Our approach also follows this overall strategy (see Section 3 for a more detailed discussion of this procedure).

all prior algorithms in this category make use of carefully specified learning rates in the SGD phase, while our approach makes use of any adaptive serial optimization algorithm, even ones that do not resemble SGD at all, such as [6,14].


More complex rollout policies can be considered, for example rollout policies that depend on the sampled model P i .

these usually incur computational overhead.


We show that the fixed point iteration method gives very fast convergence and accurate results for some classical directed graphical models.

for more challenging cases the fixed point method does not converge and the gradient based approach, which is guaranteed to converge, is preferable.


Hill and Doucet [11] derive a class-size independent risk guarantee of form O 1/n .

their bound is based on a delicate definition of margin, which is not commonly used in mainstream multi-class literature.


For example, the χ 2 kernel [8] and the histogram intersection kernel [5] are designed to compare histograms, hence they can be used as local kernels, if the features are histograms.

no random Fourier features can approximate them.


It was also observed that applying APPA/Catalyst reductions to SVRG one can achieve accelerated convergence rates [13,29], and this approach is commonly known as AccSVRG.

accSVRG requires some careful parameter tuning of its inner loops, and thus is a logarithmic-factor slower than KatyushaX and also less practical [3].


Many techniques have been proposed to speed up the softmax layer in training, such as hierarchical softmax [20] and sampling-based approaches [21][22][23][24].

most of them cannot be directly applied to decoding at inference because they rely on knowing the words to be predicted and they still need to calculate the probability of all words to find the most likely prediction.


This statistic is based on the spectral analysis of the graph, and in particular on the smallest non-zero eigenvalue of the Laplacian matrix, which is known as Fiedler value [6,7].

we use the Fiedler delta statistic to define a Boltzmann distribution over graphs, leading to the Fiedler random field (FRF) model.


We emphasize that sLDA accommodates various types of response: unconstrained real values, real values constrained to be positive (e.g., failure times), ordered or unordered class labels, nonnegative integers (e.g., count data), and other types.

the machinery used to achieve this generality complicates the presentation.


In the extreme case when κ " min a,� 2Δ ´1 � pa � q the regret guarantees are fairly weak.

in many specific cases mentioned in the previous section, κ is typically small or even 1.


This effect can be observed wherever shadows are found, such as the crevices of figure 2a.

the effect appears strongest in complex objects with many shadows and concavities, like folds of cloth, or foliage.


From a theoretical point of view, Sparse PCA is one of the simplest examples where we observe a gap between the number of samples needed information theoretically and the number of samples needed for a polynomial time estimator: It has been well understood [39,40,41] that information theoretically, given n = O(k log p) samples 1 , one can estimate v up to constant error (in euclidean norm), using a non-convex (therefore not polynomial time) optimization algorithm.

all the existing provable polynomial time algorithms [36,42,34,43], which use either diagonal thresholding (for the single spiked model) or semidefinite programming (for general covariance), first introduced for this problem in [44], need at least quadratically many samples to solve the problem, namely n = O(k 2 ).


Our theory will show that our selection scheme provably improves the convergence rate of training overall, by explicitly quantifying the benefit over uniform sampling.

existing work [2,7] only showed that the linear convergence rate on SVMs is preserved asymptotically, but not necessarily improved.


For Precision@20, there does exists some cases where GRANGER-BUSCA is outperformed by baseline methods.

the majority of these cases exist in the region where both models are below the efficacy of a Null model.


It is possible that the doubling trick or the analysis of Auer et al [2] (who encountered the same problem for the Exponential Weights algorithm) can remove the extra log T factor while still preserving the regret minimization property.

we also remark that R max is small when R 1 is instead the p-norm; hence, the use of this regularizer avoids the extraneous logarithmic in T factor while still preserving the logarithmic dependence on n and m. projection onto the simplex under the p-norm is not as elegant as the Exponential Weights update.


One way to avoid missing features is to learn only with those items that have at least ratings in the training set.

in very sparse settings this would force us to discard some of the valuable training data.


Gaussian process (GP) models are powerful tools for regression, function approximation, and predictive density estimation.

despite their power and flexibility, they suffer from several limitations.


It can be shown that in the case of extreme non-uniformity among {A i (w)} n i=1 , uniform sampling might require Ω(n) samples to capture the Hessian information appropriately.

we show that our non-uniform sampling schemes result in sample sizes completely independent of n and immune to such non-uniformity.


Naively training one detector per pose would require solving s large ns×ns systems (either with RR or SVR).

our method learns jointly all detectors using s much smaller n×n subproblems.


If implemented naively, this requires q inferences of the latent state from the activity of q−1 neurons.

this computation can be sped up by an order of magnitude by first finding the most likely state given all neurons, and then performing one Newton-update for each held out neuron from this initial state.


All the aforementioned algorithms essentially demonstrate that a signal of dimension n can be perfectly recovered through m > Cn amplitude-only measurements, where C > 1 is a constant that depends on the algorithm as well as the measurement vectors.

many interesting signals in practice contain fewer degrees of freedom than the ambient dimension (sparse signals, low-rank matrices, finite alphabet signals, etc.).


Many authors have numerically shown that spatiotemporal patterns can be stored in neural networks.

the mathematical mechanism for storage of the spatio-temporal patterns is still unknown, especially the effects of LTD.


Traditionally, MST algorithms use this rule in conjunction with a greedy "blue-edge" rule, which chooses edges for inclusion in the tree.

we will repeatedly use the red-edge rule until all but n − 1 edges have been eliminated.


Throughout the years, several extensions of PCA have been proposed to address the problems of outliers and missing data, see [6] for a review.

it still remains unclear how to generalize those extensions to KPCA; since directly migrating robust PCA techniques to KPCA is not possible due to the implicitness of the feature space.


TD learning uses the following update rule θ t+1 = θ t + α t δ t φ t , where α t is the stepsize.

tD is only guaranteed to converge in the on-policy setting, although in many off-policy situations, it still has satisfactory performance [21].


Then, a simple modification of Lemma 3 shows that, as p → +∞, any given change-point is a.s. found.

this modification, easy to do for the first change-point, is not obvious to extend to successive change-points detected by group LARS.


Nonuniform sampling is motivated by the observation that (in this and other applications) larger window sizes w generally result in significant performance improvements.

a large w results in much bigger networks, which take longer to train and, at least in theory, require larger amounts of training data to retain their generalization ability.


1 If we could evaluate P (Z (1:i−1) , Y|X (1:i−1) ), we could obtain weighted samples (or "particles") from P (Z (1:i) , Y|X (1:i) ) using importance sampling with a proposal distribution of and taking w ℓ ∝ P (X (i) |Z as the weight associated with the ℓ th particle.

we could also use a similar scheme to approximate P (Z (1:i−1) , Y|X (1:i−1) ) if we could evaluate P (Z (1:i−2) , Y|X (1:i−2) ).


We followed the prescription given in [2] for generating random cubic polynomial data.

here we use five levels of relevance instead of six, a label distribution corresponding to real datasets, and more data, all to more realistically approximate a Web search application.


On one hand, it avoids tilted ground, walls or tables and provides better visual appearance.

the shapes of objects will be simple due to not considering the angle of pitch or yaw.


Indeed, wherever complex time-series are found, Kalman filters have been successfully applied for Bayesian state estimation.

in practice, time series often have an unknown dynamical structure, and they are high dimensional and noisy, violating many of the assumptions made in established approaches for state estimation.


That is, if the loss on example z t is zero then w t+1 = w t . In case the loss is positive the update rule is w t+1 = w t + τ t v t where v t is the same as in the realizable case.

the scaling factor τ t is modified and is set to, The following theorem provides a loss bound for the online algorithm relative to the loss of any fixed weight vector w .  The proof of the theorem is based on a reduction to the realizable case (cf.


CODRNA was the only dataset where PASS-LR2 performed poorly.

this performance was expected based on the y n x n • θ MAP histogram (Fig.


The Clebsch-Gordan transform has recently appeared in two separate preprints discussing neural networks for learning physical systems [21,22].

to the best of our knowledge, it has never been proposed as a general purpose nonlinearity for covariant neural networks.


Hence, the risk of the portfolio is determined by the volatility of the time series of the assets.

insisting on the existence of useful forecast models, we propose to derive the covariance matrix from the forecast model residuals, i. e. the risk-matrix is determined by the covariances of the model errors.


It then clusters these directions to construct a series of masks that can isolate a large number of spatially separated sounds given only two side-by-side recordings.

this approach is limited in that it assumes a noiseless environment, and can only accurately estimate time differences for low frequencies due to the aliasing effect of high-frequency waveforms.


Many works in graph property testing focus on testing plain graphs that contain only the pure combinatorial information.

most graphs that model real data contain some additional information that may, for example, indicate the type of an atom, the bandwidth of a data link or spatial information of an object that is represented by a vertex or an edge, respectively.


Due to the enormous number of facts that could be asserted about our world and the difficulty in accessing and storing all these facts, KGs are incomplete.

it is possible to predict new links in a KG based on the existing ones.


We can similarly identify each graph with a bit string by "unraveling" the adjacency matrix into a long string of zeros and ones.

if we know beforehand an ordering on the nodes of our graph to which all directed graphs must stay consistent (to enforce acyclicness), then there are only n 2 possible edges, and the size of our metagraph drops to 2 ( n 2 ) .


Denoting J \k = {x j } j =k and M \k = L −1 J \k the full conditional can be simplified using Schur's determinantal equality [22]: In general, sampling directly from this full conditional is difficult.

for a wide range of kernel functions, including those which can be handled by the Nyström approximation in Sec.


