graphVite: graph embedding system
Text
Language
Speech
Image
Games
Unassisted control of vehicles
Medical diagnosis, fraud detection, network intrusion
classification
regression
Ranking
Clustering
Dimensionality reduction
Theoretical questions:
what can be learned, under what conditions?
are there learning guarantees?
analysis of learning algorithms.
Algorithms:
more efficient and more accurate algorithms
deal with large-scale problmes
handle a variety of different learning problems
Probability tools, concentration inequalities.
PAC learning model, Rademacher complexity, VC-dimension.
SVMs, margin bounds, kernel methods.
ensemble methods, boosting.
Logistic regression and conditional maximum entropy models.
On-line learning, weighted majority algorithm, Perceptron algorithm, mistake bounds.
Regression, generalization, algorithms.
Ranking, generalization, algorithms.
Reinforcement learning, MDPs, bandit problems and algorithms.
Definition: let X be a convex set. A function $\mathcal f: X \to \Bbb R$ is said to be convex if for all $x, y \in \mathrm X$ and $\alpha \in \lbrack 0,1 \rbrack$ \(f(\alpha x + (1 - \alpha)y) \le \alpha f(x) + (1 - \alpha)f(y)\)
with a strict inequality, $f$ is said to be strictly convex.
$f$ is said to be concave when $-f$ is convex.
Very large feature spaces have two potential issues:
Overfitting we handle with regularization.
“Kernel methods” can (sometimes) help with memory and computational costs.
complex measure of functions must be less than fixed r
choose r using validation data or cross-validation
each r corresponds to a different hypothesis spaces.
add complex measure with $\lambda$ to minimizer
choose $\lambda$ using validation data and cross-validation
Tikhonov Form
Ivanov Form
For identical features, l2 regularization spreads weights evenly
For linear related features, l2 regularization prefer viarables with larger scale - spreads weight proportional to scale
Tikhonov Form
Ivanov Form
For identical features, l1 regularization spreads weights arbitrarily
For linear related features, l1 regularization chooses viariable with larger scale, 0 to others
why l1(Lasso) get sparsity?
why l1(lasso) weights distribution is unstable?
combine lasso and ridge penalities.
with uncorrelated features, we get sparsity.
among correlated features(same scale), we spread it evenly.
kd tree
make a decision at each node on a single feature
for continuous value, split in the form of $x \leq t$
for discrete value, partition values into two groups
Loss function for regression Given the partition ${ R_1,…,R_M }$, prediction function is \(f(x) = sum_{m=1}^{M}c_m 1(x \in R_m)\)
for $l_2$ loss,
for $l_1$ loss,
assign each category a number, the proportion of class 0
find optimal split as though it were a numeric feature
empirical entropy
empirical conditional entrpy
information gain is defined as:
\[g(D,A) = H(D) - H(D \vert A)\]maximize g
ID3
C4.5
pruning
CART
A method is kernelized if every feature vector ψ(x) only appears inside an inner product with another feature vector ψ(x′). In particular, this applies to both the optimization problem and the prediction function.
a measure of how correct we are
we want to maximize the margin
use proper classification loss
Hinge Loss
Zero One Loss
Logistic Loss
turn local minima to non-local deep or trivial, leave it to the reader to judge
Hypothesis space
$\mathcal l2$ regularization
Loss function $\mathcal{l}(m) = max{1-m, 0}$
solution to
equivalent to a quadratic problem
differentiable
n + d + 1 unknowns and 2n affine constraints
can be solved by Quadratic solver
$$\mathsf w^\ast = sum_{i=1}^n \alpha_i x_i
Input space: $\mathbf{X}$
Feature space: $\mathbf{H}$
Feature map: $\mathbf{\psi} : \mathbf{X} \rightarrow \mathbf{H}$
\[k\left(x,x'\right) = \langle\psi(x),\psi(x')\rangle\]The kernel function corresponding to $\psi$ is
where $\langle \cdot, \cdot \rangle$ is the inner product associated with $\mathbf{H}$.
What are the Benefits of Kernelization?
It is often useful to think of the kernel function as a similarity score. But this is not a mathematically precise statement.
r(n+d) NP-hard reformulated
Gamma function Dirichlet distribution
Viterbi algorithm (dp with max)
https://spaces.ac.cn/archives/4695/
vapnik chervonenkis growth function: expression power of hypothesis space dichotomy shatter Given a set S of examples and a concept class H, we say that S is shattered by H if for every A ⊆ S there exists some h ∈ H that labels all examples in A as positive and all examples in S \ A as negative.
