sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We evaluate our framework along a number of different axes pertaining to its ability to learn disentangled representations through the provision of partial graphical-model structures for the latents and weak supervision.
Sentence 2: we evaluate its ability to (i) function as a classifier/regressor for particular latents under the given dataset, (ii) learn the generative model in a manner that preserves the semantics of the latents with respect to the data generated, and (iii) perform these tasks, in a flexible manner, for a variety of different models and data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The t w triples are chosen by first uniformly selecting a group g and then uniformly selecting three indices i, j, and k from group g and finally assigning a weight of w g . For the t a triples, the sampling procedure first selects an index i from group g i with a probability proportional to the weights of the group.
Sentence 2: indices in group g are chosen proportional to w g . Two indices j and k are then selected uniformly at random from groups g j and g k other than g i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: (9) has an insightful interpretation: each factor template F learns a multiclass SVM given by parameters w F from factors f ∈ F , while each factor is augmented with messages λ f passed from all variables related to f . Despite the insightful interpretation, formulation (9) does not yield computational advantage directly.
Sentence 2: the non-smooth loss L f (.)
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let AE ´ µ Õ ¼ be the set of players that can be directly affected by player 's behavior.
Sentence 2: the transfer risk parameter Õ of player with respect to player is either zero, in which case the player has no direct effect on player 's behavior; or it is constant, in which case, the public safety 1 ´½ Ü µAE µ of player with respect to player ¾ AE ´ µ is the same as for any other player in AE ´ µ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our construction instead relies on designing F and g such that when S is "large", then we can make use of the concentration result of Lemma 1 and when S is "small", functions in F and g are deterministically close to each other.
Sentence 2: we introduce for H ⊆ N of size h: The value of the parameters α and h will be set later in the analysis.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The system is trained by reducing the error shown below (c and s are the cover and secret images respectively, and β is how to weigh their reconstruction errors): It is important to note where the errors are computed and the weights that each error affects, see Figure 3.
Sentence 2: note that the error term ||c − c || does not apply to the weights of the reveal-network that receives the container image and extracts the secret image.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, in order to build the DAG, we draw a directed edge from l ij of depth k − 1 to l ij of depth k if and only if p ij is a sub-path of p ij .
Sentence 2: all ascendants of l ij consist of the compressed labels obtained from sub-paths of p ij of size < k. Similar to Weisfeiler-Lehman kernel, we assign a uniform weight between parents and children.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we apply the KDR method to classification problems, for which many conventional methods of dimensionality reduction are not suitable.
Sentence 2: sIR requires the dimensionality of the effective subspace to be less than the number of classes, because SIR uses the average of X in slices along the variable Y . CCA and PLS have a similar limitation on the dimensionality of the effective subspace.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is because according to (10), when the primal-dual parameters (w, α) converge to the optimal parameters (w * , α * ), we have (∇φ i (w) + λnα i ) → 0, which means that even if the learning rate η t stays bounded away from zero, the procedure can converge.
Sentence 2: this is the same effect as SVRG, in the sense that the variance goes to zero asymptotically: as w → w * and α → α * , we have SDCA is also a variance reduction method for SGD, which is similar to SVRG.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the other hand, now consider the Fredholm kernel from Eqn 6 (or similarly from Eqn 7): k F (x e , z e ) = k(x e , u)p(u) • k H (u, v) • k(z e , v)p(v)dudv, and set the outer kernel k to be the Gaussian kernel, and the inner kernel k H to be the same as target kernel k target H . We can think of k F (x e , z e ) as an averaging of k H (u, v) over all possible pairs of data u, v, weighted by k(x e , u)p(u) and k(z e , v)p(v) respectively.
Sentence 2: points that are close to x e (resp.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To improve mixing rates, we first integrate out the Dirichlet variables Φ, by exploring the conjugacy between a Dirichlet prior and multinomial likelihood.
Sentence 2: we can integrate out Φ and perform Gibbs sampling for the marginalized distribution: where C t k is the number of times topic k being assigned to the term t over the whole corpus; is a function defined with the Gamma function Γ(•).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we study how the recovery error changes as the sparsity of W varies.
Sentence 2: we applied MMV prox on the data sets of size d = 100, m = 400, and n = 10 with k (the number of nonzero rows of W) varying from 0.05d to 0.7d, and used W − W p 2 F /(dn) as the recovery quality measure.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Algorithm 1 and Theorem 1 extend readily to the Bayesian setting in which f ⇠ GP(0, k(x, x 0 )).
Sentence 2: since the proof of Theorem 1 is based on confidence bounds, the only change required is selecting t to be that used for the Bayesian setting in [31].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inspired by SDA's ability to learn noise specific features in denoising tasks, we argue that in unsupervised feature learning problems the type of noise used can also affect the performance.
