This monograph portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

/pdf/introduction-to-online-convex-optimization-2dcnz959r2.pdf

Introduction to Online Convex Optimization

(1966). Mathematical Theory of Reliability. Journal of the Operational Research Society: Vol. 17, No. 2, pp. 213-215.

https://link.springer.com/content/pdf/10.1057%2Fjors.1966.43.pdf

Mathematical Theory of Reliability

The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or "price of total anarchy"). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games --- most notably, congestion games with cost functions restricted to an arbitrary fixed set --- that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first structural characterization of atomic congestion games that are universal worst-case examples for the POA.

/pdf/intrinsic-robustness-of-the-price-of-anarchy-6n0hpoe2v7.pdf

Intrinsic robustness of the price of anarchy

In this paper we study bid optimisation for real-time bidding (RTB) based display advertising. RTB allows advertisers to bid on a display ad impression in real time when it is being generated. It goes beyond contextual advertising by motivating the bidding focused on user data and it is different from the sponsored search auction where the bid price is associated with keywords. For the demand side, a fundamental technical challenge is to automate the bidding process based on the budget, the campaign objective and various information gathered in runtime and in history. In this paper, the programmatic bidding is cast as a functional optimisation problem. Under certain dependency assumptions, we derive simple bidding functions that can be calculated in real time; our finding shows that the optimal bid has a non-linear relationship with the impression level evaluation such as the click-through rate and the conversion rate, which are estimated in real time from the impression level features. This is different from previous work that is mainly focused on a linear bidding function. Our mathematical derivation suggests that optimal bidding strategies should try to bid more impressions rather than focus on a small set of high valued impressions because according to the current RTB market data, compared to the higher evaluated impressions, the lower evaluated ones are more cost effective and the chances of winning them are relatively higher. Aside from the theoretical insights, offline experiments on a real dataset and online experiments on a production RTB system verify the effectiveness of our proposed optimal bidding strategies and the functional optimisation framework.

/pdf/optimal-real-time-bidding-for-display-advertising-3briokkrqa.pdf

Optimal real-time bidding for display advertising

We propose a new approach for constructing P2P networks based on a dynamic decomposition of a continuous space into cells corresponding to processors. We demonstrate the power of these design rules by suggesting two new architectures, one for DHT (Distributed Hash Table) and the other for dynamic expander networks. The DHT network, which we call Distance Halving allows logarithmic routing and load, while preserving constant degrees. It offers an optimal tradeoff between the degree and the dilation in the sense that degree d guarantees a dilation of O(log dn). Another advantage over previous constructions is its relative simplicity. A major new contribution of this construction is a dynamic caching technique that maintains low load and storage even under the occurrence of hot spots. Our second construction builds a network that is guaranteed to be an expander. The resulting topologies are simple to maintain and implement. Their simplicity makes it easy to modify and add protocols. A small variation yields a DHT which is robust against random faults. Finally we show that, using our approach, it is possible to construct any family of constant degree graphs in a dynamic environment, though with worst parameters. Therefore we expect that more distributed data structures could be designed and implemented in a dynamic environment.

Novel architectures for P2P applications: the continuous-discrete approach

We study a general sub-class of concave games which we call socially concave games. We show that if each player follows any no-external regret minimization procedure then the dynamics will converge in the sense that both the average action vector will converge to a Nash equilibrium and that the utility of each player will converge to her utility in that Nash equilibrium. We show that many natural games are indeed socially concave games. Specifically, we show that linear Cournot competition and linear resource allocation games are socially-concave games, and therefore our convergence result applies to them. In addition, we show that a simple best response dynamics might diverge for linear resource allocation games, and is known to diverge for linear Cournot competition. For the TCP congestion games we show that "near" the equilibrium the games are socially-concave, and using our general methodology we show the convergence of a specific regret minimization dynamics.

On the convergence of regret minimization dynamics in concave games

Ad auctions in sponsored search support "broad match" that allows an advertiser to target a large number of queries while bidding only on a limited number. While giving more expressiveness to advertisers, this feature makes it challenging to optimize bids to maximize their returns: choosing to bid on a query as a broad match because it provides high profit results in one bidding for related queries which may yield low or even negative profits.We abstract and study the complexity of the {\em bid optimization problem} which is to determine an advertiser's bids on a subset of keywords (possibly using broad match) so that her profit is maximized. In the query language model when the advertiser is allowed to bid on all queries as broad match, we present a linear programming (LP)-based polynomial-time algorithm that gets the optimal profit. In the model in which an advertiser can only bid on keywords, ie., a subset of keywords as an exact or broad match, we show that this problem is not approximable within any reasonable approximation factor unless P=NP. To deal with this hardness result, we present a constant-factor approximation when the optimal profit significantly exceeds the cost. This algorithm is based on rounding a natural LP formulation of the problem. Finally, we study a budgeted variant of the problem, and show that in the query language model, one can find two budget constrained ad campaigns in polynomial time that implement the optimal bidding strategy. Our results are the first to address bid optimization under the broad match feature which is common in ad auctions.

/pdf/bid-optimization-for-broad-match-ad-auctions-fns0k6gbvn.pdf

Bid optimization for broad match ad auctions

We show a formal duality between certain equilibrium concepts, including the correlated and coarse correlated equilibrium, and analysis frameworks for proving bounds on the price of anarchy for such concepts. Our first application of this duality is a characterization of the set of distributions over game outcomes to which "smoothness bounds" always apply. This set is a natural and strict generalization of the coarse correlated equilibria of the game. Second, we derive a refined definition of smoothness that is specifically tailored for coarse correlated equilibria and can be used to give improved POA bounds for such equilibria.

/pdf/the-limits-of-smoothness-a-primal-dual-framework-for-price-4cxuur5wk6.pdf

The limits of smoothness: a primal-dual framework for price of anarchy bounds

The holy grail of online advertising is to target users with ads matched to their needs with such precision that the users respond to the ads, thereby increasing both advertisers' and users' value The current approach to this challenge utilizes information about the users: their gender, their location, the websites they have visited before, and so on Incorporating this data in ad auctions poses an economic challenge: can this be done in a way that the auctioneer's revenue does not decrease (at least on average)? This is the problem we study in this paper Our main result is that in Myerson's optimal mechanism, additional data leads to additional revenue However in simpler auctions, namely the second price auction with reserve prices, there are instances in which additional data decreases the revenue, albeit by only a small constant factor

/pdf/ad-auctions-with-data-56cq01ndd9.pdf

Ad auctions with data

We seek to understand behavior of selfish agents accessing a broadcast channel. In particular, we consider the natural agent utility where costs are proportional to delay. Access to the channel is modelled as a game in extensive form with simultaneous play.Standard protocols such as Aloha are vulnerable to manipulation by selfish agents. We show that choosing appropriate transmission probabilities for Aloha to achieve equilibrium implies exponentially long delays. We give a very simple protocol for the agents that is in Nash equilibrium and is also very efficient --- other than with exponentially negligible probability --- all n agents will successfully transmit within cn time, for some small constant c.

Uri Nadav

Papers

On the convergence of regret minimization dynamics in concave games

Bid optimization for broad match ad auctions

The limits of smoothness: a primal-dual framework for price of anarchy bounds

Ad auctions with data

Efficient contention resolution protocols for selfish agents