The VC-dimension of H is the size of the largest set shattered by H.
| Given a set S of examples and a concept class H, let H[S] = {h ∩ S : h ∈ H}. That is, H[S] is the concept class H restricted to the set of points S. For integer n and class H, let H[n] = max | S | =n | H[S] | ; this is called the growth function of H. |
Estimation, maximization
likelihood is usually defined on exponential function, thus use ln() to iterate EM to get latent variables, compute its expectation
efficient properly sample complexity m >= poly(,,,)
Definition:
concept class C is weakly PAC-learnable if there exists a (weak) learning algorithm L and > 0 such that:
for all $\delta > 0$, for all $c \in C$ and all distributions D, \({Pr \atop {S \sim D}} \lbrack R(h(s)) \leq \frac{1}{2} - \gamma \rbrack \ge 1 - \delta,\)
for sample S of size $m = poly(\frac{1}{\delta})$ for a fixed polymonial.
\[\sigma_i \approx \frac{1}{\sqrt{\nabla^2_{i,i} \hat{L}+ \rho N_{\gamma, \epsilon}(w_i)} },\]identifying-generalization-properties-in-neural-networks
- model local smoothness related to generalization property of classifier
- The sharp minimizers, which led to lack of generalization ability, are characterized by a significant number of large positive eigenvalues in loss function Hessian
- how to perturb the model, not at all directions, noise should be put along the “flat” directions. proportion to
$N_{\gamma, \epsilon}(w_i)=\gamma |w_i| + \epsilon$
uncertainty should be equally distributed conditional entropy
cycleGAN
pixel to pixle
pairs
Church numerals
scc
realbool
churchbool
realeq
realnat
recursion
fix
group
half group
ring
field
prior distribution
posterior distribution
likelihood model
shrinkage
\[( 1 - frac{N - 2}{\Vert z \Vert^2})\]$\eta$ : natural parameter
$\psi$ : cumulant generating function
training, dev, test segmentation is based on the i.i.d. assumption
a set of vectors is linearly independent if there exist m scalar coefficients, which are not all equal to zero and \(sum_{i=1}^m \alpha_i x_i = 0\)
all possible linear combinations of a set of vectors
the span of any set of vectors in $\mathcal V$ is a subspace of $\mathcal V$
closest
dim(col(A)) = dim(row(A))
sum of diagonal elements
map vectors from vector space V to vector space R
$A\vec{x}$ is a linear combination of the columns of $A$
linear map can be represent by a matrix
when matrix is fat, it projects vectors onto a lower dimensional space
when matrix is tall, it lifts vectors onto a higher dimensional space
the adjoint of a linear map satisfies
\[\langle f(\vec{x}), \vec{y} \rangle_{\mathcal R} = \langle \vec{x},f^{\ast}(\vec{y})_{\mathcal V}\]the range of a matrix is the range of its associated linear map
the set of vectors that are mapped to zero
null space of a linear map is a subspace
null space is perpendicular to row space \(null(A) = row(A)^{\perp}\)
if the column spaces of two matrix are orthogonal, then inner product is 0
\[\langle A, B \rangle\]the range is the column space
\[range(A) = col(A)\]orthogonal matrix is a square matrix s.t.
\[U^T U = U U^T = I\]the columns form an orhonormal basis
orthogonal matrices change the direction of vectors, not their magnitude
in the neural net world, the encoder is a neural network that outputs a representation zz of data xx. In probability model terms, the inference network parametrizes the approximate posterior of the latent variables zz. The inference network outputs parameters to the distribution q(z \vert x)q(z∣x).credit
in deep learning, the decoder is a neural net that learns to reconstruct the data xx given a representation zz. In terms of probability models, the likelihood of the data xx given latent variables zz is parametrized by a generative network. The generative network outputs parameters to the likelihood distribution p(x \vert z)p(x∣z).credit
in neural nets, inference usually means prediction of latent representations given new, never-before-seen datapoints. In probability models, inference refers to inferring the values of latent variables given observed data. credit
isomorphism
Homotopy
Jan 25: Introduction to graph theory, approximation algorithm, Max-Cut approximation. Chapter 8 on Lecture Notes.
Feb 01: Max-Cut approximation. Lifting / SDP relaxations technique in mathematical signal processing, phase retrieval and k-means SDP.
Feb 08: Unique Games Conjecture, Sum-of-Squares interpretation of SDP relaxation. Chapter 8 of Lecture Notes.
Feb 15: Shannon Capacity, Lovasz Theta Function. Section 7.3.1. on Lecture Notes and “On the Shannon Capacity of a Graph” by Laszlo Lovasz. See also Section 6.5.3.
