals in the Permissionless Setting

Consensus protocols have traditionally been studied in a setting where all participants are known to each other from the start of the protocol execution. In the parlance of the 'blockchain' literature, this is referred to as the permissioned setting. What differentiates Bitcoin from these previously studied protocols is that it operates in a permissionless setting, i.e. it is a protocol for establishing consensus over an unknown network of participants that anybody can join, with as many identities as they like in any role. The arrival of this new form of protocol brings with it many questions. Beyond Bitcoin, what can we prove about permissionless protocols in a general sense? How does recent work on permissionless protocols in the blockchain literature relate to the well-developed history of research on permissioned protocols in distributed computing? To answer these questions, we describe a formal framework for the analysis of both permissioned and permissionless systems. Our framework allows for "apples-to-apples" comparisons between different categories of protocols and, in turn, the development of theory to formally discuss their relative merits. A major benefit of the framework is that it facilitates the application of a rich history of proofs and techniques in distributed computing to problems in blockchain and the study of permissionless systems. Within our framework, we then address the questions above. We consider the Byzantine Generals Problem as a formalisation of the problem of reaching consensus, and address a programme of research that asks, "Under what adversarial conditions, and for what types of permissionless protocol, is consensus possible?" We prove a number of results for this programme, our main result being that deterministic consensus is not possible for decentralised permissionless protocols. To close, we give a list of eight open questions.

cs.DC↗
arXiv2022

Are You Smarter Than a Random Expert? The Robust Aggregation of Substitutable Signals

The problem of aggregating expert forecasts is ubiquitous in fields as wide-ranging as machine learning, economics, climate science, and national security. Despite this, our theoretical understanding of this question is fairly shallow. This paper initiates the study of forecast aggregation in a context where experts' knowledge is chosen adversarially from a broad class of information structures. While in full generality it is impossible to achieve a nontrivial performance guarantee, we show that doing so is possible under a condition on the experts' information structure that we call \emph{projective substitutes}. The projective substitutes condition is a notion of informational substitutes: that there are diminishing marginal returns to learning the experts' signals. We show that under the projective substitutes condition, taking the average of the experts' forecasts improves substantially upon the strategy of trusting a random expert. We then consider a more permissive setting, in which the aggregator has access to the prior. We show that by averaging the experts' forecasts and then \emph{extremizing} the average by moving it away from the prior by a constant factor, the aggregator's performance guarantee is substantially better than is possible without knowledge of the prior. Our results give a theoretical grounding to past empirical research on extremization and help give guidance on the appropriate amount to extremize.

cs.GT↗
arXiv2021

FPT Algorithms for Finding Near-Cliques in $c$-Closed Graphs

Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of $c$-closed graphs and proved that maximal clique enumeration is fixed-parameter tractable with respect to $c$. In practice, due to noise in data, one wishes to actually discover "near-cliques", which can be characterized as cliques with a sparse subgraph removed. In this work, we prove that many different kinds of maximal near-cliques can be enumerated in polynomial time (and FPT in $c$) for $c$-closed graphs. We study various established notions of such substructures, including $k$-plexes, complements of bounded-degeneracy and bounded-treewidth graphs. Interestingly, our algorithms follow relatively simple backtracking procedures, analogous to what is done in practice. Our results underscore the significance of the $c$-closed graph class for theoretical understanding of social network analysis.

math.CO↗
arXiv2021

How Does Blockchain Security Dictate Blockchain Implementation?

Blockchain protocols come with a variety of security guarantees. For example, BFT-inspired protocols such as Algorand tend to be secure in the partially synchronous setting, while longest chain protocols like Bitcoin will normally require stronger synchronicity to be secure. Another fundamental distinction, directly relevant to scalability solutions such as sharding, is whether or not a single untrusted user is able to point to *certificates*, which provide incontrovertible proof of block confirmation. Algorand produces such certificates, while Bitcoin does not. Are these properties accidental? Or are they inherent consequences of the paradigm of protocol design? Our aim in this paper is to understand what, fundamentally, governs the nature of security for permissionless blockchain protocols. Using the framework developed in (Lewis-Pye and Roughgarden, 2021), we prove general results showing that these questions relate directly to properties of the user selection process, i.e., the method (such as proof-of-work or proof-of-stake) which is used to select users with the task of updating state. Our results suffice to establish, for example, that the production of certificates is impossible for proof-of-work protocols, but is automatic for standard forms of proof-of-stake protocols. As a byproduct of our work, we also define a number of security notions and identify the equivalences and inequivalences among them.

cs.CR↗
arXiv2021

Strictly Proper Contract Functions Can Be Arbitrage-Free

We consider mechanisms for truthfully eliciting probabilistic predictions from a group of experts. The standard approach -- using a proper scoring rule to separately reward each expert -- is not robust to collusion: experts may collude to misreport their beliefs in a way that guarantees them a larger total reward no matter the eventual outcome. Chun and Shachter (2011) termed any such collusion "arbitrage" and asked whether there is any truthful elicitation mechanism that makes arbitrage impossible. We resolve this question positively, exhibiting a class of strictly proper arbitrage-free contract functions. These contract functions have two parts: one ensures that the total reward of a coalition of experts depends only on the average of their reports; the other ensures that changing this average report hurts the experts under at least one outcome.