A standard Bayesian treatment of the hierarchical model requires selecting a fixed λ, equivalent to a predetermined weighted or unweighted average of the models of a BayesOpt run.

we found that fixing λ is not competitive with approaches to hyperparameter tuning that involve some maximization.


the same minimizer x minimizes all components f i (x) (though of course it need not be a unique minimizer of any of them); this yields linear convergence to x , with a graceful degradation as σ 2 > 0.

in the linear convergence regime, the number of required iterations scales with the expected squared conditioning EL 2 i /µ 2 .


Indeed, assuming a common kernel, such as squared exponential or Matérn, is very limiting for robotic problems that involve discontinuity and non-stationarity.

with our approach of setting the prior and posterior parameters, BO outperforms all other methods in the task and motion planning benchmark problems.


Even in such a case, the MLE can take all possible patterns (2 10 = 1024) into account through the normalization term log Z θ ≃ Const + 1 2 ||θ|| 2 that works like a regularizer.

the proposed method genuinely uses only the observed examples, and the asymptotic analysis would not be relevant in this case.


Inference in this flat MRF is not easier, and is likely harder, because it requires an exponentially-large set of labels and the hard constraints of the grammar must be enforced explicitly.

this formulation will prove useful for our parsing algorithm.


In the extreme case where Xl = X 2 , the two cells are completely redundant and provide the same information about the stimulus, yielding RSpairs = I(Xl' X 2 ; S) -I(Xl ; S) -I(X2 ; S) = -I(Xl; S), which is always non-positive.

positive RSpairs values testify for synergistic interaction between Xl and X 2 ([8, 7, 4]).


The proposed algorithms feature a single variable that needs to be set.

the variable has a clear meaning -the allowed probability that an update moves in the wrong direction.


The images are manually verified to contain the concept of interest.

we did not pose any restriction about the view-point of the objects.


However, this type of image is difficult to model with a small number of layers due to the large number of objects present.

our model can infer any number of overlapping objects, without any explicit modelling of layers or depth.


Since the number of documents is usually very large, this produces enough parallelism to make full use of the GPU's cores.

this may not produce sufficient parallelism when drawing the φ k , because the number of topics is usually small compared to the number of cores.


For small problem sizes, interior point methods offer superior accuracy and convergence rates.

off-the-shelf interior point solvers become impractical for data matrices larger than about 70 × 70, due to the O(m 6 ) complexity of solving for the step direction.


Without loss of generality, assume that the instances are ordered in the sequence {x 11  We index instances by a function Non-convex optimizations are undesirable because few algorithms effectively converge even to a local optimum.

if both objective function and constraints take the form of a difference between two convex functions, then a non-convex problem can be solved efficiently by the constrained concave-convex procedure [14].


Probabilistic graph matching [2] makes the rank-1 approximation by assuming that each second-order element of M ia;jb is the product of the probability of feature i being matched to a and feature j being matched to b, independently.

instead of maximizing the quadratic score function, they use this probabilistic interpretation of the pair-wise terms and find the solution by looking for the closest rank-1 matrix to M in terms of the KL-divergence.


Counting FLOPs (the number of multiply-accumulates) is widely used to measure the computational cost.

it cannot replace the speed test on real devices, considering that there are many other factors that may influence the actual time cost, e.g.


Experimenal results indicate existance of differences between the reconstructions obtained by centralized factorization approach and that of D-PPCA.

the differences are small, depend on the object in question, and almost always include, within their confidence, the factorization result.


The early empirical results obtained for computer Go are very promising.

more practical experience with these algorithms is needed.


Like the Neural Statistician, we also produce a summary statistic for each class.

ours is a discriminative model, as befits our discriminative task of few-shot classification.


In the limiting case, one could consider a separate feature (e.g., an indicator) over each s, a, s triple.

similarly to rectangular solutions, having separate constraints for different state-action pairs is likely to lead to very conservative solutions in the presence of limited data.


In deep generative models, the BNs may be much larger than the examples shown here.

typically at the macro-level, where we collapse each vector to a single node, they are quite simple.


DC programming [21] is a popular approach to solve non-convex problems whose objective functions can be expressed as the difference of two convex functions.

a potentially non-trivial convex subproblem is required to solve at each iteration, which is not practical for large-scale problems.


The bounds shown there depend on the SVD spectrum of a matrix.

if we consider a random matrix, then a much stronger lower bound can be obtained with high probability: Corollary 4 For a random matrix M ∈ {−1, 1} n×n and 0 < r < n, almost certainly the minimum number of sign-non-preserving changes to a matrix in R n×n that is needed to reduce the rank of the matrix to r is at least Note that the rigidity bound given in [FS06] also applies to our problem, if we use the Hadamard matrix as the problem matrix.


Therefore, some strongly correlated variables tend to be in or out of the model together.

in practice, feature variables are often correlated.


To circumvent the doublesampling problem, these papers proposed multi-time-scale stochastic approximation algorithms, the former a policy gradient algorithm and the latter an actor-critic algorithm that uses simultaneous perturbation methods [Bhatnagar et al., 2013].

as discussed in Section 2.1, multi-time-scale stochastic approximation approach suffers from several weaknesses such as no available finite-sample analysis and difficult-to-tune stepsize schedules.


Simulation results also suggest that No Learning agents have a significantly lower belief of stationarity than Expert learners, but not different from WSLS.

the comparison of model accuracy between R-W and DBM (Fig 3d ) shows DBM outperforms R-W in predicting Expert and WSLS behavior (p = .000), but it does not perform as well in No Learning behavior where R-W has significantly better performance (p = .000).


To optimize an objective, it uses the noisy gradient computed from the random data samples instead of the true gradient computed from the entire dataset.

when the variance of the noisy gradient is large, the algorithm might spend much time bouncing around, leading to slower convergence and worse performance.


By reformulating the problem in Lagrange form, [8] proposed the spectrum Lasso along with an iterative convex minimization algorithm.

( 2) is difficult to analyze when the sample fraction π 0 = n/(d 1 d 2 ) is small, due to the ill-posedness of the quadratic term n i=1 M 2 ωi .


[n] k instead, we have This shows that, when using mini-batches of appropriate size, the sparsification of the gradient does not hurt convergence.

by increasing the mini-batch size, we increase the computation by a factor of d k . These two observations seem to indicate that the factor d k is inevitably lost, either by increased number of iterations or increased computation.


By using t he rela tion (2.4) , it follows that R,, (8 ) -L* = Op( nl / 2 ) and R(f 0) -L * = Op (nl ).5 Therefore, t he TOP kernel h as a much b etter convergen ce rate in R(f 0)' which is a strong motiva tion to use the TOP kernel instead of plug-in estimate.

5Fo r detail ed disc ussion a bout t he conve rgence orders of classificatio n e rror, see C ha pte r we must notice that this fast rate is possible only when the optimal linear classifier is combined with the TOP kernel.


[8] recommended choosing the kernel width as 10% to 20% of the range of the distance between samples.

in our experiment, we found that this is not always a good choice, and in many situations it produces poor results.


There has been an impressive progress on SoS degree lower bounds (via beautiful techniques) for a variety of combinatorial optimization problems [28,12,29,30].

for machine learning problems relatively few such lower bounds (above SDP level) are known [26,31] and follow via reductions to the above bounds.


The determinantal point process (DPP) has recently emerged as a powerful method for selecting a diverse subset from a "ground set" of items [13], with applications including document summarization [14] and information retrieval [15].

existing DPP techniques have a fatal modeling flaw if applied to video (or documents) for summarization: they fail to capture their inherent sequential nature.


The complexity is exponential to the tree degree (d).

in practice, we can enforce an upperbound of the tree degree in the model estimation stage.


In one sense, the model selection criterion is more demanding, since given knowledge of the edge set E(Θ * ), one could restrict estimation procedures to this subset, and so achieve faster rates.

theorem 1 requires incoherence conditions [A1] on the covariance matrix, which are not required for Frobenius norm consistency [1].


Inverting these operators allows to recover values of signal pairs but not their locations.

recombining these values on enough overlapping sets allows one to recover their locations and hence the original signal x.


In that region, a shape recipe will produce an incorrect shape estimate, however, the stereo algorithm will often succeed at finding those occlusion edges.

stereo often fails to provide the shape of image regions with complex shape details, where the shape recipes succeed.


An easy to decrease the variance is by using larger minibatches.

this comes at a considerable computational cost, undermining the whole benefit of using SGLD.


One way to address this is to add the positive-semidefinite constraint in the formulation, which keeps the problem convex.

in all of our experiments with model (2) the spectral radius resulted positive, so we decided to use this simpler formulation due to the power of the available solvers.


Interestingly, using a prior model with Gamma distribution adds to the barrier function a linear regularization term w 1 that encourages sparsity.

a prior model with a truncated Gaussian adds a square regularization term which drifts the nonnegative parameters away from zero.


Precision medicine aims for personalized prognosis and therapeutics by utilizing recent genome-scale high-throughput profiling techniques, including next-generation sequencing (NGS).

translating NGS data faces several challenges.


This has primed a line of research on algorithms to reduce the hyperparameter dependence in SGD-see Section 2 for an overview on the related literature.

all previous algorithms resort on adapting the learning rates, rather than removing them, or rely on assumptions on the shape of the objective function.


Once computed, we then take a step in the direction of v xi,x0 .

prior to taking this step, we compare the objective function value for x i and its negation x i .


This paper is primarily focused on the second order tensors (or, matrices).

the algorithm and analysis presented here can also be applied to higher order tensors.


We observe that when more samples are revealed, both OR-PCA and GRASTA steadily improve the subspace recovery.

our proposed OR-PCA converges much faster than GRASTA, possibly because in each iteration OR-PCA obtains the optimal closed-form solution to the basis updating subproblem while GRASTA only takes one gradient descent step.


