Information Theory for ML15 / 16

Module 15. Information Geometry: Fisher, Natural Gradient, and Distribution Flows

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Question 15.2.1

15.2. Fisher information and local KL geometry

What does the Fisher–Rao metric measure?

One correct optionALocal statistical distinguishability of distributionsBThe global cost of mass transportCThe Euclidean length of weightsDOnly score variance under observed labels

A) Local statistical distinguishability of distributions

B) The global cost of mass transport

C) The Euclidean length of weights

D) Only score variance under observed labels

Answer and explanation

Correct answer: A. The local Fisher quadratic form describes distinguishability between nearby models.

Question 15.2.2

15.2. Fisher information and local KL geometry

What is the quadratic term in local KL?

One correct optionA

d\theta^\top G\,d\theta

\tfrac12 d\theta^\top G\,d\theta

d\theta^\top H^2 d\theta

\|d\theta\|_1

A) $d\theta^\top G\,d\theta$

B) $\tfrac12 d\theta^\top G\,d\theta$

C) $d\theta^\top H^2 d\theta$

D) $\|d\theta\|_1$

Answer and explanation

Correct answer: B. Under regularity, D_KL=1/2 dθᵀ G dθ+o(‖dθ‖²).

Question 15.2.3

15.2. Fisher information and local KL geometry

Why is KL not a Riemannian distance?

One correct optionAIt is asymmetric and does not satisfy the triangle inequalityBIt is always infiniteCIt depends only on parameter coordinatesDIt equals Fisher only on the simplex

A) It is asymmetric and does not satisfy the triangle inequality

B) It is always infinite

C) It depends only on parameter coordinates

D) It equals Fisher only on the simplex

Answer and explanation

Correct answer: A. Fisher is the local second-order term of KL, but KL itself is not a metric.

Question 15.2.4

15.2. Fisher information and local KL geometry

What is required for the Fisher matrix to be positive definite?

One correct optionAA large batchBLocal identifiabilityCA uniform priorDConvexity of the neural network

A) A large batch

B) Local identifiability

C) A uniform prior

D) Convexity of the neural network

Answer and explanation

Correct answer: B. Redundant parameterizations can make Fisher singular.

Question 15.2.5

15.2. Fisher information and local KL geometry

How does Fisher transform under $\theta=\theta(\phi)$ , with $J=\partial\theta/\partial\phi$ ?

One correct optionA

G_\phi=J^\top G_\theta J

G_\phi=JG_\theta J^\top

G_\phi=G_\theta+J

G_\phi=J^{-1}G_\theta

A) $d\theta^\top G\,d\theta$

B) $G_\phi=JG_\theta J^\top$

C) $G_\phi=G_\theta+J$

D) $G_\phi=J^{-1}G_\theta$

Answer and explanation

Correct answer: A. This is the transformation law for a metric tensor.

Question 15.3.1

15.3. Natural gradient: a step in distribution space

The natural-gradient direction for minimization is…

One correct optionA

Gg

H^{-1}g

\operatorname{diag}(G)g

G^\dagger g

A) $d\theta^\top G\,d\theta$

B) $H^{-1}g$

C) $\operatorname{diag}(G)g$

D) $G^\dagger g$

Answer and explanation

Correct answer: D. The descent direction is -G† g; the natural gradient itself is G† g.

Question 15.3.2

15.3. Natural gradient: a step in distribution space

What is correctly invariant under a smooth reparameterization?

One correct optionAThe natural-gradient vector fieldBEvery finite Euler stepCThe learning-rate scheduleDThe momentum buffer

A) The natural-gradient vector field

B) Every finite Euler step

C) The learning-rate schedule

D) The momentum buffer

Answer and explanation

Correct answer: A. A finite coordinate step is invariant only to first order.

Question 15.3.3

15.3. Natural gradient: a step in distribution space

Natural gradient and Newton coincide…

One correct optionAAlways for cross-entropyBOnly in reinforcement learningCNeverDOnly under additional conditions

A) Always for cross-entropy

B) Only in reinforcement learning

C) Never

D) Only under additional conditions

Answer and explanation

Correct answer: D. Fisher and the objective Hessian are distinct, though they coincide in special likelihood settings.

Question 15.4.1

15.4. Which “curvature” is actually used in deep learning

Where do labels in the inner expectation of the conditional model Fisher come from?

One correct optionAThe current model

p_\theta(y\mid x)

BThe validation setCThe hard argmaxDEmpirical class frequencies

A) The current model $d\theta^\top G\,d\theta$

B) The validation set

C) The hard argmax

D) Empirical class frequencies

Answer and explanation

Correct answer: A. The inner expectation is over y~p_θ(·|x).