cs.GT↗
arXiv2021

Smoothed Analysis with Adaptive Adversaries

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time an adversary chooses an input distribution with density function bounded above by $\tfrac{1}σ$ times that of the uniform distribution; nature then samples an input from this distribution. Crucially, our results hold for {\em adaptive} adversaries that can choose an input distribution based on the decisions of the algorithm and the realizations of the inputs in the previous time steps. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of adaptive adversaries to the simpler case of oblivious adversaries. We apply this technique to prove strong smoothed guarantees for three problems: -Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of $σ$-smooth distributions and the hypothesis class has VC dimension $d$. We bound the regret by $\tilde{O}\big(\sqrt{T d\ln(1/σ)} + d\sqrt{\ln(T/σ)}\big)$. This answers open questions of [RST11,Hag18]. -Online discrepancy minimization: We consider the online Komlós problem, where the input is generated from an adaptive sequence of $σ$-smooth and isotropic distributions on the $\ell_2$ unit ball. We bound the $\ell_\infty$ norm of the discrepancy vector by $\tilde{O}\big(\ln^2\!\big( \frac{nT}σ\big) \big)$. -Dispersion in online optimization: We consider online optimization of piecewise Lipschitz functions where functions with $\ell$ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is $\big( σ/{\sqrt{T\ell}}, \tilde O\big(\sqrt{T\ell} \big)\big)$-dispersed. This matches the parameters of [BDV18] for oblivious adversaries, up to log factors.

cs.LG↗
arXiv2021

Approximately Efficient Two-Sided Combinatorial Auctions

Mechanism design for one-sided markets has been investigated for several decades in economics and in computer science. More recently, there has been an increased attention on mechanisms for two-sided markets, in which buyers and sellers act strategically. For two-sided markets, an impossibility result of Myerson and Satterthwaite states that no mechanism can simultaneously satisfy individual rationality (IR), incentive compatibility (IC), strong budget-balance (SBB), and be efficient. On the other hand, important applications to web advertisement, stock exchange, and frequency spectrum allocation, require us to consider two-sided combinatorial auctions in which buyers have preferences on subsets of items, and sellers may offer multiple heterogeneous items. No efficient mechanism was known so far for such two-sided combinatorial markets. This work provides the first IR, IC and SBB mechanisms that provides an O(1)-approximation to the optimal social welfare for two-sided markets. An initial construction yields such a mechanism, but exposes a conceptual problem in the traditional SBB notion. This leads us to define the stronger notion of direct trade strong budget balance (DSBB). We then proceed to design mechanisms that are IR, IC, DSBB, and again provide an O(1)-approximation to the optimal social welfare. Our mechanisms work for any number of buyers with XOS valuations - a class in between submodular and subadditive functions - and any number of sellers. We provide a mechanism that is dominant strategy incentive compatible (DSIC) if the sellers each have one item for sale, and one that is bayesian incentive compatible (BIC) if sellers hold multiple items and have additive valuations over them. Finally, we present a DSIC mechanism for the case that the valuation functions of all buyers and sellers are additive.

cs.GT↗
arXiv2021

A General Framework for the Security Analysis of Blockchain Protocols

Blockchain protocols differ in fundamental ways, including the mechanics of selecting users to produce blocks (e.g., proof-of-work vs. proof-of-stake) and the method to establish consensus (e.g., longest chain rules vs. Byzantine fault-tolerant (BFT) inspired protocols). These fundamental differences have hindered "apples-to-apples" comparisons between different categories of blockchain protocols and, in turn, the development of theory to formally discuss their relative merits. This paper presents a parsimonious abstraction sufficient for capturing and comparing properties of many well-known permissionless blockchain protocols, simultaneously capturing essential properties of both proof-of-work (PoW) and proof-of-stake (PoS) protocols, and of both longest-chain-type and BFT-type protocols. Our framework blackboxes the precise mechanics of the user selection process, allowing us to isolate the properties of the selection process that are significant for protocol design. We demonstrate the utility of our general framework with several concrete results: 1. We prove a CAP-type impossibility theorem asserting that liveness with an unknown level of participation rules out security in a partially synchronous setting. 2. Delving deeper into the partially synchronous setting, we prove that a necessary and sufficient condition for security is the production of "certificates," meaning stand-alone proofs of block confirmation. 3. Restricting to synchronous settings, we prove that typical protocols with a known level of participation (including longest chain-type PoS protocols) can be adapted to provide certificates, but those with an unknown level of participation cannot. 4. Finally, we use our framework to articulate a modular two-step approach to blockchain security analysis that effectively reduces the permissionless case to the permissioned case.

cs.DC↗
arXiv2020

Simple versus Optimal Contracts

We consider the classic principal-agent model of contract theory, in which a principal designs an outcome-dependent compensation scheme to incentivize an agent to take a costly and unobservable action. When all of the model parameters---including the full distribution over principal rewards resulting from each agent action---are known to the designer, an optimal contract can in principle be computed by linear programming. In addition to their demanding informational requirements, such optimal contracts are often complex and unintuitive, and do not resemble contracts used in practice. This paper examines contract theory through the theoretical computer science lens, with the goal of developing novel theory to explain and justify the prevalence of relatively simple contracts, such as linear (pure commission) contracts. First, we consider the case where the principal knows only the first moment of each action's reward distribution, and we prove that linear contracts are guaranteed to be worst-case optimal, ranging over all reward distributions consistent with the given moments. Second, we study linear contracts from a worst-case approximation perspective, and prove several tight parameterized approximation bounds.

cs.GT↗
arXiv2020

Approximately Optimal Mechanism Design

Optimal mechanism design enjoys a beautiful and well-developed theory, and also a number of killer applications. Rules of thumb produced by the field influence everything from how governments sell wireless spectrum licenses to how the major search engines auction off online advertising. There are, however, some basic problems for which the traditional optimal mechanism design approach is ill-suited---either because it makes overly strong assumptions, or because it advocates overly complex designs. This survey reviews several common issues with optimal mechanisms, including exorbitant communication, computation, and informational requirements; and it presents several examples demonstrating that passing to the relaxed goal of an approximately optimal mechanism allows us to reason about fundamental questions that seem out of reach of the traditional theory.

econ.TH↗