Based on the group structures of p and w, we introduce their corresponding group penalties as follows: One can see the symmetry here: group penalty (4) on p is a squared, unweighted L 2,1 -norm, which is designed to secure a unique solution that favors more randomness in p for the sake of diversity in sonority generation [9]; group penalty (5) on w is a weighted L 1,2 -norm (group lasso), which enables rule selection.

there is a pitfall of the group lasso penalty when deployed in Problem (3): the problem has multiple global optima that are indefinite about the number of rules to pick (e.g.


So the capacity of these types of models is limited by the parametric form.

kernel methods are non-parametric models.


Standard projected subgradient ascent could be applied to solve (6), where at the t th iterate the (matrix-valued) subgradient is W M pB ptq q .

this approach requires solving optimal transport problems with large n ˆm matrices at each iteration.


In [28] an efficient spatio-temporal branch-and-bound approach was proposed as alternative.

the assumption of single fixed axis aligned volumetric representation is limiting and only applicable for well defined and relatively static actions.


There are some learning paradigms related to the L2MT framework, including multitask learning, transfer learning [27], lifelong learning [12], and learning to transfer [33].

there exist significant differences between the L2MT framework and these related paradigms.


As expected, increasing the amount of unlabeled data tends to improve the performance of SSL techniques.

we found that performance levelled off consistently across algorithms once 80,000 unlabeled examples were available.


As the learning rate parameter is reduced so the timescale for escape is also reduced.

the choice of optimal learning rate is non-trivial and cannot be determined by considering only the leading order terms in f as above, because although small F will result in a quicker escape from the unstable fixed point near f 0 x , this in turn will lead to a very slow learning transient after escape.


One complication is that solving this ODE requires the knowing value of z(t) along its entire trajectory.

we can simply recompute z(t) backwards in time together with the adjoint, starting from its final value z(t 1 ).


Theoretically, Newton's method has a quadratic convergence rate in terms of iteration, but with a cubic algorithmic complexity at each iteration.

we manage to lower the computation in each iteration to linear complexity.


We can see that the max-margin based multi-view MMH performs consistently better than any other competitors.

the likelihood-based TWH does not show any conclusive improvements compared to the unsupervised DWH.


Note that the total domain distinctiveness TDD(K) increases as K increases -presumably, in the extreme case, each domain has only a few instances and their distributions would be maximally different from each other.

such tiny domains would offer insufficient data to separate the categories of interest reliably.


Then for any S, c > 0 and n 2 N, we have We remark that the right-hand side of the above bound does not depend on the number of trading periods n. The key parameter is ⇣, which determines the size of the largest price jump of the stock.

we expect that as the trading frequency increases, the size of the largest price jump will shrink.


The simulations can be viewed as individual samples taken from a completely specified MJP, and can be very useful to reveal possible steady states.

it is not clear how observed data can be incorporated in a principled way, which renders this approach of limited use for posterior inference and parameter estimation.


This graph is well-represented by a latent class model in which edges occur with high probability between pairs having one member in each of A and B or in B and C, and among pairs having both members in C (in models of stochastic equivalence, nodes within each class are not differentiated).

representation of this type of graph with a latent distance model would require the dimension of the latent characteristics to be on the order of the class membership sizes.


Specifically, we show positive results for a large subclass of HMMs whose transition matrices are sparse, well-conditioned, and have small probability mass on short cycles.

we show that learning is impossible given only a polynomial number of samples for HMMs with a small output alphabet and whose transition matrices are random regular graphs with large degree.


As described in the full paper [19], this EM-GAMP method corresponds to a special case of the Adaptive GAMP method for a particular choice of the adaptation functions H t x and H t z .

in this work, we consider an alternate parameter estimation method based on ML adaptation.


Note that if the exact distribution was loop-free then SP would obviously be the optimal method when the error is measured in number of symbols.

since the BP is applied to an approximated distribution the superiority of the SP is not straightforward.


Impressive results have been obtained using loopy belief propagation for super-resolution (Freeman and Pasztor 1999) and for infering layered representations of scenes (Frey 2000) .

despite several theoretical advances in our understanding of loopy belief propagation (c.f.


To see this, first, it is not hard to prove that L K in the sense of usual the order on the positive semi-definite cone.

second, one can prove that the variance is matrix-increasing, hence the variance will decrease when replacing K by L. the bias (while not matrix monotone in general) can be proven to not increase too much when replacing K by L. This latter statement will be the main technical difficulty for obtaining a bound on R( fL ) (see Appendix A).


For the class of single Gaussians in d dimensions, this approach leads to the optimal sample complexity upper bound of O(d 2 /ε 2 ).

for mixtures of Gaussians and axis-aligned Gaussians in R d , the best known VC-dimension bounds [1,Theorem 8.14]  Another approach is to first approximate the mixture class using a more manageable class such as piecewise polynomials, and then study the associated Yatracos class, see, e.g., [5].


A generalization of attractive interactions to the non-pairwise case is presented in [2]: for all α ∈ A, ψ α , when appropriately normalized, has non-negative central moments.

the relationship between this generalization and log-supermodularity remains unclear.


It should be noted that Jennrich's algorithm [13,15,17] could recover {µ i } k i=1 directly from the 3rd order tensor M 3 alone when {µ i } k i=1 is linearly independent.

we still adopt the above simultaneous diagonalization framework because the intermediate vectors {W µ i } k i=1 play a vital role in the recovery procedure of the linear regression model η.


Some AUs happen together to show a meaningful facial expression: AU6 (cheek raiser) tends to occur together with AU12 (lip corner puller) when someone is smiling.

some AUs may be mutually exclusive: AU25 (lips part) never happens simultaneously with AU24 (lip presser) since they are activated by the same muscles but with opposite motions.


These latent variable perspectives are a powerful tool for understanding and designing neural networks.

they often overlook the fundamental process taking place, where successive layers successively warp the coordinate representation of the data manifold with nonlinear transformations into a form where the classes in the data manifold are linearly separable by hyperplanes.


In principle, one can implement AdaptReg by setting the termination criteria of the oracle in the inner loop as precisely suggested by the theory, such as setting the number of iterations for SDCA to be exactly T = O(n + L σt ) in the t-th epoch.

in practice, it is more desirable to automatically terminate the oracle whenever the objective distance to the minimum has been sufficiently decreased.


Examples alone may be sufficient when the distribution of data points are 'clean' -in the sense that there exists a set of prototypical examples which sufficiently represent the data.

this is rarely the case in real world data.


Variational inference [17] has also been used to explicitly compress the network.

the priors and posteriors used in these approaches will typically zero out individual weights.


Given a generative model and the restriction of DSTUMP to using root-level impurity reduction, the general thrust of our result is straightforward: impurity reduction due to active variables concentrates at a significantly higher level than that of inactive variables.

there are significant technical challenges in establishing this result, mainly deriving from the fact that the splitting procedure renders the data in the child nodes non-i.i.d., and hence standard concentration inequalities do not immediately apply.


Video datasets: Existing video datasets capture YouTube clips [12], human motion [10], synthetic video game frames [20], and driving [8].

to investigate learning visual physics prediction, we need data that exhibits rich object motion, collisions, and interaction information.


However, several Grids benchmarks were compilable to AC1, but failed to compile to AC2 or PSDD, given the time and space limits.

we were able to compile some of the relational benchmarks to PSDD, which did not compile to AC1 and compiled partially to AC2.


In addition, in a DBN containing only Gaussian units, one would only be able to model Gaussian data.

combining Gaussian with other types of units could be interesting.


We will then formulate the interface problem the solution of which will allow us to combine the embeddings on the two subdomains together to obtain an embedding over the whole domain.

it is not always feasible to carry out the procedure described above.


Therefore, larger regions provide higher saving (in bits) in the description length of R. A region R induces "average savings per point" for every point q ∈ R, namely, (where |R| is the number of points in R).

a point q ∈ R may also be contained in other regions generated by ref , each with its own local evidence score.


However, by drawing samples from all intensities, the computational complexity would not improve.

when the network is sparse, whenever we sample a new node (or link) event from the model, only a small number of intensity functions, in the local neighborhood of the node (or the link), will change.


Such representations can then be included as components in larger models which may be responsible for several tasks.

it can often be expensive to acquire multi-modal data even when large amounts of unsupervised data may be available.


Since many different people recorded the digits used in one of the types of audio CAP TCHAs we tested, PLP and RAS TA-PLP were needed to extract the features that were most useful for solving them.

in [4]- [5], the authors conducted experiments on recognizing isolated digits in the presence of noise using both PLP and RAS TA-PL P. the noise used consisted of telephone or microphone static caused by recording in different locations.


We also noted that the number of faces with an attribute was often several times smaller than that without the attribute in the dataset.

coGAN learning was not hindered by the mismatches.


Notably, with higher values of β, the mutual information penality in β-VAE is too strong and this hinders the usefulness of the latent variables.

β-TCVAE with higher values of β consistently results in models with higher disentanglement score relative to β-VAE.


All other architecture details are identical to the original model.

[9] down-weight the KL-divergence by a factor of 1e-6 at all steps.


The data structure internally maintains a clustering D 1 , .

. . , D k . when it queries some cluster C, it is not clear which of D 1 , .


[4]) as a special case when each set A j is reduced to a singleton and the order graph is equal to a full graph.

as illustrated in Figure 1, the formulation is more general and does not require a total ordering of the sets of training samples A j , i.e.


However, most of these algorithms are limited to the case of right-censored and un-censored data.

in the interval regression setting, the data are either left, right or interval-censored.


For instance, Empirical Bayes requires the maximization of marginal likelihood which is difficult to compute in hierarchical logistic models [9] in general, and cross-validation requires reducing the number of free parameters for computational reasons, potentially losing the flexibility to capture the desired phenomena.

we propose a principled way to set the hyper-parameters directly from data using an approximation to the observed Fisher Information Matrix.