Sentence 2: instead of corrupting the input with arbitrarily chosen noise, more sophisticated corruption process that agrees to the true noise distribution in the data can improve the quality of the learned features.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We performed two further experiments to answer the following questions: • What happens if we continue training HCIAE-D-NP-ATT in an adversarial setting?
Sentence 2: we continue training by maximizing the score of the ground truth answer a gt t and minimizing the score of the generated answer ât , effectively setting up an adversarial training regime L D = −L G . The resulting discriminator HCIAE-GAN1 has significant drop in performance, as can be seen in Table .
Relationship: entailment
Domain: NeurIPS
Sentence 1: More formally, if the product is the graph G = G 1 × G 2 , then the vertex set of G is the Cartesian product of the vertex sets of G 1 and G 2 .
Sentence 2: in other words, for any vertex v 1 in G 1 and any vertex v 2 in G 2 , there exists a vertex (v 1 , v 2 ) in G. Additionally, the edge set of G is such that, for any edge the set of hypercube graphs (or, identically, the set of Hamming cubes) can be derived using the Cartesian product operator.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This defines a probabilistic mapping to the topic variables, which can be seen as an economical representation encoding essential information of the pose.
Sentence 2: from a image I new , we estimate the topic proportions θ(I new ) as:  The computation of this score requires only a dot product between low dimensional unit vectors θ, so our approach represent an efficient method for matching and clustering poses in large datasets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the current work we build on these advances by considering, not how search might operate on a pre-specified semantic representation, but rather how the representation itself can be learned from data (i.e., human-produced semantic fluency lists) given a specified model of the list-generation process.
Sentence 2: we model search as a random walk on a set of states (e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We compared the performance of the CRM algorithm to a simple unconstrained no-regret algorithm that treats both the true-positive and false positive probabilities similarly, but with different weight.
Sentence 2: the reward at stage n of this algorithm is g n (w) = f n (a n ) I {b n = 1} − wf n (a n ) I {b n = −1} for some weight parameter  w ≥ 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Somewhat counter-intuitively, we show that despite being computationally tractable to optimize, submodular functions cannot be minimized from samples to within a desirable guarantee, even when these functions are learnable.
Sentence 2: we show that there is no algorithm for minimizing a submodular function from polynomially-many samples drawn from any distribution that obtains an additive approximation of 1/2 o(1), even when the function is PAC-learnable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then the sequence of primal and dual variables {w t , α t } t≥0 of Quartz (Algorithm 1) satisfies: where θ is defined in (7).
Sentence 2: if we fix ≤ P (w 0 ) − D(α 0 ), then for In order to put the above result into context, in the rest of this section we will specialize the above result to two special samplings: a serial sampling, and the τ -nice sampling.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is much easier to do for ⇣ m,j as there are much fewer of these: individually they have higher correlation with the backpropagated gradient @L SGVB D /@b m,j , so the covariance is easier to estimate.
Sentence 2: measuring the effect of ⇣ m,j on @L SGVB D /@b m,j is easy as ⇣ m,j is the only random variable directly influencing this gradient via b m,j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We believe that our method can also be extended so that to model the case where the entries of the regression vector have a finite support, even if irrationally valued, and the entries of Y are only observed through their magnitude.
Sentence 2: we expect that the method of the present paper applies to the phase retrieval problem at least in some of the cases and this is one of the current directions we are exploring.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the case where g is strongly convex it is possible to obtain linear convergence rates that depend on the ratio γ = µ/L as opposed to the sublinear convergence rates discussed above.
Sentence 2: we obtain the following convergence rate on the iterates of the basic proximal-gradient method: Proposition 3 (Basic proximal-gradient method -Strong convexity) Assume (H), that g is µstrongly convex, and that we iterate recursion (3) with y k = x k . Then, for all k 1, we have: A consequence of this proposition is that we obtain a linear rate of convergence even in the presence of errors, provided that e k and √ ε k decrease linearly to 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we reverse the idea of training parameters of the network given fixed inputs to instead optimizing the inputs (i.e., actions) subject to fixed parameters (effectively the transition and reward parameterization assumed a priori known in planning).
Sentence 2: that is, as shown in figure 2, given transition T (s t , a t ) and reward function R(s t , a t ), we want to optimize the input a t for all t to maximize the accumulated reward value V . we want to optimize all actions a = (a 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This example is an enhanced version of the fused lasso [12], with some graph structure exploited to improve feature selection in biostatistic applications [6].
