sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We propose to search a policy in P that minimizes the following Bellman residual: Notice that, as for the maximization of the mean value, we used a sampling distribution ν, possibly different from the distribution of interest µ.
Sentence 2: from the basic properties of the Bellman operator, for any policy π we have that T * v π ≥ v π . Consequently, the ν-weighted 1 -norm of the residual is indeed the expected Bellman residual: there is naturally no bias problem for minimizing a residual here, contrary to other residual approaches [1].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the exact factors ψ ik with i = 1, . . . , n and k = y i , corresponding to the likelihood, (2), only depend on f k (x i ), f yi (x i ) and z i .
Sentence 2: the beta part of the corresponding approximate factors can be removed and the multivariate Gaussian distributions simplify to univariate Gaussians.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If γ-feasibility holds, then we will denote by y * ∞ (y * ∞ is not necessarily unique) the minimizer of the following minimization problem: (1) and we define the γ-feasible optimal value as 1) is a convex minimization problem over Y().
Sentence 2: the problem is equivalent to finding the saddle point of the Lagrangian function [15], namely, where the Lagrangian is We denote the optimal dual by λ * ∞ and assume that L can be maximize by a unique F ∞ -measurable function, λ * ∞ ().
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Definition 2 (Submodularity ratio) Let f be a non-negative set function.
Sentence 2: the submodularity ratio of f with respect to a set U and a parameter k ≥ 1 is it captures how much f can increase by adding any subset S of size k to L, compared to the combined benefits of adding its individual elements to L. In particular, [5] defines the submodularity ratio for the R 2 function and relates it to the smallest eigenvalue of the covariance matrix of the data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since by definition of v t , subtracting ⌘t v t from x t can only decrease positive entries in x t (see proof of Observation 2), and both v t , v + t are vertices of P (and thus in {0, 1} n ), and ⌘t = 2 t , it follows that each entry i in x t+1 is given by: Thus, x t+1 can also be written in the form 2 t st+1 for some st+1 2 N n . By definition of t and the monotonicity of {⌘ t } t 1 , we have that 2 t 2 t+1 is a positive integer.
Sentence 2: setting s t+1 = 2 t 2 t+1 st+1 , the induction holds also for t + 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Gaussian kernel To obtain the approximation of the Gaussian kernel, we use the following property [30].
Sentence 2: given a smooth, shift-invariant kernel k(x − x ) = k(z) with Fourier transform p(w), then: Using the Gaussian distribution p(w) = N (0, σ −1 ), we obtain the Gaussian kernel the kernel value k(x, x ) can be approximated by the inner product between φ(x) and φ(x ), where φ is defined as and W B ∈ R d×D/2 as a random matrix with its entries drawn from N (0, σ −1 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that Z is query dependant, i.e., the values of Z depend on the query vector w, and n inner product values can be obtained by taking the column-wise summation of Z > .
Sentence 2: in particular, for each j we have h > j w = P k t=1 z jt , j = 1, . . . , n. the ranking induced by the n inner product values can be characterized by the marginal ranking ⇡(j|w) defined on the implicit matrix Z as follows: ) As mentioned earlier, it is hard to design an ideal candidate screening procedure generating C(w) based on the marginal ranking.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The goal is thus similar to the usual distributed averaging problem (4), with the  key difference that each local value h i is itself an average depending on the entire data sample.
Sentence 2: our algorithms will combine two steps at each iteration: a data propagation step to allow each node i to estimate h i , and an averaging step to ensure convergence to the desired value Ûn (H).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, with δ := max i δ i , we have the following result: Theorem 3 (Structural Coherence (SC) Condition) Let δ := max i δ i with δ i as defined in (14).
Sentence 2: if δ < 1, then there exists a ρ > 0 such that for any ∆ i ∈ C i , i = 1, . . . , k, the SC condition in (9) holds, i.e., the SC condition is satisfied in the general case as long as the reflection −C i of any cone C i does not intersect, i.e., share a ray, with the Minkowski sum j =i C j of the other cones.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: κ (2) (W rec ) can be written and calculated in the matrix form: where for any i, j, k we have The only terms that require extra computation are powers of W rec which can be done in O(dT H 3 ) and the rest of the matrix computations need O(dT 2 H 2 ).