Please note that for DQN and A2C, we choose their noisy net variants for comparison instead of the parameter noise ones, as the former variants lead to relatively better performance.

we select parameter noise DDPG as our baseline rather than the noisy net variant, as the authors of [8] do not provide their implementations.


Similar bounds are provided for cyclic BCD in [7,Theorem 6.1].

it is well known that when applying the classical gradient descent (GD) method to problem (1) with the constant stepsize 1/L, we have the following rate estimate [11, Corollary 2.1.2] Note that unlike (6), here the constant in front of the 1/(r + 4) term is independent of the problem dimension.


However, when modeling the image, the entire image embedding is used to obtain the answer.

our proposed encoder HCIAE (Section 4.1) localizes the region in the image that can help reliably answer the question.


While the sample complexity bounds match ours, the analysis for the Nystrom method has focused on positive-semidefinite kernel matrices and requires incoherence of both the row and column spaces.

entrywise subsampling is applicable, but the guarantees are weaker than ours.


To obtain these weights we averaged the covariance matrices C z (x), C τ (x) over all points and applied the proposed algorithm in Section 4, but independent of x.

a locally adapted weighting (c) outperforms the above methods and using more than one eigenvector (d-f) further enhances the correspondence.


EM-MLIRL failed to group the trajectories that align well with the players, even though we restarted it 100 times in order to mitigate the convergence to bad local optima.

dPM-BIRL was incorrect on only one trajectory, assigning a coin collector's trajectory to the expert player cluster.


STRIPES provably found the subsequent top 50 solutions within 4.5 hours for all but one of these cases (up to 8 spanning trees per calculation), and BMMF found the same 50 solutions for each case within 0.5 hours; note that only STRIPES provides a certificate of optimality for these solutions.

only for 146 of the 315 problems was the Nilsson method able to complete within five days; thus, we do not compare its performance here.


An initial step size must be selected, but the adaptive step size adjustment rules make this choice less important.

the BADMM method requires choosing an initial weight on the augmented Lagrangian term, an adjustment schedule for this term, a step size on the dual variables, and a step size for local policies, all of which have a substantial impact on the final performance of the method (the reported results are for the best setting of these parameters, identified with a hyperparameter sweep).


More recently, there have been attempts to predict the transmission probabilities from infection times, either by learning node representations [13], or by learning diffusion representations using the underlying network structure [13,14,15].

it remains an open problem to provide a generative probabilistic model for the underlying network from partial observations.


The update of θ is based on the samples generated from q φ (H|V), and is identical to (10).

in contrast to (11), the recognition parameters φ are estimated from samples generated by the model, i.e., This update does not optimize the same objective as in (10), hence the wake-sleep algorithm is not guaranteed to converge [13].


Many clustering algorithms have been proposed in literature, including K-means, spectral clustering, Gaussian mixture models and hierarchical clustering, to solve problems with respect to a wide range of cluster shapes.

much research has pointed out that these methods all suffer from instabilities [3,20,16,15,13,19].


This is especially important in the task of molecular graph generation, whose goal is to discover novel molecules with desired properties such as drug-likeness and synthetic accessibility, while obeying physical laws such as chemical valency.

designing models to find molecules that optimize desired properties while incorporating highly complex and non-differentiable rules remains to be a challenging task.


As gets smaller the runtime slightly increases, but it decreases for = 0 since the ∞ norm is computed efficiently using the max function.

= 0 is less accurate than = 0.01; When the approximated structured SVM converges, the gap between the primal and dual objectives was 1.3, and only 10 −5 for > 0.


It was experimentally confirmed the chip operates at 1000 frames/sec.

the operation is limited by the integration time of PD's and typical motion images are processed at about 200 frames/sec.


We train a separate reconstruction network for each noise level (positing that a camera could select between these based on the ISO settings).

since it is impractical to employ different sensors for different settings, we learn a single spatial multiplexing pattern, optimized for reconstruction under moderate noise levels with standard deviation (STD) of 0.01 (with respect to intensity values in x(n) scaled to be between 0 and 1).


We observe that regardless of which norm we use to optimize the critic, there are improvements brought by applying ACKTR to the actor compared to the baseline A2C.

the improvements brought by using the Gauss-Newton norm for optimizing the critic are more substantial in terms of sample efficiency and episode rewards at the end of training.


A deep ResNet with one iteration and 10 RBs captures a cleaner image, but still blurs out fine texture details such as vessels.

when using 10 unrolled iterations with a single RB for proximal modeling, more details of the liver vessels are visible, and the texture appears to be more realistic.


Machine learning (ML) is a powerful and widely-used tool for making potentially important decisions, from product recommendations to medical diagnosis.

despite ML's impressive performance, it makes mistakes, with some more costly than others.


due to the overhead of computing connected components.

due to better cache locality and convergence properties, CYCLADES is able to reach a lower objective value in less time than HOGWILD!.


On the other hand, convergence of the correlation does not necessarily imply ( order to guarantee the left hand side converges to zero.

with the normalization f n , ( Σ We show only the outline of the proof in this paper.


We used a fairly standard ConvNet architecture and the standard stochastic gradient descent learning procedure.

we note that standard feedforward face detectors seek to model p(u|I), while completely ignoring the canonical face v. here we take v into account as well.


Artifacts such as eye blinks are independent of brain source activity and ICA has been able in many cases to successfully separate the two types of signals into distinct groups of output variables.

iCA techniques have several shortcomings.


It is expected that the performances of WBS, PELT, and SML deteriorate when the errors do not follow a normal distribution, because they all rely upon parametric model assumptions and thus are not robust to model misspecifications.

both BMS and NOT behave well under various error distributions.


On the computational front, DPMBQ costs O(n 3 ).

this cost is de-coupled from the often orders-of-magnitude larger costs involved in both evaluation of f (x) and p(dx), which form the main computational bottleneck.


Our formulation requires inferring alignments during training.

our results indicate that this can be done relatively fast, and with little loss of accuracy, even on a small dataset where no effort was made at regularization.


Recently, many heuristics for tensor completion have been developed such as the weighted least squares [11], Gauss-Newton [12], alternating least-squares [13,14], trace norm minimization [15].

no theoretical guarantees are known for these approaches.


The attribute space [67,61] is the most widely used.

for large-scale problems, annotating attributes for each class becomes very difficult.


This procedure has similarities with model-based RL and with the common approach in control to first perform system identification and then find a controller using the resulting model.

the key insight from [3] is that this procedure is guaranteed to converge to expert performance in a polynomial number of iterations.


ICD-9 codes, for instance, lack sensitivity and their use as a gold standard could lead to correctly positive predictions being labeled as false positives [5].

given that the label space is often large (as in our examples) it is a moderate assumption that erroneous false positives should not skew results significantly.


Recently, there are several works trying to decouple and parallelize the backpropagation algorithm.

all of them suffer from severe accuracy loss or memory explosion when the neural network is deep.


The generalised portraits are, like the decision-images, another useful visualisation of the categorisation algorithm of the machine classifier.

the central result of our paper is the corroboration of the machine-learningpsychophysics research methodology.


SPGP-DR works well for large training set sizes, since there is enough information for it to avoid overfitting and the advantage of using a prior on W is reduced.

2 Available at http://theoval.cmp.uea.ac.uk/ ˜gcc/competition/ Temp: 106 dimensions 7117/3558 training/testing data, SO2: 27 dimensions 15304/7652 training/testing data.


Their regret bound (for our setting) depends linearly on the dimension of the ambient space, and quadratic in the rank of the loss matrix.

our regret bounds do not depend on the dimension of the ambient space and are linear in the rank of the loss matrix -two properties that are necessary in order to achieve optimal regret bound for the low rank expert problem.


This reduces computation time to O(m 2 n).

active set selection interferes with hyperparameter learning, due to its non-smooth nature (see [1,3]).


[3,4] recently provide a novel view of the ranking problem by modeling intransitivity in pairwise comparisons (intransitivity means a > b, b > c, c > a).

all these papers consider pairwise comparisons, while we consider the problem of group comparisons in this paper.


The problem is that the Chow-Liu algorithm is based on the (2-dimensional) marginal distributions.

finding the marginal distribution of the probability function ( 9) is, unfortunately, NP hard and it is (equivalent to) our final target.


Recently, there has been significant progress in understanding reinforcement learning in discounted infinite-horizon Markov decision processes (MDPs) by deriving tight sample complexity bounds.

in many real-world applications, an interactive learning agent operates for a fixed or bounded period of time, for example tutoring students for exams or handling customer service requests.


For most applications, we expect the preprocessing cost is not a major issue because the preprocessing can be conducted off-line (or combined with the data-collection step) and is easily parallelizable.

even if the speed is not a concern, the energy consumption might be an issue, especially considering (b-bit) minwise hashing is mainly used for industry applications.


There it was shown that, given a fixed labeling of a graph, the number of mistakes made by an algorithm similar to the kernel perceptron [9] with a kernel that was the pseudoinverse of the graph Laplacian, could be bounded by the quantity [8, Theorems 3.2, 4.1, and 4.2] Here u ∈ {−1, 1} n is a binary vector defining the labeling of the graph, Φ G (u) is the cut size 1 defined as Φ G (u) := |{(i, j) ∈ E(G) : u i = u j }|, that is, the number of edges between positive and negative labels, D G is the diameter of the graph and bal(u This bound is interesting in that the mistakes of the algorithm could be bounded in terms of simple properties of a labeled graph.

there are a variety of shortcomings in this result.


The MotionNet is relatively light-weighted and can accept a sequence of multiple images.

this is only used to substitute the pre-calculated optical flow while maintaining the conventional two-stream architecture.


This property is not held for our polytope as the LP contains inequalities of the form a, x ≥ 0.

we prove that both 0 and v 0 are feasible solution to the LP that do not share a common facet.