Feb 22: Stochastic Block Model and Phase Transitions on graphs. Chapter 9 of Lecture Notes
Mar 01: Recovery in the Stochastic Block Model with Semidefinite relaxations. Chapter 9 of Lecture Notes
Jan 24: review of linear algebra and probability
Jan 31: discussion of homework 1
Feb 07: graph Laplacian and Cheeger’s inequality
Feb 14: pseudo distribution for maxcut, derivation of primal and dual program for Maxcut, SOS4
Feb 21: introduction to Grothendieck inequality and a proof of an upper bound of Grothendieck constant (Krivines bound)
Feb 28: calculate the Lovasz theta function for n-cycle and discuss connection with Grothendieck constant on graph
A clique of a graph G is a subset S of its nodes such that the subgraph corresponding to it is complete. In other words S is a clique if all pairs of vertices in S share an edge. The clique number c(G) of G is the size of the largest clique of G.
An independence set of a graph G is a subset S of its nodes such that no two nodes in S share an edge. Equivalently it is a clique of the complement graph Gc := (V, Ec). The independence number of G is simply the clique number of Sc.
A natural question is whether it is possible to have arbitrarily large graphs without cliques (and without its complement having cliques), Ramsey answer this question in the negative in 1928 [Ram28]
It is easy to show that R(3) = 6
We say an event happens with high probability if its probability is ≥ 1 − n−Ω(1)
“Erdos asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6). In that case, he believes, we should attempt to destroy the aliens.”
Given n and p, the random Erdos-Renyi graph G(n,p) is a random graph on n vertices where each possible edge appears, independently, with probability p.
For every r >= 2, \(R(r) \geq 2^{\frac{r-1}{2}}\)
For any finite graph H, there exists a constant $\delta H > 0$ such that any graph on n nodes that does not contain H as a subgraph (is a H-free graph) must have
\[r(G) \geq n^{\delta^H}\]to design polynomial algorithms that, in any instance, produce guaranteed approximate solutions.
Given a graph G = (V, E) with non-negative weights wij on the edges, find a set S ⊂ V for which cut(S) is maximal.
Goemans and Williamson [GW95] introduced an approximation algorithm that runs in polynomial time and has a randomized component to it, and is able to obtain a cut whose expected value is guaranteed to be no smaller than a particular constant αGW times the optimum cut. The constant αGW is referred to as the approximation ratio.
a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition.
sparse-cut
Given a graph and a set of k colors, and, for each edge, a matching between the colors, the goal in the unique games problem is to color the vertices as to agree with as high of a fraction of the edge matchings as possible.
For any ε > 0, the problem of distinguishing whether an instance of the Unique Games Problem is such that it is possible to agree with a ≥ 1 − ε fraction of the constraints or it is not possible to even agree with a ε fraction of them, is NP-hard.
GM = Multivariate Statistics + Structure
It is a smart way to write/specify/compose/design exponentially-large probability distributions without paying an exponential cost, and at the same time endow the distributions with structured semantics
It refers to a family of distributions on a set of random variables that are compatible with all the probabilistic independence propositions encoded by a graph that connects these variables
two types:
Bayesian network
Markov random field
an undirected graphical model, or Markov random filed, represents a distribution $P(X_1, …, X_n)$ defined by and undirected graph H, and a set of positive potential functions $\Psi_c$ associated with cliques of H, s.t.
\[P(x_1, ..., x_n) = \frac{1}{\mathbf Z} prod_{c \in C}\Psi_c(\mathbf X_c)\]where Z is known as the partition function
contingency function of its arguments assigning “pre-probabilistic” score of their joint configuration.
LSTM
GRU
Elman network就是指现在一般说的RNN（包括LSTM、GRU等等）。一个recurrent层的输出经过时延后作为下一时刻这一层的输入的一部分，然后recurrent层的输出同时送到网络后续的层，比如最终的输入层。一个Jordan network说的是直接把整个网络最终的输出（i.e. 输出层的输出）经过时延后反馈回网络的输入层
隐层被分成了g个模块，每个模块的大小是k，每个模块内部是全连接的。模块j到i的recurrent仅当Ti小于Tj时才会存在。根据增长的阶段对模块进行分类，模块之间的传递在隐层之间是从右向左的，从慢的模块到快的模块。
CW-RNN跟ＲＮＮ的主要不同之处在于在每个时间点ｔ，只有模块ｉ满足ｔ%Ti=0,才会执行，产生输出。Ｔｉ的是任意的，这篇论文取得是Ｔi=２^（ｉ－１）
Restraint and unrestrained variables Reject the null hypothesis Use Degree freedom1 and df2 and critical level 0.05 to look for the value, ssr r - ssr ur / ssr ur
https://ourworldindata.org/
for the negative log-likelihood loss, ERM and MLE are equivalent convex opt review
\text{PMI}(w_1,w_2)=\log \frac{P(w_1,w_2)}{P(w_1)P(w_2)}\tag{2}
At time t=1 you’re shown a picture of either a zero or a one. You have two possible actions: press a button marked “zero” or press a button marked “one.”The “correct” thing to do at t=1 is to press the button that corresponds to the image you’ve been shown. Pressing the correct button leads t01=0; the incorrect leads to1=1. Now, at time t=2 you are shown another image, again of a zero or one. The correct thing todo in this time step is the xor of (a) the number written on the picture you see right now, and (b) the correct answer from the previous time step. This holds in general for t>1
single mistake never recover
dagger
data aggregation
do something unusual.We put the human expert in the car, and record their actions, but the car behaves not according to the expert’s behavior, but according to f0. That is,f0is in control of the car, and the expert is trying to steer,but the car is ignoring them and simply recording their actions as training data
http://ciml.info/dl/v0_99/ciml-v0_99-ch18.pdf
russian tank problem
the classifier is being told to pay more attention to training examples that have high probability under the new distribution, and less attention to training that have low probability under the new distribution.