Sentence 2: the ratio of time complexity of calculating the first term and second term with respect to the gradient over mini-batch is O(1/B) and O((T + H)/B) respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: At a high level, our proof of Theorem 1 (as well as our new bounds on non-private estimation) follow from the fact that for all B and π, the expected score E[Score(B, π; G)] is equal to the score Score(B, π; Q), combined with a concentration argument.
Sentence 2: the maximizer B of Score(B; G) will approximately minimize the L 2 -distance δ2 (B, Q), which in turn will approximately minimize 1 ρ W [B] − W 2 , thus relating the L 2 -error of our estimator B to the "oracle error" (O) k (W ) defined in (4).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The last equality in (23) follows from the definition of Θ * ∈ T , since it gives T ∆ .
Sentence 2: thus, by the second inequalities of ( 8) and ( 9), Since Θ * = ∆ * − Θ ∈ T and the singular values of Θ is no smaller than (π ) by the second inequality in (7), Proposition 1 and (22) imply It follows from ( 23), ( 24) and (25) that the error bound (10) follows from (21), ( 22) and (26).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Intracellular recordings give clean signals on a spike train from a specific neuron, giving accurate spike times for that neuron.
Sentence 2: if we detect a spike in a nearby extracellular recording within a close time period (<.5ms) to an intracellular spike, we assume that the spike detected in the extracellular recording corresponds to the known neuron's spikes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In order to simplify these problems and exploit existing theorems, we will consider in this section that p ij = p is fixed for each edge {i, j} ∈ E. Infection dynamics thus only depend on p, the set of influencers A, and the structure of the underlying network.
Sentence 2: for a star shaped network centered around a given node v 1 , and A = {v 1 }, the exact influence is computable and writes σ({v Proposition 1 states that σ({v 1 }) ≤ 1 + (n − 1)γ 1 where γ 1 is the solution of equation It is worth mentionning that, when p = 1 (17) and therefore the bound is σ({v 1 }) ≤ 1 + √ n − 1 which is tight.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let L + and Q − be the Laplacian and the signless Laplacian matrices of the graphs G + and G − , respectively.
Sentence 2: we note that the matrix L SR blends the informations from G + and G − into (twice) the arithmetic mean of L + and Q − , namely the following identity holds ) as an alternative to the normalization defining L SN from L SR , it is natural to consider the arithmetic mean of the normalized Laplacians In the next section we introduce the geometric mean of L + sym and Q − sym and propose a new clustering algorithm for signed graphs based on that matrix.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Both models are particularly suited for search contexts, where the user is assumed to be looking for something relative to a query.
Sentence 2: the learner expects explicit feedback: in the Cascade Model each valid observation sequence must be either all zeros or terminated by a one, such that no ambiguity is left on the evaluation of the presented items, while multiple clicks are allowed in the DCM thus leaving some ambiguity on the last zeros of a sequence.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The log likelihood in this case reduces to a quadratic form: Substituting this into Eq.
Sentence 2: (2) gives: T ∂f ∂ x for these models, the Fisher information matrix induces a locally adaptive Euclidean metric on the space of images, as specified by the Jacobian matrix, ∂f /∂ x. pixel 1 pixel 2 For unit vectors : (human) Figure 1: Measuring and comparing model-derived predictions of image discriminability.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To determine the RG flow of the dimensionless coupling parameters the highfrequency components are integrated out from ω = λ − dλ to ω = λ to give a new effective action S over the low frequency modes ω < λ.
Sentence 2: to accomplish this the function α and the new action is obtained by integrating over a > (ω), where the averaging is defined by In the weak-coupling limit, we may expand the exponential term in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here we obtain a slight generalization of the result in Kakade and Ng [2004] as a special case.
Sentence 2: namely, the familiar linear model -with u(x) = θ • x, θ, x ∈ R d and Gaussian prior θ ∼ N (0, I) -can be seen as a GP model with the linear kernel: and so log |I + cK | = log I + cX T X our result gives an input-dependent version of the result of Kakade and Ng [2004].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the only component that explicitly depends on the neural parameters θ i is the conditional probability P (F|X, θ).
Sentence 2: the reward is conditionally independent of the neural parameters θ i given the coding feature F. if we want to optimize the expected reward by employing a gradient ascent method, we get a learning rule of the form If we choose a small learning rate η, the average over presynaptic patterns X and coding features F can be replaced by a time average.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: i where e i = y i − x T i βn , and holds, the excess risk of β for approximating βn is bounded in probability 1 − δ for δ > R log d 3nλmin(Σn) where Theorem 1 indicates that, β − βn can be bounded by Cr −1/2 .