Question 15.4.2

15.4. Which “curvature” is actually used in deep learning

What is the empirical Fisher?

One correct optionAThe second moment of observed per-example gradientsBFisher after whiteningCThe exact HessianDThe covariate matrix

A) The second moment of observed per-example gradients

B) Fisher after whitening

C) The exact Hessian

D) The covariate matrix

Answer and explanation

Correct answer: A. It uses observed labels and need not equal the model Fisher.

Question 15.4.3

15.4. Which “curvature” is actually used in deep learning

How is Adam best described?

One correct optionAAn inverse-Fisher solverBExact diagonal natural gradientCAdaptive diagonal scaling based on gradient momentsDNewton without a Hessian

A) An inverse-Fisher solver

B) Exact diagonal natural gradient

C) Adaptive diagonal scaling based on gradient moments

D) Newton without a Hessian

Answer and explanation

Correct answer: C. Squared gradients may resemble empirical moments, but they are not the model Fisher.

Question 15.4.4

15.4. Which “curvature” is actually used in deep learning

What does K-FAC usually approximate?

One correct optionAThe Wasserstein metricBLayerwise Fisher/GGN blocks by Kronecker productsCOnly a scalar gradient varianceDThe full Hessian without assumptions

A) The Wasserstein metric

B) Layerwise Fisher/GGN blocks by Kronecker products

C) Only a scalar gradient variance

D) The full Hessian without assumptions

Answer and explanation

Correct answer: B. K-FAC uses Kronecker factorization and damping.

Question 15.5.1

15.5. Policy geometry: NPG, TRPO, PPO, and KL in RLHF

Which KL orientation is normally used in practical TRPO?

One correct optionA

D_{\mathrm{KL}}(\pi_{\mathrm{new}}\|\pi_{\mathrm{old}})

D_{\mathrm{KL}}(\pi_{\mathrm{old}}\|\pi_{\mathrm{new}})

CThe symmetric Jensen–Shannon divergenceD

D_{\mathrm{KL}}(\pi_{\mathrm{ref}}\|\text{data})

A) $d\theta^\top G\,d\theta$

B) $D_{\mathrm{KL}}(\pi_{\mathrm{old}}\|\pi_{\mathrm{new}})$

C) The symmetric Jensen–Shannon divergence

D) $D_{\mathrm{KL}}(\pi_{\mathrm{ref}}\|\text{data})$

Answer and explanation

Correct answer: B. The practical constraint usually uses D_KL(π_old‖π_new).

Question 15.5.2

15.5. Policy geometry: NPG, TRPO, PPO, and KL in RLHF

Why does PPO clipping not guarantee a global trust region?

One correct optionAIt uses no gradientsBIt requires an exact HessianCIt controls ratios of sampled actions, not the whole policyDIt always increases KL

A) It uses no gradients

B) It requires an exact Hessian

C) It controls ratios of sampled actions, not the whole policy

D) It always increases KL

Answer and explanation

Correct answer: C. Probability mass may move among actions absent from the minibatch.

Question 15.5.3

15.5. Policy geometry: NPG, TRPO, PPO, and KL in RLHF

How does fixed-reference KL in RLHF differ from the TRPO KL?

One correct optionAIt is always symmetricBIt compares the current policy with a fixed base/SFT policyCIt does not use probabilitiesDIt is a PAC-Bayes bound

A) It is always symmetric

B) It compares the current policy with a fixed base/SFT policy

C) It does not use probabilities

D) It is a PAC-Bayes bound

Answer and explanation

Correct answer: B. The reference anchor limits accumulated drift, whereas TRPO constrains the next local update.

Question 15.5.4

15.5. Policy geometry: NPG, TRPO, PPO, and KL in RLHF

What is the ideal KL-regularized policy optimum?

One correct optionA

\pi^*=\pi_{\mathrm{ref}}+\beta r

\pi^*\propto\exp(-r)

\pi^*\propto\pi_{\mathrm{ref}}\exp(r/\beta)

\pi^*=\arg\max r

, with no reference policy

A) $d\theta^\top G\,d\theta$

B) $\pi^*\propto\exp(-r)$

C) $\pi^*\propto\pi_{\mathrm{ref}}\exp(r/\beta)$

D) $\pi^*=\arg\max r$ , with no reference policy

Answer and explanation

Correct answer: C. This is an exponential tilt of the reference policy.

Question 15.6.1

15.6. Another geometry: Wasserstein, CNFs, and Flow Matching

What does Wasserstein-2 measure?