It will remain open whether the algorithm can be amended to remove this blemish.

as the nodes do not enter the GP regression formulation, their choice does not directly affect the probabilistic interpretation.


With the quadratic cost, only local displacements are permissible.

the harmonic-invariant cost additionally permits larger displacements to divisor frequencies, improving robustness to variations of timbre besides to inharmonicities.


Minimax rates, but only for d = 1, were obtained in [Gen12,AG14] for models of purely random forests such as Toy-PRF (where the individual partitions corresponded to randomly shifts of the regular partition of [0, 1] in k intervals) and PURF (Purely Uniformly Random Forests, where the partitions were obtained by drawing k random thresholds at random in [0, 1]).

for d = 1, tree partitions reduce to partitions of [0, 1] in intervals, and do not possess the recursive structure that appears in higher dimensions and makes their precise analysis difficult.


Previous studies using HMMs have assumed that all neural activities were independent of one another given the hidden states; hence, the models could not discriminate states whose firing rates were almost the same but whose correlations among neurons were different.

there has been reports that shows the correlation between neurons changes within a fraction of a second without modulating the firing rate (e.g., [4]).


If τ = 0, a sp-tree turns back to a metric tree with depth O(log n).

if τ ≥ ||v.rpv − v.lpv||/2, then N(v.lc) = N(v.rc) = N(v).


In consequence, the assumption of uniform prior taken by SIBA is closer to the real one.

when P(T) differs from the uniform distribution (see for instance Votes in Figure 1 right histogram) the results of SIBA are rather different from the expected disagreement rates.


Assuming independent outputs (no correlation between each output dimension) and given a test input xk = (x k , u k ) at time step k, the one-step predictive mean and variance of the state transition are specified as The state distribution at k = 1 is p(x 1 ) ∼ N (µ 1 , Σ 1 ) where the state mean and variance are  Generally, this predictive distribution cannot be computed analytically because the nonlinear mapping of an input Gaussian distribution lead to a non-Gaussian predictive distribution.

the predictive distribution can be approximated by a Gaussian p(dx k ) ∼ N (dµ k , dΣ k ) [16].


The optimum solutions for the three MLNs were found in (a) 20 minutes, (b) 6 hours and (c) 8 hours respectively.

our new approach P-IPP yields close to optimal solutions in a fraction of the time, and for relatively small values of k (≈ 5 − 10).


Let Π f be the uniform distribution over the random draw of the labelled subgraph Sn to be small.

since, the labelled nodes are drawn randomly, there is a small probability that one gets an unrepresentative subgraph S n . for large n, we want -regret to be close to zero with high probability 4 .


On the other hand, there is a number of recent works establishing the convergence of play in potential games with finite action sets under different assumptions for the number of players involved (continuous or finite) and the quality of the available feedback (perfect, semi-bandit/imperfect, or bandit/payoff-based) [5,11,14,19].

these works focus on games with finite action sets and feedback is assumed to be instantly available to the players (i.e.


Local time constraints on the priors are insufficient to pick one interpretation over another.

nonlocal time constraints, induced by a 1/f noise spectrum of the priors, is shown to permit learning of a specific model parameter even when there are infinitely many equally plausible interpretations of the data.


Note that when the parameters are known, one can take e ⇥ = {✓ 0 }.

when the parameters are unknown, one takes e ⇥ = ⇥ to prove a lower bound on the sample complexity of the estimate-then-explore algorithm, which is tasked with deciding whether or not samples are coming from a mixture of distributions or just a single distribution within the family.


according to Theorem 2, if the game has a unique Nash equilibrium, then this equilibrium is a minimax solution of the zero-sum game defined by the joint cost function of Equation 17.


In practical implementations, it is natural to choose option I, or take ws to be the average of the past t iterates.


In general cases, the sum of positively skewed random variables is not necessarily a positively skewed random variable.

the sum of regular log-normal random variables can be approximated by another log-normal [23], thus is still positively skewed.


As we show in this paper, alpha-divergences have the disadvantage of inducing high-variance gradients, since the ratio between posterior and variational distribution enters the bound polynomially instead of logarithmically.

our approach leads to a more stable inference scheme in high dimensions.


lated quantity, the delayed mutual information (dMI) provides an indication of instantaneous dependence: dM I(s) = I(X t ; X t+s ), where X t is a binned spike train, and I(X; Y ) = H(X)−H(X|Y ) denotes the mutual information.

this quantity ignores any temporal dependences in the intervening times: X t+1 , .


First, they pertain to parametric estimation in R d , where the communication budget per machine is much smaller than what is needed to even specify the answer with constant accuracy (O(d) bits).

our results pertain to simpler detection problems, where the answer requires only O(log(d)) bits, yet lead to non-trivial lower bounds even when the budget size is much larger (in some cases, much larger than d).


Given weakly annotated training images, they infer latent segmentation masks based on Multiple Instance Learning (MIL) [7,9] or Expectation-Maximization (EM) [8] framework based on the CNNs for supervised semantic segmentation.

performance of weakly supervised learning approaches except [10] is substantially lower than supervised methods, mainly because there is no direct supervision for segmentation during training.


Our work was influenced by these publications especially in the way the decoupling of joint and task-specific parameters is performed.

our method uses a different optimization and learns metrics rather than separating hyperplanes.


On some datasets, matrix completion has 100%accuracy.

on other datasets, either because of issues a fragile spike-sorting, or of other noise, the approach does not do as well.


For example, one subject thought the shape categories were determined by whether the shape "faces" downward; another thought the word categories indicated whether the word contains the letter T. Such beliefs, though helpful in learning the particular training samples, amount to over-fitting the noise.

when n = 40, about half the participants indicated that they believed the labels to be random, as spurious "rules" are more difficult to find.


[3], due to neglecting model uncertainty, suffering model bias.

the other methods consider uncertainty and have relatively unbiased predictions, judging by the similarity between predictive-vs-empirical performances.


( 2) and the exchangeable representation due to Kallenberg (2005).

for greater generality, we will let the latent trait be a general continuous parameter u i ∈ [0, 1] and later show that block-structured models can be obtained as a special case.


When γa > 0, log , producing power law behavior which accurately predicts observed object frequencies.

the closest fitting DP model (γ a = 0) significantly underestimates the number of rare labels.


If subtype alone determined trajectory, then we could cluster individuals.

other unobserved individual-specific factors such as behavior and prior exposures affect health and can cause different trajectories across individuals of the same subtype.


Pham and Pagh then showed that TENSORSKETCH can also be used for statistical learning with the polynomial kernel [16].

it was unclear whether TENSORSKETCH can be used to approximately preserve entire subspaces of points (and thus can be used as an OSE).


The examples given are for sources without temporal correlations.

this derivation can easily be extended to treat temporal correlations.


Consequently, the adversarial objective discourages generating responses that demonstrate less variation than human responses.

while GANs help reduce the level of blandness, the technique was not developed for the purpose of explicitly improving either informativeness or diversity.


An admissible choice of regularizer here would be the (semi-convex) function V (w, θ) = |w| • |θ| [4].

as shown in Appendix D.4, the differential dΦ has discontinuities: this prevents altogether from defining gradient flows, even in the finite-particle regime.


The f-GAN, however, only considers a single divergence.

our proposed method combines KL and reserve KL divergences.


We were surprised with the mileage we got out of the very simple prior we used: even though it only looks at two operators (gradients, and cornerness) it can generate surprisingly powerful predictions.

our experiments with real images show that this simple prior is not powerful enough.


The marginal loglikelihood tends to be larger, as it is easier for a zero-mean GP prior to fit the data.

this behavior is not desirable for classification purposes.


In conclusion, probably neither the linear nor the maximum combination rule does represent the exact model for local visual component combinations.

while linear component combinations have extensively been studied in the context of image statistics, the investigation of other combination rules has been limited to relatively small scale applications [17,16,35,19].


All previous models assume that a portion of the recommendation list is explicitly examined by the user and hence that the learning algorithm eventually has access to rewards corresponding to the unbiased user's evaluation of each item.

we propose to analyze multiple-play bandits in the Position-based model (PBM) [5].


If so, then the extremely low Bonferroni corrected α-threshold crossings effectively become mutually exclusive, which makes the Union Bound (on which Bonferroni correction is based) nearly tight.

when variables are highly dependent -and indeed even without smoothing there are many sources of strong non-Gaussian dependencies between voxels, the true FWER threshold can be much more relaxed, and it is precisely this phenomenon which drives the search for alternatives to Bonferroni correction.


This difference explains the improvement on the 1-1 mapping, where VEM is assigning larger size states to the most frequent tags.

vEM achieves this power law distribution at the expense of the mutual information with the gold labels as we see in the rightmost plot.


Divisive normalization has previously been found [33,34] to be useful as part of a multi-stage object classification pipeline.

it was introduced as a processing stage [33], rather than a part of unsupervised feature learning (pretraining).


Despite the advantages of the LLDE, it requires the bandwidth to be data independent and vanishingly small (sublinearly in sample size) for consistency almost everywhere -both of these are impediments to practical use since there is no obvious systematic way of choosing these hyperparameters.

if we restrict our focus to functionals of the density, then both these issues are resolved: this is the focus of the next section where we show that the bandwidth can be chosen to be based on fixed k-NN distances and the resulting universal bias easily corrected.


Recent compressed sensing research has provided scaling laws on numbers of measurements that guarantee good performance of these methods [15][16][17].

these scaling laws are in general conservative.


Because of this, the obtained factors may be suboptimal measured by the difference between the target matrix V and their product.

there exist much fewer works directly solving the mfNMF problem, one example is a multiplicative algorithm based on the general Bregmann divergence [9].


This captures the relatively uselessness of these large RFs in the high load condition.

and importantly, their mean values are unaffected by the non-target stimuli, since the non-targets are balanced between positive and negative values, preferring neither sign of target.


