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Lower bounds for propositional proofs and independence results in bounded arithmetic (abstract) [chapter]

Alexander A. Razborov
1995 Lecture Notes in Computer Science  
Then we survey some known lower bounds on the complexity of proofs in various propositional proof systems, paying special attention to recent attempts on reducing such bounds to some purely complexity  ...  We begin with a highly informal discussion of the role played by Bounded Arithmetic and propositional proof systems in the reasoning about the world of feasible computations.  ...  Roughly speaking, if lower bounds for propositional proofs must necessarily be at least as hard as complexity lower bounds (and the latter are currently inaccessible), let us at least show that they are  ... 
doi:10.1007/3-540-60246-1_117 fatcat:4nthzsfwwramnj2bugsxm25rrm

P NP, propositional proof complexity, and resolution lower bounds for the weak pigeonhole principle [article]

Ran Raz
2003 arXiv   pre-print
After a short introduction to the problem of P NP and to the research area of propositional proof complexity, I will discuss the above mentioned lower bounds for the weak pigeonhole principle and the connections  ...  Recent results established exponential lower bounds for the length of any Resolution proof for the weak pigeonhole principle.  ...  An exponential lower bound for any Regular Resolution proof was proved in [8] , and an exponential lower bound for any Resolution proof was finally proved in [9] .  ... 
arXiv:cs/0304041v1 fatcat:jykfzldpy5hoxohe63iyx7cedi

Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

Fu Li, Iddo Tzameret, Zhengyu Wang, Marc Herbstritt
2015 Computational Complexity Conference  
In this work we show that Frege lower bounds in fact follow from corresponding size lower bounds on non-commutative formulas computing certain polynomials (and that such lower bounds on non-commutative  ...  The field of propositional proof complexity aims to understand and analyze the computational resources required to prove propositional statements.  ...  circuit class: both explicit exponential lower bounds and a deterministic PIT algorithms are known for non-commutative formulas.  ... 
doi:10.4230/lipics.ccc.2015.412 dblp:conf/coco/LiTW15 fatcat:hhbf52sujbehvgumh62uolm4y4

2-Exp Time lower bounds for propositional dynamic logics with intersection

Martin Lange, Carsten Lutz
2005 Journal of Symbolic Logic (JSL)  
We then sharpen our lower bound, showing that it even applies to IPDL without the test operator interpreted on tree structures.  ...  Since then, the exact complexity of IPDL has remained an open problem: the best known lower bound was the ExpTime one stemming from plain PDL until, in 2004. the first author established ExpSpace-hardness  ...  He also has found a simpler reduction using nondeterministic, exponential space bounded Turing Machines.  ... 
doi:10.2178/jsl/1129642115 fatcat:jq6alyg5f5gwjghla54bz6bfly

The Cost of Privacy in Generalized Linear Models: Algorithms and Minimax Lower Bounds [article]

T. Tony Cai, Yichen Wang, Linjun Zhang
2020 arXiv   pre-print
The lower bounds are obtained via a novel technique, which is based on Stein's Lemma and generalizes the tracing attack technique for privacy-constrained lower bounds.  ...  We show that the proposed algorithms are nearly rate-optimal by characterizing their statistical performance and establishing privacy-constrained minimax lower bounds for GLMs.  ...  Cai was supported in part by the National Science Foundation grants DMS-1712735 and DMS-2015259 and the National Institutes of Health grants R01-GM129781 and R01-GM123056. The research of L.  ... 
arXiv:2011.03900v2 fatcat:ynnepozcf5cn7jv2somcuhkx3u

Algorithm-Dependent Generalization Bounds for Overparameterized Deep Residual Networks [article]

Spencer Frei and Yuan Cao and Quanquan Gu
2019 arXiv   pre-print
Our optimization and generalization guarantees require overparameterization that is only logarithmic in the depth of the network, while all known generalization bounds for deep non-residual networks have  ...  By showing (i) we are able to demonstrate that deep residual networks trained with gradient descent have a small generalization gap between training and test error, and together with (ii) this guarantees  ...  The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agencies.  ... 
arXiv:1910.02934v1 fatcat:q5y7ulwlnvgqpav7txyyg6vciq

Sublinear Algorithms and Lower Bounds for Estimating MST and TSP Cost in General Metrics [article]

Yu Chen, Sanjeev Khanna, Zihan Tan
2022 arXiv   pre-print
We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on  ...  We design an algorithm that performs Õ(n^1.5) distance queries and achieves a strictly better than 2-approximation when either the metric is known to contain a spanning tree supported on weight-1 edges  ...  Part I Streaming Algorithms and Lower Bounds In this part we present our results on streaming algorithms and lower bounds for MST and TSP cost estimation.  ... 
arXiv:2203.14798v1 fatcat:hr5ch22k3fhkhagzxzsv6ytwse

Generalization Bounds and Algorithms for Learning to Communicate over Additive Noise Channels [article]

Nir Weinberger
2021 arXiv   pre-print
empirical error and generalization error, as well as a high probability generalization bound, are stated.  ...  In addition, a Gibbs-based algorithm that gradually expurgates an initial codebook from codewords in order to obtain a smaller codebook with improved error probability is proposed, and bounds on its average  ...  Proposition 2.  ... 
arXiv:2111.08253v1 fatcat:dbgzhkm63bf27drqpq5wsy5vs4