One correct optionAA local Fisher normBParameter-space curvatureCOnly support overlapDThe minimum expected squared transport cost

A) A local Fisher norm

B) Parameter-space curvature

C) Only support overlap

D) The minimum expected squared transport cost

Answer and explanation

Correct answer: D. Wasserstein uses a ground metric and a coupling.

Question 15.6.2

15.6. Another geometry: Wasserstein, CNFs, and Flow Matching

When does Flow Matching define a Wasserstein geodesic?

One correct optionAAlwaysBOnly for a diffusion pathCWith an optimal coupling and displacement interpolationDOnly for discrete data

A) Always

B) Only for a diffusion path

C) With an optimal coupling and displacement interpolation

D) Only for discrete data

Answer and explanation

Correct answer: C. General Flow Matching permits other paths and couplings.

Question 15.6.3

15.6. Another geometry: Wasserstein, CNFs, and Flow Matching

What is true of a probability-flow ODE and its corresponding SDE?

One correct optionAThe ODE is always OT-optimalBThey have identical sample pathsCWith an exact score, they have the same one-time marginalsDThe SDE has no density

A) The ODE is always OT-optimal

B) They have identical sample paths

C) With an exact score, they have the same one-time marginals

D) The SDE has no density

Answer and explanation

Correct answer: C. Their couplings and trajectories differ, while their marginals agree.

Question 15.6.4

15.6. Another geometry: Wasserstein, CNFs, and Flow Matching

The score $\nabla_x\log p(x)$ is…

One correct optionAThe natural gradient in parameter spaceBA Fisher-vector productCA global direction to the nearest modeDA gradient with respect to the observation coordinate

A) The natural gradient in parameter space

B) A Fisher-vector product

C) A global direction to the nearest mode

D) A gradient with respect to the observation coordinate

Answer and explanation

Correct answer: D. The score is local in sample space and is not a parameter natural gradient.

Question 15.7.1

15.7. Exponential families, dual coordinates, and mirror descent

For an exponential family, Fisher equals…

One correct optionA

A(\theta)

BEntropy under all conditionsC

A^*(\eta)

without derivativesD

\nabla^2A(\theta)

A) $d\theta^\top G\,d\theta$

B) Entropy under all conditions

C) $A^*(\eta)$ without derivatives

D) $\nabla^2A(\theta)$

Answer and explanation

Correct answer: D. The Hessian of the log-partition function is the covariance of sufficient statistics.

Question 15.7.2

15.7. Exponential families, dual coordinates, and mirror descent

What is the Bregman argument order in $D_{\mathrm{KL}}(p_{\theta_1}\|p_{\theta_2})$ ?

One correct optionA

B_A(\theta_1,\theta_2)

B_A(\theta_2,\theta_1)

CA symmetric averageD

B_{A^*}(\theta_2,\theta_1)

under all conditions

A) $d\theta^\top G\,d\theta$

B) $B_A(\theta_2,\theta_1)$

C) A symmetric average

D) $B_{A^*}(\theta_2,\theta_1)$ under all conditions

Answer and explanation

Correct answer: B. KL orientation reverses the Bregman arguments in natural coordinates.

Question 15.7.3

15.7. Exponential families, dual coordinates, and mirror descent

What does dually flat mean?

One correct optionAKL becomes symmetricBFisher–Rao curvature is zero in every senseCThe parameter norm is EuclideanDThe e- and m-affine connections are flat

A) KL becomes symmetric

B) Fisher–Rao curvature is zero in every sense

C) The parameter norm is Euclidean

D) The e- and m-affine connections are flat

Answer and explanation

Correct answer: D. The Levi–Civita curvature of the Fisher–Rao metric may still be nonzero.

Question 15.7.4

15.7. Exponential families, dual coordinates, and mirror descent

Which update is induced by a negative-entropy mirror map?

One correct optionAExponentiated or multiplicative weightsBLinear interpolationCA Wasserstein barycenterDOrdinary Newton

A) Exponentiated or multiplicative weights

B) Linear interpolation

C) A Wasserstein barycenter

D) Ordinary Newton

Answer and explanation

Correct answer: A. The KL-Bregman proximal step normalizes p_i exp(-η g_i).

Question 15.8.1

15.8. Practical map: which geometry should you choose?

Which statement is most accurate?

One correct optionAShampoo is always a Fisher approximationBOne geometry fits every ML taskCChoose geometry by the object and meaning of a small changeDPPO, Adam, and Flow Matching are one algorithm

A) Shampoo is always a Fisher approximation

B) One geometry fits every ML task

C) Choose geometry by the object and meaning of a small change

D) PPO, Adam, and Flow Matching are one algorithm

Answer and explanation

Correct answer: C. First determine what object should change little and how that change is measured.