Sentence 2: from (6), the choice of sampling method has no effect on the decreasing rate of the bound, r −1/2 , but influences the constant C. a theoretical measure of efficiency for some sampling method is whether it can make the constant C attain its minimum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It possesses a lower and upper envelope around its global optimum that are equivalent to x 2 and √ x; and therefore have different smoothness.
Sentence 2: for a standard partitioning, there is no semi-metric of the form (x, y) = ||x − y|| α for which the near-optimality dimension is d = 0, as shown by Valko et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While deriving unbiased estimators is very useful for ensuring stable convergence of stochastic gradient methods, it is not sufficient: convergence rates of stochastic gradient descent methods depend on the variance of the stochastic gradients, and this can be rather large for naïve choices of degree distributions.
Sentence 2: our main contribution is in establishing a provably optimal degree distribution minimizing the estimators' variances with respect to the Chebyshev series.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Assume we conduct the update (7) T iterations, the mirror descent algorithm outputs z T x,ξ;θ for each pair of (x, ξ).
Sentence 2: it naturally establishes another nonparametric function that maps from R d × Ξ to R r to approximate the sampler of the variational distribution point-wise, i.e., Since such an approximation function is generated by the mirror descent algorithm, we name the corresponding function class as optimization embedding.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One can check that the contribution based on these single-edge walks can be computed as: This weight provides a measure of the error-reduction capacity of edge {u, v} by itself at iteration n. These single-edge walks for edges in S n are a subset of all the walks in S n , and consequently provide a lower-bound on φ(|h (n−1) |; S n ).
Sentence 2: the maximization  is equivalent to minimizing a looser upper-bound than ( 8).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It can be easily checked that if the marginal distribution of z C for each clique C ∈ C is defined by (2) then these distributions are consistent as they yield the same distribution (2) over the separators.
Sentence 2: the unique Markov distribution over G with Dirichlet multinomial distribution over the cliques is defined by [8] Pr(z where for each complete set B ⊆ G, we have Pr(z B ) given by (2).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let M π j,k denote a normalised single-reward Markov chain (NSR-MC) of M π , which has exactly the same S, A, γ and P π as M π , but all rewards are trivially 0 except r j,k = 1.
Sentence 2: for an NSR-MC M π j,k , the equation above becomes the next step of analysis is to examine the state values in NSR-MCs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is easy to verify that this is equivalent to replacing the optimal h ⋆ by an approximate ĥ ∈ H that satisfies g ĥ p ≥ ν g h ⋆ p . Sparse case.
Sentence 2: when Ω(•) = • 1 , we need to solve It is well known that the optimal solution of max h∈H |h T Γ c h| is the dominant eigenvector of Γ c . we simply need to find the dominant eigenvector h c of each Γ c and select ĥ as the h c with largest singular value |h T c Γ c h c |.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: With τ → ∞, the reliability of the momentary evidence becomes overwhelming, such that the latent variable z is again immediately known, resulting in V (g, τ ) → V d (1) (= V d (0)) for all g. For τ = 0, the infinitesimal moments are dg = dg 2 = dτ 2 = 0, and dτ = ηµdt, such that g remains unchanged and τ drifts deterministically towards positive values.
Sentence 2: there is no leakage of V towards τ < 0, which makes this lower boundary well-defined.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, by Lemma 6, the assumption of non-emptiness of A cccp (x) for any x ∈ Ω ensures that A cccp is closed on Ω and so satisfies condition (3) in Theorem 2.
Sentence 2: by Theorem 2, all the limit points of {x (l) } ∞ l=0 are the generalized fixed points of A cccp and lim l→∞ (u(x , where x * is some generalized fixed point of A cccp .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In section 3.1, we describe the effects on the document-specific conditional posterior distributions, or predictive distributions, of replacing u with a fixed asymmetric (i.e., nonuniform) base measure m. In section 3.2, we then treat m as unknown, and take a fully Bayesian approach, giving m a Dirichlet prior (with a uniform base measure and concentration parameter α ) and integrating it out.