Algorithm-Dependent Generalization Bounds for Multi-Task Learning

Tongliang Liu, Dacheng Tao, Mingli Song, Stephen J. Maybank
2017 IEEE Transactions on Pattern Analysis and Machine Intelligence  
Based on this observation, in this paper, we present novel algorithm-dependent generalization bounds for MTL by exploiting the notion of algorithmic stability.  ...  Thus, the corresponding algorithm for learning the common parameter is also uniformly stable with respect to the domains of the multiple tasks, and its generalization bound is of the order O(1/T ), where  ...  Acknowledgment The work was supported in part by Australian Research Council Projects DP-140102164 and FT-130101457.  ... 
doi:10.1109/tpami.2016.2544314 pmid:27019472 fatcat:3dstuyzecbbnfgxyvhnbn5t23e

Practical Implementation and Error Bound of Integer-Type Algorithm for Higher-Order Differential Equations

Fuminori Sakaguchi, Masahito Hayashi
2011 Numerical Functional Analysis and Optimization  
We also give some results of numerical experiments and compare them with the corresponding exact analytical solutions, which show that the proposed algorithm is successful in yielding solutions with high  ...  (where p_m is a polynomial with rational-number-valued coefficients), by using only the four arithmetical operations on integers, and we proved its validity.  ...  useful in many applications, and it is very convenient for the interpretation of the basis systems used in this paper (for this, the relationship between these wavepackets and the Fourier series is shown  ... 
doi:10.1080/01630563.2011.595602 fatcat:ozs7bso27bezxojnjv47ayrnwi

Compression of Propositional Resolution Proofs by Lowering Subproofs [chapter]

Joseph Boudou, Bruno Woltzenlogel Paleo
2013 Lecture Notes in Computer Science  
This paper describes a generalization of the LowerUnits algorithm [10] for the compression of propositional resolution proofs.  ...  The generalized algorithm, called LowerUnivalents, is able to lower not only units but also subproofs of non-unit clauses, provided that they satisfy some additional conditions.  ...  and useful discussions on this topic.  ... 
doi:10.1007/978-3-642-40537-2_7 fatcat:2u2ilbcxjbb5tb6psk7lntfdby

General Bounds and Finite-Time Improvement for the Kiefer-Wolfowitz Stochastic Approximation Algorithm

Mark Broadie, Deniz Cicek, Assaf Zeevi
2011 Operations Research  
We consider the Kiefer-Wolfowitz (KW) stochastic approximation algorithm and derive general upper bounds on its meansquared error.  ...  The bounds are established using an elementary induction argument and phrased directly in the terms of tuning sequences of the algorithm.  ...  Acknowledgments This work was supported by Credit Suisse and NSF grants DMII-0447652 and DMS-0914539.  ... 
doi:10.1287/opre.1110.0970 fatcat:hncu5mxl3nggxf2pdu2u35i3my

Generalized feedback vertex set problems on bounded-treewidth graphs: chordality is the key to single-exponential parameterized algorithms [article]

Édouard Bonnet and Nick Brettell and O-joung Kwon and Dániel Marx
2019 arXiv   pre-print
Formally, for a class P of graphs, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that  ...  Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible  ...  Lower bound for fixed d We showed that Bounded P-Component Vertex Deletion and Bounded P-Block Vertex Deletion admit single-exponential time algorithms parameterized by treewidth, when P is a class of  ... 
arXiv:1704.06757v2 fatcat:hkees4ry6nebdno3lknpzxvw3y

Dynamic Algorithms Against an Adaptive Adversary: Generic Constructions and Lower Bounds [article]

Amos Beimel, Haim Kaplan, Yishay Mansour, Kobbi Nissim, Thatchaphol Saranurak, Uri Stemmer
2021 arXiv   pre-print
In the latter case our separation result draws from lower bounds in adaptive data analysis.  ...  We give a general reduction transforming a dynamic algorithm against an oblivious adversary to a dynamic algorithm robust against an adaptive adversary.  ...  However, our analysis differs significantly as our focus is on runtime lower bounds, while the focus of [KMNS21] was on space lower bounds.  ... 
arXiv:2111.03980v1 fatcat:ovd2fifhjvdurjqtpgosnc3nb4

A Globally Convergent Augmented Lagrangian Pattern Search Algorithm for Optimization with General Constraints and Simple Bounds

Robert Michael Lewis, Virginia Torczon
2002 SIAM Journal on Optimization  
We give a pattern search adaptation of an augmented Lagrangian method due to Conn, Gould, and Toint. The algorithm proceeds by successive bound constrained minimization of an augmented Lagrangian.  ...  So far as we know, this is the first provably convergent direct search method for general nonlinear programming.  ...  Moreover, if the feasible region for the original problem (6.1) is compact (e.g., if there are upper and lower bounds on all the components of y), then the feasible region for (6.4) is also compact, so  ... 
doi:10.1137/s1052623498339727 fatcat:osqo7iknf5aofdghmd3jajmyte
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