For various types of data such as text, this might cause concerns.

random projections [15] or low-rank approximations [2,28] can alleviate this problem and are already frequently employed in machine learning.


[17] decomposed a video sequence into short-term segments and characterized segments by the dynamics of their attributes.

since attributes are defined over the entire action video instead of short-term segments, different decomposition of video segments may obtain different attribute dynamics.


This system outperforms baselines that draw on previous approaches in both machine vision and cognitive science, coming closer to human performance than any of these approaches.

there is still significant room to improve performance on this task, and we present our visual concept learning dataset as the basis for a new challenge problem for machine vision, going beyond assigning labels to individual objects.


A nice property of Risk 1,n and Risk 2,n is that they are not sensitive to outliers of R in the sense that a small perturbation of R will not change the risk too much.

the Hausdorff distance is very sensitive to outliers.


In this case, the policy could react to a response time flucutation by setting an extreme powercap value in an attempt to quickly drive the response time back to the goal value, and then backing off to a less extreme value 5 seconds later.

such behavior would be quite poor in the actual system, in which the extreme powercap setting is held fixed for 30 seconds.


Our model is clearly more abstract and less biologically detailed than [4].

our intent was to ask whether the important insights and capabilities of that model could be captured using a four-parameter, pure RL model with a clear normative basis.


In realworld applications, obtaining correctly paired data samples for the entire dataset is expensive or even impossible.

in some situations obtaining paired data for a very small subset of the observations may be feasible.


However, the probability of picking up the correct element by chance is tiny as we restrict s ≤ d 1−α .

we in fact believe that the result holds generally for any s and d. proving it turns out to be quite intricate since it requires finer results about the the behavior of the order statistics of independent Gaussian variables.


But since the fixed-point iterations always generate hpd iterates, no extra eigenvalue computation is needed, which leads to substantial computational advantages.

a nonlinear solver must perform constrained optimisation, which can be unduly expensive.


We found that by using the mixture model we were able to reduce the KL divergence between a subject's average self-transition statistics and the observed transitions by 32% on average.

in the mixture model we have added extra degrees of freedom, and hence tested whether the better fit was statistically significant by using the F-test.


Prior work on multi-class learning in generative hierarchies either learns separate hierarchies for each class [14,15,16,10,17], trains jointly [7,18,9,19,20,11], whereas work on sequential learning of classes has been particularly scarce [6,13].

to the best of our knowledge, no work has dealt with, evaluated and compared multiple important learning concepts under one hierarchical framework.


Alternating direction method of multipliers (ADMM) has received tremendous interest for solving numerous problems in machine learning, statistics and signal processing.

it is known that the performance of ADMM and many of its variants is very sensitive to the penalty parameter of a quadratic penalty applied to the equality constraints.


For each task t i and each algorithm A ∈ A we define the expected and empirical error of applying this algorithm as follows: where ).The goal of the learner is to find A that minimizes er n+1 given the history of the observed tasks.

if the task environment would change arbitrarily from step to step, the observed tasks would not contain any relevant information for a new task.


The gradient of L(W) is the following: The first set of terms are easy to compute.

to compute the the normalizer, Z W (x ), and the expected value of the features under the current weight vector, E Pr(s |W) G(x , s ), we must use the Markov property of G to construct a dynamic programming algorithm, similar for forward-backward.


Basically, we want to move n/k line gadgets to the exact position of n/k other line gadgets such that in the resulting graph, n/k pairs of line gadgets share the same coordinates.

we have to make sure that the algorithm is oblivious of this relocation with sufficiently high probability.


As a result, oscillations were mainly transient with suitably chosen learning parameter values.

a commonality among all the algorithms was that the relevant matrices became quickly highly ill-conditioned.


Indeed, for general M * (x), such flipping would give different values t * λ (x) ∈ M * (x) for λ → 0, and hence would result in significantly different sets M low (f P,λ ) and M up (f P,λ ).

as a consequence, it seems hard to show that, for probability measures P whose conditional distributions P( • |x), x ∈ X, have no discrete components, we always have there are situations in which this equality can easily be established.


Our current model cannot capture any of these dependencies.

we believe that our approach is quite general and can be extended to more complex models.


This happens for example if s is an isolated point of some S ⊂ R n and µ is the Lebesgue measure, in which case B(s, ϑ t ) = ∅.

under an additional regularity assumption on the measure µ we can avoid such degenerate situations.


The Gaussian assumption of f t can be interpreted hierarchically as the result of central limit theorem because these factors themselves represent the ensemble effects of numerous factors from the whole environment.

the disturbances ϵ it are local factors that describe the intrinsic behaviors of the observed variables [4].


Inverse reinforcement learning corresponds to the case when we assume the expert triggers an event at all timesteps (the ALL query), in which case we require full demonstrations.

if we assume the expert is optimal under an ANY or AT query, full demonstrations are not required because the event is not assumed to be triggered at each timestep.


A uniform prior was placed on V 2 for both people, reflecting the fact that they have no prior expectations about the spending in the bill.

the priors placed on V 1 for Alice and Bob reflect their different beliefs about the best economic policy.


In CYK, each span of a sentence is explicitly parsed as each production, making it straightforward to have multiple instances of a symbol.

since there are an exponential number of subregions of an image, SFG-PARSE instead constructs a parse of the entire image for each production and reuses that parse for each of these subregions.


The red and blue groups can be easily separated by their complete trajectories.

given very similar score values at the time points of t0 and t3, any of the aforementioned summarized values may not be sufficient to identify the group difference.


Convergence can sometimes then be assured by existing results on the convergence of on-policy methods (Tsitsiklis & Van Roy 1997;Tadic 2001).

the importance sampling weights are cumulative products of (possibly many) target-to-behavior-policy likelihood ratios, and consequently they and the corresponding updates may be of very high variance.


The EPM adopts a gamma process (ΓP) prior to automatically shrink the number of active atoms.

we empirically found that the model shrinkage of the EPM does not typically work appropriately and leads to an overfitted solution.


In summary, our primary objective is improving the ability of the trained classifier to usefully identify the spatial pattern of discriminative information.

incorporating spatial information into boosting may also improve classification accuracy.


the identity from one and the pose from another, and so disentangling was sufficient.

our analogy method enables us to perform more challenging analogies by learning the manifold of character animations, defined by the sequence of frames in each animation.


The probability of clicking on the second position is larger in a than in a .

the probability of clicking on the fourth position is larger in a than in a.


The EEG and fMRI data points described in the previous section are extremely high dimensional.

both EEG and fMRI signals are the result of sparse neuronal activity.


The optimal MSE code over noisy channels was examined previously in [6] for Ndimensional data.

the capacity constraint was defined for a population and only examined the case of undercomplete codes.


Finally, E is a noise-plus-interference term where we assume, for simplicity, that columns are drawn independently from N (0, Σ ).

temporal correlations can easily be incorporated if desired using a simple transformation outlined in [3].


Given a data matrix Y ∈ R n×p and an arbitrary unit vector in x ∈ R p , an intuitive way to remove the contribution of x from Y is to project Y onto the orthocomplement of the space spanned by x: Ŷ = Y (I − xx T ).

if A is the sample covariance matrix of Y , then the sample covariance of Ŷ is given by Â = (I − xx T )A(I − xx T ), which leads to our formulation for projection deflation: Projection deflation Note that when x t is a true eigenvector of A t−1 with eigenvalue λ t , projection deflation reduces to Hotelling's deflation: in the general case, when x t is not a true eigenvector, projection deflation maintains the desirable properties that were lost to Hotelling's deflation.


The mean terms simply represent a clamped LGSSM with averaged parameters.

the extra contributions from the fluctuations mean that Eq.


If the learner performs active learning and an oracle provides noiseless labels, then the error reduces faster | θ − θ| = O( 1 2 n ) since the optimal strategy is binary search.

a helpful teacher can simply teach with n = 2 items (θ − /2, white), (θ + /2, black) to achieve an arbitrarily small error .


A naïve approach to deal with this bias is to reduce .

a reduced have three dire effects: (i) based on Theorem 3.1, any change in the generator parameters would result in large changes in the optimal discriminator parameters.


They outperform GLM-based approaches by fitting a multi-layer neural network consisting of parameterized difference-of-Gaussian filters in the first layer, followed by two fully-connected layers.

because they do not use a convolutional architecture, features are shared only locally.


For problems in which there is a single best solution, the KL-divergence measure seems to work well.

we would like to be able to extend the load-unload result to the situation where the agent has a memory bit.


In [13], a classification algorithm is also used to produce a reward function.

instead of associating actions to states, as we do, it associates optimal policies (labels) to MDPs (inputs), which is how the structure is incorporated.


For linearly separable binary-classification problems, both Perceptron and Winnow are able to find a weight that separate the in-class vectors from the out-of-class vectors in the training set within a finite number of steps.

the number of mistakes (updates) before finding a separating hyperplane can be very different [10,9].


We'll let s be the state, defined as the vector of cumulative losses of all the experts.

the proposed reduction almost works, except for one key issue: this only allows cost vectors of the form t = M e it = e it , since by definition Nature chooses columns of M . as shown in Abernethy et al, this is not a problem.


The gradient-based methods seem to often fall into this trap and then fail to recover as the gradient information from the new state has no guarantees of undoing the last gradient update.

eRL with its population provides built in redundancies which selects against this deceptive trap, and eventually finds a direction for learning which avoids it.


The model (1) appears in an entirely separate context, as the model for learning a regression function as an ensemble of weak learners f θ , or boosting [4].

the problem of fitting a mixture model and the problem of fitting an ensemble of weak learners have several important differences.


In [13], a convex programming method for estimating the sparse parameters of logistic regression model is proposed: where s is the sparseness prior parameter on β.

this method is not robust to corrupted covariate matrix.


