Post author response: I thank the author(s) for their response and commenting on my discussion points. As those would need additional work, I for now keep my original score: this is a solid paper. While the proof for Lemma 4 & 5 is described very well in the main text, it would be helpful to have a short explanation how this is used to achieve Lemma 6. If necessary, I suggest to drop the proof of Lemma 3 from the main text as this result is standard. Quality: I have verified the proof in the main text and individual lemmas in the appendix.

PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. To do that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modeling parts, and the worst-case analysis is performed numerically via a standard solver.

Edit distance--a classical problem in computer science--has received ongoing attention from both practitioners and theoreticians. Given two strings A and B, the edit distance is the minimum number of substitutions, insertions, and deletions needed to transform A into B. For example, the edit distance between apf leee and rapleet is 3: . The edit distance problem is widely known, as it is often taught as part of the undergraduate curriculum to illustrate two-dimensional dynamic programming. Theoreticians have studied the problem starting from as early as 196624 and 1974,43 but it very much remains an active topic of research today (for example, Boroujeni et al.8). Simultaneously, bioinformatics practitioners continue to actively develop14,15,40 and apply9,28,38,39,42,45 fast edit distance solvers ubiquitously. Given its status as a classic problem and its importance to both theoreticians and practitioners, edit distance provides a unique opportunity to study the interaction between theory and practice. Theoreticians develop abstract algorithms that have superior theoretical performance; practitioners develop implemented algorithms that have superior empirical performance. In an ideal world, practitioners would implement and analyze the empirical performance of the abstract algorithms developed by theoreticians, while theoreticians would analyze the theoretical performance of the implemented algorithms developed by practitioners. In the real world, there is often a wide gap between the practical and theoretical communities; understanding how to close this gap is critical to making theoretical computer science more relevant to applications. The edit distance problem is then an excellent lens through which to understand how the theoretical analysis of algorithms impacts practical implementations. There are many ways to approach the practice/theory gap in a problem like edit distance.

The worst-case resource usage of a program can provide useful information for many software-engineering tasks, such as performance optimization and algorithmic-complexity-vulnerability discovery. This paper presents a generic, adaptive, and sound fuzzing framework, called DSE-SMC, for estimating worst-case resource usage. DSE-SMC is generic because it is black-box as long as the user provides an interface for retrieving resource-usage information on a given input; adaptive because it automatically balances between exploration and exploitation of candidate inputs; and sound because it is guaranteed to converge to the true resource-usage distribution of the analyzed program. DSE-SMC is built upon a key observation: resource accumulation in a program is isomorphic to the soft-conditioning mechanism in Bayesian probabilistic programming; thus, worst-case resource analysis is isomorphic to the maximum-a-posteriori-estimation problem of Bayesian statistics. DSE-SMC incorporates sequential Monte Carlo (SMC) -- a generic framework for Bayesian inference -- with adaptive evolutionary fuzzing algorithms, in a sound manner, i.e., DSE-SMC asymptotically converges to the posterior distribution induced by resource-usage behavior of the analyzed program. Experimental evaluation on Java applications demonstrates that DSE-SMC is significantly more effective than existing black-box fuzzing methods for worst-case analysis.

A case study reveals the theoretical analysis of algorithms is not always as helpful as standard dogma might suggest.

We provide a worst-case analysis of selective sampling algorithms for learning linear threshold functions. The algorithms considered in this paper are Perceptron-like algorithms, i.e., algorithms which can be effi- ciently run in any reproducing kernel Hilbert space. Our algorithms ex- ploit a simple margin-based randomized rule to decide whether to query the current label. We obtain selective sampling algorithms achieving on average the same bounds as those proven for their deterministic coun- terparts, but using much fewer labels. We complement our theoretical findings with an empirical comparison on two text categorization tasks.

The theoretical study of algorithms and data structures has been bolstered by worst-case analysis, where we prove bounds on the running time, space, approximation ratio, competitive ratio, or other measure that holds even in the worst case. Worst-case analysis has proven invaluable for understanding aspects of both the complexity and practicality of algorithms, providing useful features like the ability to use algorithms as building blocks and subroutines with a clear picture of the worst-case performance. More and more, however, the limitations of worst-case analysis become apparent and create new challenges. In practice, we often do not face worst-case scenarios, and the question arises of how we can tune our algorithms to work even better on the kinds of instances we are likely to see, while ideally keeping a rigorous formal framework similar to what we have developed through worst-case analysis. A key issue is how we can define the subset of "instances we are likely to see."

Value function approximation is a crucial module for policy evaluation in reinforcement learning when the state space is large or continuous. The present paper takes a generative perspective on policy evaluation via temporal-difference (TD) learning, where a Gaussian process (GP) prior is presumed on the sought value function, and instantaneous rewards are probabilistically generated based on value function evaluations at two consecutive states. Capitalizing on a random feature-based approximant of the GP prior, an online scalable (OS) approach, termed {OS-GPTD}, is developed to estimate the value function for a given policy by observing a sequence of state-reward pairs. To benchmark the performance of OS-GPTD even in an adversarial setting, where the modeling assumptions are violated, complementary worst-case analyses are performed by upper-bounding the cumulative Bellman error as well as the long-term reward prediction error, relative to their counterparts from a fixed value function estimator with the entire state-reward trajectory in hindsight. Moreover, to alleviate the limited expressiveness associated with a single fixed kernel, a weighted ensemble (E) of GP priors is employed to yield an alternative scheme, termed OS-EGPTD, that can jointly infer the value function, and select interactively the EGP kernel on-the-fly. Finally, performances of the novel OS-(E)GPTD schemes are evaluated on two benchmark problems.

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the "best" for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its worst performance on any input of a given size, implicitly advocating for the algorithm with the best-possible worst-case performance. Strong worst-case guarantees are the holy grail of algorithm design, providing an application-agnostic certification of an algorithm's robustly good performance. However, for many fundamental problems and performance measures, such guarantees are impossible and a more nuanced analysis approach is called for. This chapter surveys several alternatives to worst-case analysis that are discussed in detail later in the book.

Comparing different algorithms is hard. For almost any pair of algorithms and measure of algorithm performance like running time or solution quality, each algorithm will perform better than the other on some inputs.a For example, the insertion sort algorithm is faster than merge sort on already-sorted arrays but slower on many other inputs. When two algorithms have incomparable performance, how can we deem one of them "better than" the other? Worst-case analysis is a specific modeling choice in the analysis of algorithms, where the overall performance of an algorithm is summarized by its worst performance on any input of a given size. The "better" algorithm is then the one with superior worst-case performance. Merge sort, with its worst-case asymptotic running time of Θ(n log n) for arrays of length n, is better in this sense than insertion sort, which has a worst-case running time of Θ(n2). While crude, worst-case analysis can be tremendously useful, and it is the dominant paradigm for algorithm analysis in theoretical computer science. A good worst-case guarantee is the best-case scenario for an algorithm, certifying its general-purpose utility and absolving its users from understanding which inputs are relevant to their applications. Remarkably, for many fundamental computational problems, there are algorithms with excellent worst-case performance guarantees. The lion's share of an undergraduate algorithms course comprises algorithms that run in linear or near-linear time in the worst case.