Sentence 2: if Θ is given an asymmetric Dirichlet prior with concentration parameter α and an known (nonuniform) base measure m, the predictive probability of topic t occurring in document d given Z is If topic t does not occur in z (d) , then N t|d will be zero, and the probability of generating z N d +1 = t will be m t . In other words, under an asymmetric prior, N t|d is smoothed with a topic-specific quantity αm t . different topics can be a priori more or less probable in all documents.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 4) that the set (A, t)(T k ∩ P k ) coincide with the feasible set of problem (5).
Sentence 2: if the system (5) has no solution, then (A, t)(T k ∩P k ) = ∅, and hence (A, t)(T k ∩ D) = ∅ (because D ⊂ P k ).
Relationship: reasoning
Domain: Networks
Sentence 1: 1 Proposed Protocol 0.9 SCA with LEACH 0.8 Packet Loss Rate (PLR) 0.7 0.6 0.5 0.4 0.3 low values of SNR, the proposed protocol performance is similar to that observed for the SCA with LEACH, because the channel quality is severely degraded.
Sentence 2: beyond 12dB, thePLR measuredin theproposedprotocoldecreases more than using SCA with LEACH, which also contributes to enhance the network lifetime.
Relationship: contrasting
Domain: Networks
Sentence 1: IPv4 address lookup techniques have received a fair amount of attention in recent years [36, 1, 2, 11, 12], and we do not attempt to be exhaustive in our survey of this sub-.eld.
Sentence 2: broadly speaking, there are two classes of lookup techniques: algorithmic approaches, exempli.ed by [12], attempt to cleverly compress for­warding tables compactly into memory, while hardware approaches rely on (ternary) content-addressable memories (CAMs) to perform fast lookup.
Relationship: contrasting
Domain: Networks
Sentence 1: CONCLUSIONS AND FUTURE WORK For SDN to support heterogeneous FEs, the common ap­proach is to just provide higher level abstractions (WHAT) for the south-bound interface and ask a smart FE to han­dle all the detailed work (HOW).
Sentence 2: every time a new feature is introduced, the FE needs to be amended to know how to handle it.
Relationship: contrasting
Domain: Networks
Sentence 1: Filemanagersprepareforrecoverybygeneratinga write-aheadloginsharedstorage.Forsystemsthat usetheshared-diskmodelwithoutfxedplacement, alloperationsexecuteatasinglemanagersite,and itisnecessaryandsufcientforthesystemtopro­videlockingandrecoveryproceduresfortheshared diskblocks[24].Forsystemswithfxedplacement, serversdonotshareblocksdirectly,butsomeoper­ationsmustupdatestateatmultiplesitesthrough apeer-peerprotocol.Thusthereisnoneedfordis­tributedlockingorrecoveryofindividualblocks,but thesystemmustcoordinateloggingandrecovery acrosssites,e.g.,usingtwo-phasecommit.
Sentence 2: formkdirswitching,theoperationsthatupdate multiplesitesarethoseinvolvingthe.orphaned" directoriesthatwereplacedondiferentsitesfrom theirparents.Theseoperationsincludetheredi­rectedmkdirsthemselves,associatedrmdirs,and anyrenameoperationsinvolvingtheorphaneden­tries.Sincetheseoperationsarerelativelyinfre­quent,asdeterminedbytheredirectionprobability parameterp,itisacceptabletoperformafulltwo­phasecommitasneededtoguaranteetheiratom­icityonsystemswithfxedplacement.namehashingrequiresfxedplacement|un­lessthedirectoryserverssupportfne-graineddis­tributedcaching|andanynamespaceupdatein­volvesmultiplesiteswithprobability(N,1)/N orhigher.Whileitispossibletoreducecommit costsbyloggingasynchronouslyandcoordinating rollback,thisapproachweakensfailureproperties becauserecentlycompletedoperationsmaybelost inafailure.
Relationship: contrasting
Domain: Networks
Sentence 1: The total network rate maximization problem can be formulated as M L max um (p) p m m=1 L s.t.
Sentence 2: pm,n = P¯m,m =1,...,M (22) m pm,n = 0 which unfortunately is a dif.cult problem, since the objective function in non-convex in p. a relaxation of this optimization problem (see in [12]) be considered a geometric programming problem [22], therefore, can be transformed into a convex optimization problem.
Relationship: contrasting
Domain: Networks
Sentence 1: In summary, our experiments on Wilocity radios show that to­day s low-cost 60GHz radios already offer online array alignment to handle moderate user motions.