PCA succeeds nicely in summarizing the population response for our data set: the first ten principal components capture more than 90% of the variance of the data.

pCA completely ignores the causes of firing rate variability.


Other methods for this problem include the relaxed Richardson iteration algorithm [6], the approximated joint diagonalization algorithm [9], and Riemannian Newton and quasi-Newton type methods, notably the limited-memory Riemannian BFGS [37].

none of these methods were shown to greatly outperform RGD, especially in data science applications where n is large and extremely small optimization error is not required.


This is feasible to test certain hypothesis, e.g., to evaluate whether there exists a causal link between two specific points within the brain.

testing pairwise causal links between more than just a few points within the brain is computationally intractable.


We can be more precise on what actually breaks in the approach we have described so far.

on the one hand, it is possible to write counterparts of Lemmas on the other hand, we did not manage to adapt Lemma 5 nor Lemma 7.


Motivated by this, SVGD approximates q with the empirical distribution of a set of particles {x i } n i=1 , and iteratively updates the particles by Liu et al.

( 2017) studied the asymptotic properties of the dynamic system underlying SVGD, showing that the evolution of the limit density of the particles when n → ∞ can be captured by a nonlinear Fokker-Planck equation, and established its weak convergence to the target distribution p. the analysis in Liu et al.


This is consistent with our observation that during learning the variational approximation is good in this case (the free energy bound appears tight).

for the LT model, the variational bound is loose during learning and localization performance is equivalent, but slightly lower than that of exact LT model.


For a general Kblockmodel, we conjecture that the number of stationary points grows exponentially with K. Similar to Figure 1-(a), a significant fraction of the random initializations converge to the ground truth.

when p, q are unknown, random initializations always converge to the uninformative stationary point (1/3, 1/3, 1/3), analogous to Lemma 3.2.


More precisely, they have proposed a method to recover a low-rank tensor with sparse errors.

there are two issues that limit the practicality of such methods: (1) The tensorial data are assumed to be well aligned and rectified.


Unlike bandit algorithms, which cannot use any side-information or context, contextual bandit algorithms can learn to map the context into appropriate actions.

contextual bandits do not consider the impact of actions on the evolution of future contexts.


The techniques described here may thus be applied unmodified to experimental data for which such channels have been blocked pharmacologically.

we should note that our methods extend directly to the case where simultaneous access to voltage and calcium signals is possible; more generally, one could develop a semi-realistic model of calcium concentration, and optimize over the parameters of this model as well.


[9], [10] and [11] who study the F-measure.

such methods frequently seek to learn a complex probabilistic model, a task arguably harder than the one at hand itself.


1-PkO is the probability that symptom Sk is active when none of the diseases are active.

exact inference computes the distribution over d given a subset of observed values in s. if Sk is not observed, the corresponding likelihood (node plus edges) may be deleted to give a new network that describes the marginal distribution over d and the remaining variables in s. So, we assume that we are considering a subnetwork where all the variables in s are observed.


Suppose there are ¡ possible objects, and that there are ¢ possible values that the instantiation parameters of any one object can take on; we will need to consider £ ¥¤ ¦¢ ¨ § ¨© combinations to explain any image.

in our approach we find one object at a time, thus avoiding the combinatorial explosion.


They study hardware implementations of the TRW-S algorithm (a coordinate-minimization algorithm very similar to the ones we study here), where some message computations can be parallelized.

their parallelization scheme is quite different from ours as it is synchronous, i.e., the messages computed in parallel have to be carefully chosen, and it is specific to grid-structured graphs.


Low-precision inference has already made its way into production hardware such as Google's tensor processing unit (TPU) [15].

reasonable successes in low-precision training have been obtained with binarized [13,16,17,5] or ternarized weights [18], or binarized gradients in the case of stochastic gradient descent [19], while accumulation of activations and gradients is usually at higher precision.


For many well-behaved graphs, this gap is rather small.

for general graphs, the difference can be huge -see the next section for details.


Using Fact 3.5. in [24], the set of global maxima solutions to the likelihood function, denoted by S D , is convex since the likelihood function is log-concave.

we also need that S D is bounded, and would further like that it provides one unique order as the estimation for the ground truth.


For x e = 1, in order to have f ∂vi (x ∂i ) < ∞, only one other x e ∈ x ∂vi should be non-zero.

we know that messages are normalized such that ν e→∂vi (0) = 0 ∀v i , e ∈ ∂v i , which means they can be ignored in the summation.


Because this computation is carried out at query time, it is not feasible for today's search engines, which need to handle tens of millions of queries per day.

pageRank computes a single measure of quality for a page at crawl time.


In this case, the learning algorithm should be robust enough to handle missing labels.

during inference, we may partially obtain high quality labels from external sources [3].


Assuming a number of arcs for each preference constraint equal to q, the dual problem in (8) involves N • q variables leading to a very large scale problem.

it can be noted that the independence of constraints among the different preference constraints allows for the separation of the variables in N disjoints sets of q variables each.


This often leads to inference results that are better than either simple variational methods or univariate Gibbs sampling (with a limited time budget).

this approach is limited to Ising models, and scales poorly in the size of the model, due to the difficulty of projecting onto the spectral norm.


the minimization of R[Y, f ] given Y or f, are convex.

the overall optimization of ( 11) is not convex.


The three cone photoreceptor types can support color vision because they are sensitive to different wavelengths of light.

each cone type responds to a range of wavelengths, and two lights with different spectra may activate the three cone types identically.


Recent GAN extensions [10,11] have sought to address this limitation by learning an inverse mapping (encoder) to project data into the latent space, achieving encouraging results on semi-supervised learning.

these methods still fail to obtain faithful reproductions of the input data, partly due to model underfitting when learning from a fully adversarial objective [10,11].


When there exist optimal solutions that are 'stable', in the sense that minor variations to these solutions will not effect their cost, then we require a high approximation ratio.

when all optimal solutions are 'unstable' then we settle for lower approximation ratios.


Each singleton concept has teaching dimension 1, while the teaching dimension for the empty concept 0 is n, because the teacher has to reveal all labels to distinguish 0 from the other concepts.

if the teacher and the learner are cooperative, every concept can be taught with one label: If the teacher reveals a "0" label, the learner can safely assume that the target concept must be 0, because otherwise the teacher would present a "1" label instead for the other concepts.


This is not a huge effect for a small dataset such as MNIST.

this speed-up might be very helpful for training large networks on large datasets, such as ImageNet.


Such environments are widely used, both as benchmark tasks for RL agents [Brockman et al., 2016], and as "cheap" simulation environments that can (ideally) allow for transfer to real domains.

despite their ubiquity, these simulation environments are in some sense poorly suited for deep learning settings: the environments are not natively differentiable, and so gradients (e.g., policy gradients for control tasks, physical property gradients for modeling fitting, or dynamics Jacobians for model-based control) must all be evaluated via finite differencing, with some attendant issues of speed and numerical stability.


Due to the optimal matching's complexity, researchers have developed approximation algorithms to compute close solutions for a fraction of the computational cost [4,8,1,7].

previous approximations suffer from distortion factors that increase linearly with the dimension of the features, and they fail to take advantage of structure in the feature space.


Guillotine partitions have wide ranging applications including circuit design, approximation algorithms and computer graphics.

the question of consistent stochastic processes over guillotine partitions, i.e.


This is because we do not perform any sort of variance normalization on the image patch, a step that will slow us down.

by now we have computed the approximated mean and variance of all the image segments and we can construct rejectors based on all of them to reject texture-less regions.


Knowledge distillation is introduced to facilitate training a smaller network by transferring knowledge from another high-capacity model, so that the smaller one obtains better performance than that trained by using labels only [10].

distillation is not an end-to-end solution due to having two separate training phases, which consume more training time.


The algorithm would always assign the individual to expert j = 1 and has a cumulative expected utility of .

given the true thresholds, the algorithm would have always assigned the individual to expert j = 2 and would have a cumulative expected utility of 0.


We report ν in Figure 2. l 1 init + refine achieves nearly the global maximum on both datasets and outperforms random init + refine, showing the effectiveness of the proposed initialization and that the iterative update ( 6) can get stuck in a bad local minimum if initialized badly.

l 1 init + refine outperforms l 1 init alone, showing the importance of iteratively refining the solution.


COCO tends to find functions with large variance for f (X) and g(Y ), which may not be the most correlated features.

kernel CCA may encounter situations where it finds functions with moderately large covariance but very small variance for f (X) or g(Y ), since Σ XX and Σ Y Y can have arbitrarily small eigenvalues.


Several approaches have been proposed for matrix completion with side information, and most of them empirically show that features are useful for certain applications [1,28,9,29,33].

there is surprisingly little analysis on the effect of features for general matrix completion.


This bound is better than that for UCT in a worst-case sense.

as their look-ahead tree is built in a uniform and non-adaptive way, this algorithm fails to benefit from a potentially favorable structure of the MDP.


In contrast to these methods, our approach accurately and seamlessly generates more detailed and precise virtual person images conditioned on different target poses.

there are some blurry results on Market-1501 [34], which might result from that the performance of the human parser in our model may be influenced by the low-resolution images.


Due to the properties of the L 1 score, in this solution, many features will have weight 0, generally leading to a sparse network structure.

it is generally impractical to simply initialize the model to include all possible features: exact inference in such a model is almost invariably intractable, and approximate inference methods such as loopy belief propagation [17] are likely to give highly inaccurate estimates of the gradient, leading to poorly learned models.


In this setting, it would be desirable to obtain zero TE between both electrodes, since there is no interaction between the sources giving rise to the EEG.

some rather tedious algebraic manipulations reveal that in this case Note that ( 4) is zero if and only if γ = 0, i.e., if s 3 represents white noise.