when the distributions agree on the value of a feature, let them share it, but when they dis-agree, allow them to learn separately
create three ver-sions of every feature: one that’s shared (for words like “awesome”),one that’s old-distribution-specific and one that’s new-distribution-specific
fairness
policy certificate: https://arxiv.org/pdf/1811.03056.pdf
regret minimization, epsilon bound, tighter bound: https://arxiv.org/pdf/1901.00210.pdf
differentially private fair learning
even with unbiased dataset, learning could still be biased
https://www.nowpublishers.com/article/DownloadSummary/TCS-042
subgroup fairness:
Adaptive Low-Nonnegative-Rank Approximation for State Aggregation of Markov Chains
Adaptive Low-Nonnegative-Rank Approximation for State Aggregation of Markov Chains
A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation
temporal difference learning
Attribute-Efficient Learning of Monomials over Highly-Correlated Variables
Kiran Vodrahalli
Nadav Cohen, wei hu
A Convergence Analysis of Gradient Descent for Deep Linear Neural…
deep linear NN, close form solution, provably converge
two condition: 1. deficiency margin c, 2. delta-balance
std of initialization has a sweet spot for convergence
Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks
a way to compute data complexity, without training, related to intrinsic dimension
Deep Generative Models with Learnable Knowledge Constraints
teacher-student learning, build loss object with external knowledge, as extrinsic reward in RL
use Posterior regularization to compute confidence on teacher distillation, reject samples with low confidence
graph convolution
Provably Efficient Maximum Entropy Exploration
maximize cross entropy with uniform distribution, to explore
a efficient appro plan, or density estimaiton algo, return a policy no less effective than optimal with epsilon
the result of the estimation is as close as to the truth distribution, given the uniform as estimation prior.
algo for unknown MDP
Approximate planning oracle
State distribution estimate oracle
https://github.com/facebookresearch/wav2letter/blob/master/docs/installation.md https://github.com/erikd/libsndfile#hacking https://github.com/kpu/kenlm#compiling
feature-wise transformation https://distill.pub/2018/feature-wise-transformations/
sparse tensor
sparse tensor in tensorflow
tf.SparseTensor(indices=np.array([fm[0],fm[1]]).T, values=fm[2], dense_shape=[weight_basis.size, total_dim])
tf.SparseTensor(indices=[[0, 0], [1, 2]], values=[1, 2], dense_shape=[3, 4])
tf.sparse_tensor_dense_matmul(ww, theta_small_norm, adjoint_a=True, adjoint_b=True)
modules = [module for k, module in model._modules.items()]
sparse matrix in Eigen
birthday paradox
mode collapse
https://openreview.net/pdf?id=BJehNfW0-
BIRTHDAY PARADOX TEST FORGANS
the training objective can approach its optimum value even if the generated distribution has very lowsupport —in other words, the training objective is unable to preventmode collapse.
cannot prevent learning meaningless codes for data
F-Integral Probability Metric, F = {all 1 Lipchitz functions}
https://arxiv.org/pdf/1702.07028.pdf
On the ability of neural nets to express distributions
Barron theorem
if a certain quantity involving the Fourier transform is small, then the function can be approximated by a neural network with one hidden layer and a small number of nodes
a generative model can be expressed asthe composition of n Barron functions, then it can be approximated by an+ 1-layer neural network
if a distribution is generated bya composition ofnBarron functions, then the distribution can be approximately generated by aneural network withnhidden layers.
Bar 93: gave a upper bound for the size of the network required in terms of a Fourier criterion. He showed that a functionf can be approximated in L2 up to error ε by a 2-layer neural network with \(O(\frac{C_f^2}{\epsilon})\) units, where Cf depends on Fourier properties of f
the numberof parameters required to obtain a fixed error increases linearly
dual-quaternion