Sentence 2: these radios are de­signed for quasi-static indoor environments, and the speed and amount  900 TCP Throughput (Mbps) 800 700 700 700 600 600 600 500 400 300 200 Link Rate (Mbps) TCP Throughput (Mbps) TCP Throughput (Mbps)   100 100 100 0 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -150 -100 -50 0 50 100 150 -8 -6 -4 -2 0 2 4 6 8 Deviation from original position (m) Circular deviation (degree) at 0.5m radius Vertical deviation from original position (m) (a) Moving closer/further(b) Moving circularly(c) Moving perpendicularly Figure 8: Measured throughput using our Wilocity testbed (2x8) when the receiver moves in three different motion patterns.
Relationship: contrasting
Domain: Networks
Sentence 1: The DT time caused by dependency between objects and retrieval, which would in turn affect whole page latency.
Sentence 2: as most common web browsers use a limited the WT time caused by limited parallelism contribute even more parallelism width for the parallel retrieval of objects, so some EOs of a page may be held in waiting state due to the unavailability of parallelism when the number of objects contained in the page is larger than the parallelism width.
Relationship: contrasting
Domain: Networks
Sentence 1: The number of used degrees of freedom is equal to the number of ongoing transmissions, which a node can learnfrompriorRTS-CTSmessages.As nodes contendfor the un­used degrees of freedom, they again use a contention window and random backoff similar to802.11.
Sentence 2: while carrier sensing, nodes need to ignore the signals from past contention winners.
Relationship: contrasting
Domain: Networks
Sentence 1: We might reasonably expect to .nd in each trace a relatively steady baseline replication level that holds for a long time (min­utes), pockmarked with brief depressions in the level caused by measurement loss.
Sentence 2: across all of the traces we .nd that the median interval length was just 3.4 seconds, the 75th percentile 32 seconds, and the 95th percentile just over 10 minutes.
Relationship: contrasting
Domain: Networks
Sentence 1: When the latency increases, the static setting of tcp rmem max fails to saturate the pipe, resulting in throughput degradation.
Sentence 2: networks 0.4 0.2 0 Throughput (Mbps) Figure 16: DRWA improves the throughput by 23%  with small latencies do not show such degradation since the in AT&#38;T HSPA+ network and 30% in Verizon LTE network when the BDP of the underlying network is large.
Relationship: contrasting
Domain: Networks
Sentence 1: When applied to our scenario, the 1 becomes C = TW iF i · cW iF i (h) + TGP RS · cGP RS (h) (2) Thus, given the time that the user is accessing the net­works (h) and the costs associated to the usage of each sin­gle network (cW iF i (h) and cGP RS (h)), by adopting a suit­able handover decision s policy (in this case, by tuning the threshold Th) the willingness to pay expressed by the user can be reasonably satis.ed.
Sentence 2: the adoption of a policy that allows the user to save money does not give guarantees that the performance of the running application sessions will be preserved.
Relationship: contrasting
Domain: Networks
Sentence 1: Note that this 2-round strategy indeed maintains the waypoint at any time during the policy update.
Sentence 2: the resulting solution may still be problematic, as it violates another desirable transient consistency property, namely loop-freedom: if the update for switch s3 arrives before the update at switch s2 , packets may be forwarded in a loop, from switch s2 to s3 and back.
Relationship: contrasting
Domain: Networks
Sentence 1: Furthermore, the system can now be entirely controlled as a signi.cant .exibility in the allocation of resources is introduced: If the system seeks to realize a fair rate sharing, it can give more slots to far users who are disadvantaged as compared to nearer users due to their small feasible rates, as for instance in WPF(1,1,4) where users in Zone 2 perceive a gain of 50% in comparison with PF.
Sentence 2: if the system wants to maximize the overall throughput without causing the starvation of distant users (as it is done in the Max SNR scheduler that always serves the user with highest rate [4]), it can give more slots to close users than to far away users.
Relationship: contrasting
Domain: Networks
Sentence 1: This entire data structure mapping scheme is essential for most of the application data struc­tures (e.g., a hash table or the virus-signature table).
Sentence 2: for certain data structures such as the trie used in LPM it is pos­sible to partition the data structure such that the most frequently accessed portions of the data structure (namely, the top-levels of the trie) are placed in fast memory and the remaining structure is stored in a larger, slower memory [9, 22].
Relationship: contrasting