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The 5th International Conference on Automatic Differentiation, Bonn, Germany, August 15, 2008

Design and Implementation of a Context-Sensitive, Flow-Sensitive Activity Analysis Algorithm for Automatic Differentiation. The 5th International Conference on Automatic Differentiation, Bonn, Germany, August 15, 2008. Outline. Activity Analysis Previous work: CSFI New CSFS Algorithm

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The 5th International Conference on Automatic Differentiation, Bonn, Germany, August 15, 2008

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  1. Design and Implementation of a Context-Sensitive,Flow-Sensitive Activity Analysis Algorithm forAutomatic Differentiation The 5th International Conference on Automatic Differentiation,Bonn, Germany, August 15, 2008

  2. Outline • Activity Analysis • Previous work: CSFI • New CSFS Algorithm • Experimental Results • Conclusion

  3. Activity Analysis • AD is applied to a function with a set of input variables and a set of output variables. • Sometimes, we are interested in the derivatives • of a subset of the output variables  dependent • with respect to a subset of the input variables  independent • An intermediate variable is • varied if it is transitively dependent on any independent variable • useful if any dependent variable is transitively dependent on it • active if it is both varied and useful. • Partial derivatives need to be computed only for active variables. • Activity analysis is nonseparable.

  4. Activity Analysis: Given f, compute dy1/dx1 f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1) y1 = t1*2 t2 = cos(x2) y2 = t2*3 y3 = x3*t1 } f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1) dt1/dx1 = cos(x1) y1 = t1*2 dy1/dt1 = 2 t2 = cos(x2) y2 = t2*3 y3 = x3*t1 dy1/dx1 = (dy1/dt1)*(dt1/dx1) }

  5. Activity Analysis: Given f, compute dy1/dx1 f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1) y1 = t1*2 t2 = cos(x2) y2 = t2*3 y3 = x3*t1 } f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1) dt1/dx1 = cos(x1) y1 = t1*2 dy1/dt1 = 2 t2 = cos(x2) dt2/dx2 = -sin(x2) y2 = t2*3 dy2/dt2 = 3 y3 = x3*t1 dy3/dx3 = t1 dy3/dt1 = x3 dy1/dx1 = (dy1/dt1)*(dt1/dx1) dy2/dx2 = (dt2/dx2)*(dy2/dt2) dy3/dx1 = (dt1/dx1)*(dy3/dt1) }

  6. Activity Analysis: varied + useful = active f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; t2 = cos(x2); y2 = t2*3; y3 = x3*t1; } f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; t2 = cos(x2); y2 = t2*3; y3 = x3*t1; } varied active useful useful f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; t2 = cos(x2); y2 = t2*3; y3 = x3*t1; } f(x1, x2, x3, y1, y2, y3){ t1 =sin(x1); y1 = t1*2; t2 = cos(x2); y2 = t2*3; y3 = x3*t1; } varied

  7. Previous work: Context-Sensitive, Flow-Insensitive Activity Analysis (VDGAA) • Graph reachability problem • Variable Dependence Graph (VDG) • Two separate (color) propagations: • Forward (coloring red) for “varied” variables • Backward (coloring yellow) for “useful” variables • Context sensitivity is supported by a stack of contexts. • Run time • Very fast in practice • Small overestimations of active variables due to • the way programs are usually written • the property of activity analysis

  8. VDGAA on the run: 1. build VDGAA f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; t2 = cos(x2); y2 = t2*3; y3 = x3*t1; } x3 x2 t2 x1 y3 t1 y2 y1 Variable Dependence Graph (VDG)

  9. VDGAA on the run: 2. Find ‘varied’ variables f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; t2 = cos(x2); y2 = t2*3; y3 = x3*t1; } x3 x2 t2 x1 y3 t1 Forward propagation y2 y1

  10. VDGAA on the run: 3. Find ‘useful’ variables f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; t2 = cos(x2); y2 = t2*3; y3 = x3*t1; } x3 x2 t2 x1 y3 t1 Backward propagation y2 y1

  11. VDGAA on the run: Context Sensitivity call foo(x, p1); foo(a, b){ b = a; } call foo(p2, y); p2 y x p1 a b Forward propagation