Sentence 2: given some graph whose nodes correspond to the feature variables, we let f ij (x) = |x i − x j | for every edge (i, j) ∈ E. For a general graph, the proximal map of the regularizer Similar as above, each f ij is 1-Lipschitz continuous w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a p-dimensional MHP, the intensity function of the i-th dimension takes the following form: where the constant µ i is the base intensity of the i-th dimension, N j (t) counts the number of arrivals in the j-th dimension within [0, t], and f i,j (t) is the triggering function that embeds the underlying causal structure of the model.
Sentence 2: one arrival in the j-th dimension at time τ will affect the intensity of the arrivals in the i-th dimension at time t by the amount f i,j (t − τ ) for t > τ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let T be a trellis over A proof of Theorem 1 can be found in the Appendix in Section E. By making a minor alteration to Algorithm 1, we are also able to compute the value of and find the clustering with maximal energy.
Sentence 2: at each vertex in the trellis, v, store the clustering of D(v) with maximal energy (and its associated energy).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each CS decision tree is trained in an iterative way and, similar to the case of standard decision trees, a decision about whether to branch new nodes or produce a leaf is taken based on a subset of the training samples Z ⊆ D. However, in contrast to the standard setting, the learning process depends on the order in which nodes of the tree are grown because split functions depend on t (<m) which in turn is affected by the node ordering.
Sentence 2: we impose an explicit ordering on the recursive calls of function g described in Equation ( 4).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Even though great care was taken in collecting the highest quality training data, there were a number of imperfections in the training data that could not be avoided: (a) The smallform-factor, low-cost cameras presented significant differences in their default settings.
Sentence 2: the white balance of the two cameras were somewhat different; (b) To maximize image quality, the automatic gain control and automatic exposure were activated.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From the first observation above, we will now introduce a new definition of margin, m, which performs a maximization over a superset Several examples are NP-hard m for H (DAGs, trees or cardinality constrained sets), but poly-time m for H being a set of binary strings.
Sentence 2: we can encode any DAG (in H) as a binary string (in H), but not all binary strings are DAGs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, under linear sparsity (k = αn for some constant α), the theorem shows that m ≍ 2α MAR • SNR n log n measurements are necessary for sparsity recovery.
Sentence 2: similarly, if m/n is bounded above by a constant, then sparsity recovery will certainly fail unless when SNR • MAR is bounded, the sparsity ratio k/n must approach zero.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Task Module Given a subtask graph G, the remaining time steps step t ∈ R, an eligibility vector e t and a completion vector x t , we compute a context embedding using recursive-reverse-recursive neural network (R3NN) [5] as follows: where [•] is a concatenation operator, b θ , t θ are the bottom-up and top-down encoding function, φ i bot,a , φ i top,a are the bottom-up and top-down embedding of i-th AND node respectively, and φ i bot,o , φ i top,o are the bottom-up and top-down embedding of i-th OR node respectively (see Appendix for the detail).
Sentence 2: the w i,j + , Child i , and P arent i specifies the connections in the subtask graph G. w i,j + = 1 if j-th OR node and i-th AND node are connected without NOT operation, −1 if there is NOT connection and 0 if not connected, and Child i , P arent i represent a set of i-th node's children and parents respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, it should be noted that these degrees should be treated differently because only "relevant" images represent a "concept" in the user's mind, while "non-relevant" images may represent a number of other concepts different from user's interest.
Sentence 2: while it is meaningful to treat the degree of relevance as a degree of membership to the class of relevant images, the same does not apply to the degree of non-relevance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper we tackle the problem of obtaining a tight characterization of the sample complexity which a particular learning rule requires, in order to learn a particular source distribution.
Sentence 2: we obtain a tight characterization of the sample complexity required for large (Euclidean) margin learning to obtain low error for a distribution D(X, Y ), for X ∈ R d , Y ∈ {±1}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An efficient approach to solving the constrained optimization problem (10) is the mirror descent algorithm (MDA) with Bregman divergence chosen to be generalized Kullback-Leibler divergence [2,18,21].
Sentence 2: let f (✓ d ) denote the cost function in (10), then the MDA updates the MAP estimate of ✓ d iteratively according to: ✓ d,`d enotes the estimate of ✓ d,`a t the `-th iteration, T d,`d enotes the step-size of MDA, and (x, y) is the Bregman divergence chosen to be (x, y) = x T ln(x/y) 1 T x + 1 T y.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The fact that F [v] < F max can be seen either by simply noting that χ 2 [v] ≥ 0, or from the data processing inequality, which applies not only to Kullback-Leibler divergence, but also to Rényi divergences [12,13,1].
Sentence 2: the variance in the firing rate explained by a given dimension F [v] cannot be greater than the overall variance in the firing rate F max .
Relationship: entailment
Domain: NeurIPS
Sentence 1: What's more, TrailBlazer exploits MDPs with specific structure by exploring only a fraction of the whole search space when either κ or d is small.