The latter may explain why dendrites can have a tuning different from the whole neuron as has been observed in Layer 2/3 pyramidal cells of the visual cortex [6].

saturating dendritic units can enhance single neuron computation through implementing all positive Boolean functions (Proposition 3), while reducing the energetic costs associated with the active ion channels required for dendritic spikes [4,13].


Even if driving times are stochastic, the annotations can be expected times, so this presents no additional challenge.

consider what happens if we would like to include public transportation in our route planning.


Notice that ρ ξ can be factorized as where ⊗ is the normal tensor product.


Here we show that they provide natural priors for Bayesian entropy estimation, due to the analytic tractability of the moments of the induced posterior distribution over entropy H. We derive formulas for the posterior mean and variance of H given data.

we show that a fixed Dirichlet or Pitman-Yor process prior implies a narrow prior on H, meaning the prior strongly determines the estimate in the under-sampled regime.


No particular problem is given that is easy to learn by say multiplicative updates and hard to learn by GD.

we give a random problem in Figure 1 that is trivial to learn by some algorithms, but hard to learn by the natural and most commonly used class of algorithms which predicts with linear combinations of instances.


Such a representation of the surface structure of music would be very useful in music information retrieval (Music-IR) and content description of musical material in large audio databases.

when operating on sampled audio data from polyphonic acoustical signals, extraction of a score-like description is a very challenging auditory scene analysis task [13].


Such an approach could be disadvantageous if observations at several different intervals were essential for discriminating the models under consideration.

increasing the number of intervals at which to test in each stage greatly increases the complexity of the design space, thus increasing the length of the computation needed to find an optimal design.


The myopic most uncertain selection method, MostUncertain, shows an overall inferior performance to Discriminative on Australian, Cleve, Crx, Heart and Hepatitis, and achieves a tie on Flare and Vote.

discriminative demonstrates weak perfor-  These empirical results suggest that selecting unlabeled instances through optimizing the classification model directly would obtain more relevant and informative instances, comparing with using heuristic scores to guide the selection.


Kernel density plug-in estimators are simple, easy to implement and widely used for estimation of entropy.

for large feature dimension d, kernel plug-in estimators suffer from the curse of dimensionality: the MSE rate of convergence is glacially slow -of order O(T −γ/d ), where T is the number of samples, and γ > 0 is a rate parameter.


Equivalently, this can be seen as letting the player use a refined partition with P + n cells.

instead of sampling one point in R p,k , the algorithm is only allowed to sample all the K points in region in the chosen R p at the same time, one uniformly in each sub-region R p,k , still using of course the same total budget of n points (and not nK).


Xu and Jordan [34] examine several aspects of EM for GMMs and counter the claims of Redner and Walker [24], who claimed EM to be inferior to generic secondorder nonlinear programming techniques.

it is now well-known (e.g., [34]) that EM can attain good likelihood values rapidly, and scales to much larger problems than amenable to secondorder methods.


We experimented with several induction algorithms including simple Linear Regression, Least Squares Support Vector Machine (LS-SVM) (both linear as well as non-linear) and Gaussian Processes (GP).

as the LS-SVM and GP showed no substantial advantage over Linear Regression, the latter was used and is presented in the sequel.


The main advantage of R 1 is that it has the form of conventional regularized error minimization, so it accommodates the boosting of all learning algorithms that minimize an error functional of this form (e.g., neural networks with weight decay).

the minimization of R 1 is suboptimal from boosting's point of view.


This intersection framework provides several advantages with respect to earlier approaches based on computing the correlations between trial-averaged psychometric performance and trial-averaged neurometric performance [13,14,18], because it quantifies associations between sensory information coding and choices within the same trial, instead of considering the similarity of trial-averaged neural stimulus coding and trial-averaged behavioral performance.

the intersection information measure proposed in Ref.


An alternative is to use the logarithm function in place of the power function.

the theory would need to be changed significantly to show that the resulting estimator remains strongly consistent.


As this parameter is effectively a variance, natural estimators exist by assessing the problem in hyperparameter (variance) space, transitioning natural ideas from Type II to solve what is much less intuitive for Type I.

for analyses of update rules and sparsity properties of local and global solutions, as well as extensions to more general likelihood models, we can leverage coefficient-space techniques developed for Type I and apply them to Type II.


In terms of run-time, KF-RTRL requires O(n 3 ), while UORO requires O(n 2 ) operations.

the faster run-time of UORO comes at the cost of worse variance and therefore worse performance.


‡ CPR appropriation problems are concerned with the allocation of the flow.

cPR provision problems concern the supply of the stock.


The k nonnegative components W ∈ W k returned by our Low Rank NNPCA algorithm on the sketch M are used as a surrogate for the desired components of the original input M. Intuitively, the performance of the extracted nonnegative components depends on how well M is approximated by the low rank sketch M; a higher rank approximation leads to better results.

the complexity of our low rank solver depends exponentially in the rank of its input.


There are a number of different reasons one might use the app: to search for a nearby business, to get directions from one point to another, or just to orient oneself.

the users of the app never specify an explicit intent, rather they swipe, type, zoom, etc., until they are satisfied.


This approach has made variational inference applicable to a large class of complex generative models.


An immediate corollary of Theorem 4 is that Hebbian learning rules, such as STDP and the selectron learning rule (4), improve the efficacy of spikes.

it also shows that naively increasing the frequency of spikes carries a cost.


The expectation (1) cannot be computed since the distribution P(x, y) is unknown.

given a training set {(Xi, Yi) h <i<n, it is common to minimize instead the empirical ri~: -- Empirical Risk Minimization (ERM) is therefore equivalent to minimizing the expectation of the loss function with respect to an empirical distribution Pemp(x,y) formed by assembling delta functions located on each example: It is quite natural to consider improved density estimates by replacing the delta functions ox, (x) by some estimate of the density in the vicinity of the point Xi, PXi (X).


Using a batch B s decreases the number of gradient evaluations required when SVRG is far from the solution, but its benefit diminishes over time.

for certain objectives we can further Algorithm 3 Heuristic for skipping evaluations of {Update the number of consecutive times f i (x) was zero.}


We find that POEM picks a hypothesis with large w , attempting to assign a probability of 1 to all training points with negative losses.

norm-POEM converges to a much shorter w . The loss of an instance relative to others in a sample D governs how Norm-POEM tries to fit to it.


Permanently small ¡ corresponds to over-confidence in estimates of `¡ , thus making adaptation of that estimate slow, similar to NE depletion.

because the NE system is still intact, the system is able to detect when ) ¡ dramatically differs from the prediction (which is often, since `is slow to adapt and leaves little room for variance), and thus to base inference of ) ¡ directly on the bottom-up information ¥ ) ¡ 0 ¡ .


[14] proposed a multi-task network containing i) a regression model that maps noisy labels and image features to clean labels ii) an image classification model that labels input.

the model in this paper is trained using a principled objective function that regularizes the inference model using extra sources of information without the requirement for oversampling clean instances.


We previously used the bias template as a classifier for recognizing objects when there is no training data available.

in some cases, there may be a few real examples available for learning.


Also note that JT-VAE performs optimization steps for each given molecule constraint.

gCPN can generalize well: it learns a general policy to improve property scores, and applies the same policy to all 800 molecules.


However, in PLDSID, only the cost O(N T pq 2 ) of calculating the Hankel-matrix scales with the data set size (assuming k is of order p).

this simple covariance calculation was much cheaper in our experiments than the moment conversion with cost O(pq 2 ) or the SVD with cost O(p 3 q 3 ), both of which are independent of the data set size N T . each iteration of EM requires at least O(N T (p 3 + pq)) time.


We investigated several simple methods to defend against these attacks, including (i) evaluating the validation accuracy for each class label, instead of a global value, to find the contaminated label, (ii) running independence tests on the distribution of attributes between each party, and (iii) performing leave-one-party-out cross validation techniques.

simple methods such as these were insufficient as a general defense.


Though non-uniform sampling is adopted in [2], it requires an additional pass over the training data to compute the sampling distribution.

our sketch based method takes only one pass of the data.


This results in the breakdown of the independency assumption between posterior probabilities for classes which Cover and Hart used implicitly when they derived the expected error of k-NN classification [4].

once we have enough data, and hence the density functions Eq.


Ignoring terms independent of the weighting parameters, its log is given by: (5) Computing ( 5) requires the quantities ICI and (VC-1rP, which for infinite dimensionality feature spaces might appear problematic.

by judicious re-writing of the terms of interest, we are able to both compute the log-likelihood (to within a constant) and optimise it with respect to the weights.


AdaBoost, as shown by Breiman [5] and others, is actually a coordinate descent algorithm on a particular exponential loss function.

minimizing this function in other ways does not necessarily achieve large margins; the process of coordinate descent must be somehow responsible.


A drawback of the objective in Equation 2 is that it depends on the input latent model and thus its accuracy.

empirically we found this objective to work well producing robust models.


Theorem 8 is proved via an inductive argument, given in Appendix C. Roughly, consider in Step 6 of Algorithm 2, setting Ŝ := S instead of S • S. By Lemma 5 and the formula in Lemma 6, the leverage score approximations l i computed in Step 7 would be good approximations to the true leverage scores, and S would satisfy Theorem 8 by a standard matrix Bernstein bound (see Lemma 9).

if we set Ŝ := S, it will have n/2 columns in expectation, and the computation in Step 7 will be expensive -requiring roughly O(n 3 ) time.


Lemma 2 gives D re (P Q 1−η ) ≥ (1/2) log(1/δ) − log 2.


Their results achieve estimation consistency in spectral norm with only one matrix observation.

their rate is slower than ours with K = 2.


This approach is related to the noisy-OR proposed in [24] for multi-view object detection.

their approach aims at decomposing a complex population into a collection of homogeneous populations, while our objective is to speed up the computation for the detection of a homogeneous population.