  12. VDGAA on the run: Context Sensitivity call foo(x, p1); foo(a, b){ b = a; } call foo(p2, y); p2 y x p1 a b Backward propagation

  13. VDGAA : Flow Insensitivity f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; y3 = x3*t1; t1 = cos(x2); y2 = t1*3; } x3 x2 x1 t1 y3 Forward propagation y2 y1

  14. VDGAA : Flow Insensitivity f(x1, x2, x3, y1, y2, y3){ t1 = sin(x1); y1 = t1*2; y3 = x3*t1; t1 = cos(x2); y2 = t1*3; } x3 x2 x1 t1 y3 Backward propagation y2 y1

  15. Context-Sensitive, Flow-Sensitive Activity Analysis (DUGAA) • Graph reachability problem • Two separate (color) propagations: • Forward for “varied” variables • Backward for “useful” variables • Context sensitivity is supported by a stack of contexts. • Definition-Use Graph (DUG) • Flow sensitivity is supported by the use of reaching definitions for nodes. • Algorithm: Build a def-use graph Forward propagation Backward propagation UD-DU Chains

  16. DUGAA on the run: 1. build a DUG 1:f(x1, x2, x3, y1, y2, y3){ 2: t1 = sin(x1); 3: y1 = t1*2; 4: y3 = x3*t1; 5: t1 = cos(x2); 6: y2 = t1*3; 7:} x2@I t1@5 x3@I y2@6 y3@4 y2@O x1@I t1@2 y3@O y1@3 y1@O Def-Use Graph (DUG)

  17. DUGAA on the run: 2. Forward propagation 1:f(x1, x2, x3, y1, y2, y3){ 2: t1 = sin(x1); 3: y1 = t1*2; 4: y3 = x3*t1; 5: t1 = cos(x2); 6: y2 = t1*3; 7:} x2@I t1@5 x3@I y2@6 y3@4 y2@O x1@I t1@2 y3@O Forward propagation y1@3 y1@O

  18. DUGAA on the run: 3. Backward propagation 1:f(x1, x2, x3, y1, y2, y3){ 2: t1 = sin(x1); 3: y1 = t1*2; 4: y3 = x3*t1; 5: t1 = cos(x2); 6: y2 = t1*3; 7:} x2@I t1@5 x3@I y2@6 y3@4 y2@O x1@I t1@2 y3@O Backward propagation y1@3 y1@O

  19. Implementation OpenAnalysis Output (Fortran) Input (Fortran) VDGAA Open64 Front end OpenAD: AD Transformation Open64 Unparser DUGAA

  20. Benchmarks

  21. Slowdowns in analysis run time: DUGAA vs. VDGAA

  22. Analysis run time

  23. Analysis run-time breakdown on MITgcm

  24. Reduction in active variables

  25. Conclusion • A new context-sensitive, flow-sensitive (CSFS) activity analysis algorithm: Def-Use Graph Activity Analysis (DUGAA) • Comparison of two activity analyses: DUGAA (CSFS) vs. VDGAA (CSFI) • Slower than VDGAA for all 8 benchmarks • by a factor > 27 • but takes less than one minute for a code larger than 27k lines • Makes fewer overestimations than VDGAA for two of the eight benchmarks. • May save human effort in managing AD code. • Future work • Comparison among CIFS, CSFI, and CSFS activity analyses • Dealing with pointers and recursion

  26. PARAM Edges • Edges between formal parameters • Summarize the connectivity among formal parameters • Transitive closure is applied to the dependence matrix of local variables • Checking for connectivity through global variables: A PARAM edge is generated from formal variable node F1 to formal variable node F2 by traversing the entire graph only when • F1 has a value flow path to a global variable node AND • F2 has a value flow path from a global variable node.

  27. Context-Insensitive, Flow-Sensitive Activity Analysis (ICFGAA) • Interprocedural Control Flow Graph (ICFG) • Iterative data-flow analysis (DFA) • Two separate DFAs: • Forward for “varied” variables • Backward for “useful” variables • Long run time • Large number of iterations • Nonseparability of activity analysis: data-flow analysis values depend on other data-flow values • Large overestimation • Due to context-insensitivity • Value propagation through unrealizable control paths

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