Sentence 2: we showed that if the set of near-optimal nodes have non-vanishing action-gaps, then the sample complexity is O(1/ε 2 ), which is the same rate as Monte-Carlo sampling.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 3 The problem and our approach Formally, we have a random variable X that takes values in a set of combinatorial objects M. For example, X could be a random tree drawn from the collection M of all trees in some graph G. We can think of the members of M as being the valid configurations among all possible sets in 2 V , so that any configuration not in M should get a probability of zero.
Sentence 2: we consider distributions over the configurations M of the general form where F : 2 V → R is the objective function and • is the Iverson bracket.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We show that our algorithm makes a conservative use of the privacy budget.
Sentence 2: if the underlying non-private learner yields a classification error of at most α ∈ (0, 1), then our construction answers more queries, by at least a factor of 1/α in some cases, than what is implied by a straightforward application of the advanced composition theorem for differential privacy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Some of these include: • Multivariate Regression with Covariance Estimation (MRCE-ℓ 2 ): With the task inverse covariance matrix Σ −1 = I K and the bias term set to zero, our model results in the ℓ 2 regularized weights variant of the MRCE model [17] which would be equivalent to minimizing the following objective: • Multitask Relationship Learning for Regression (MTRL): With the output inverse covariance matrix Ω −1 = I K and the sparsity constraint on Σ −1 dropped, our model results in the regression version of the multitask relationship learning model proposed in [22].
Sentence 2: the corresponding objective function would be: In [22], the − log |Σ −1 | term is dropped since the authors solve their cost function in of Σ and this term is concave in Σ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We scale up the algorithm to terabytes of real-world SS and BT data that contains hundreds of millions of users and hundreds of thousands of features, by leveraging the scalability characteristics of the algorithm and the inherent structure of the problem including data sparsity and locality.
Sentence 2: we demonstrate two somewhat orthogonal philosophies of scaling algorithms to large-scale problems, through the SS and BT implementations, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We first study the effectiveness of the algorithm, measured by the EV value of its output after the last sample, and compare it to the nuclear norm based online RPCA (OR-PCA) algorithm [8].
Sentence 2: we vary the intrinsic dimension d from 0.02p to 0.5p, with a step size 0.04p, and the corruption fraction ρ from 0.02 to 0.5, with a step size 0.04.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given access to a vector u = m k=1 α k (u)A k + P A ⊥ u (where P A ⊥ is the projection onto the orthogonal complements of the range of A), some recovered columns A 1 , .
Sentence 2: . . , A r , and corresponding rows of A † , we may zero out the components of u corresponding to the recovered columns of A. Let- u is orthogonal (in the •, • C space) to the previously recovered columns of A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This assumption originates from the fact that the convergence of MP-and FW-type algorithms fundamentally depends on an alignment assumption of the search direction returned by the LMO (i.e., z t in Algorithm 1) and the gradient of the objective at the current iteration (see third premise in [39]).
Sentence 2: for Algorithm 1, the LMO is assumed to select a descent direction, i.e., ∇f (x t ), z t < 0, so that the resulting weight (i.e., γ for Variant 0) is always positive.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The resulting state of applying a sequence of complete actions π = a 1 , ..., a n in s with respects to D i is defined as: A planning problem with incomplete domain D is P = D, I, G where I ⊆ F is the set of propositions that are true in the initial state (and all the remaining are false), and G is the set of goal propositions.
Sentence 2: an action sequence π is considered a valid plan for P if π solves the problem in at least one completion of D . Given that D can be exponentially large in terms of possible preconditions and effects, validity is too weak to guarantee on the quality of the plan.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We consider a pseudo-Euclidean inner product defined as follows: Let C = ADA T where D is a diagonal matrix with non-zero diagonal entries, and define •, • C by u, v C = u T C † v. When D contains negative entries, this is not a proper inner product since C is not positive definite.
Sentence 2: gives that the columns of A are orthogonal in this space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To limit the contribution of the top gate we force its main weight to be equal to +1 but its bias to be arbitrary (this is necessary also to avoid cases where the input to the top threshold gate is equal to 0).
Sentence 2: if the lower gate as weights w i and activation S = N i=0 w i x i , then the final output is given by O = sign(sign S + b) in the pure threshold gate case, and by O = sign(ReLU (S) + b) in the ReLU case.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We need to see if − a n , 1 ∈ W . Note that W is the convex set formed by starting with the graph of x → e ηx on the domain [−1, 1], including the line segment connecting this curve's endpoints (−1, e −η ) to (1, e ηx ), and including all of the points below this line segment but above the aforementioned graph.
Sentence 2: w is precisely the set It remains to check that 1 is sandwiched between the lower and upper bounds at x = − a n . Clearly the lower bound holds.
Relationship: entailment